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Kevin	Carlberg	
Sandia	Na(onal	Laboratories	
SAMSI	MUMS	Opening	Workshop	
Duke	University	
August	21,	2018
Advances	in	nonlinear	model	reduction:

least-squares	Petrov–Galerkin	projection	and		
machine-learning	error	models
Kevin	Carlberg	
iCME	*Talk	
Stanford	University	
May	22,	2017
Breaking	computaDonal	barriers	
Using	data	to	enable	extreme-scale	simulaDons	for	many-query	problems
-8 -6 -4 -2 0 2
-8
-6
-4
-2
0
2
support	vector	machine	
error	predicDon
R2
= 0.990
error
ROM h-refinement
e), b) prolongation (fine)
reduced-order	model
high-fidelity	model
Sandia	Na(onal	Laboratories	is	a	mul(mission	laboratory	managed	and	operated	by	Na(onal	Technology	and	Engineering	Solu(ons	of	
Sandia,	LLC.,	a	wholly	owned	subsidiary	of	Honeywell	Interna(onal,	Inc.,	for	the	U.S.	Department	of	Energy’s	Na(onal	Nuclear	Security	
Administra(on	under	contract	DE-NA-0003525.
⇡HFM
post(µ|qmeas)
true
prior
⇡HFM
post (µ | qmeas)
⇡
]HFM
post (µ | qmeas)
⇡
]HFM
post (µ | qmeas)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
High-fidelity simulation
2
computa4onal	barrier
+Indispensable	across	science	and	engineering	
- High	fidelity:	extreme-scale	nonlinear	dynamical	system	models
/38
Kevin	CarlbergBreaking	computa5onal	barriers
High-fidelity simulation
computaDonal	barrier
๏ uncertainty	propagaDon ๏ mulD-objecDve	opDmizaDon
+Indispensable	across	science	and	engineering	
- High	fidelity:	extreme-scale	nonlinear	dynamical	system	models
Many-query problems
๏ Bayesian	inference ๏ stochasDc	opDmizaDon
Magnetohydrodynamics	
courtesy	J.	Shadid,	Sandia
Turbulent	reac5ng	flows		
courtesy	J.	Chen,	Sandia
Antarc5c	ice	sheet	modeling		
courtesy	R.	Tuminaro,	Sandia
2
/38High-fidelity simulation
computaDonal	barrier
๏ uncertainty	propagaDon ๏ mulD-objecDve	opDmizaDon
+Indispensable	across	science	and	engineering	
- High	fidelity:	extreme-scale	nonlinear	dynamical	system	models
Many-query problems
๏ Bayesian	inference ๏ stochasDc	opDmizaDon
Magnetohydrodynamics	
courtesy	J.	Shadid,	Sandia
Turbulent	reac5ng	flows		
courtesy	J.	Chen,	Sandia
Antarc5c	ice	sheet	modeling		
courtesy	R.	Tuminaro,	Sandia
/38High-fidelity simulation
computaDonal	barrier
๏ uncertainty	propagaDon ๏ mulD-objecDve	opDmizaDon
+Indispensable	across	science	and	engineering	
- High	fidelity:	extreme-scale	nonlinear	dynamical	system	models
Many-query problems
๏ Bayesian	inference ๏ stochasDc	opDmizaDon
Magnetohydrodynamics	
courtesy	J.	Shadid,	Sandia
Turbulent	reac5ng	flows		
courtesy	J.	Chen,	Sandia
Antarc5c	ice	sheet	modeling		
courtesy	R.	Tuminaro,	Sandia
๏ uncertainty	propagaGon ๏ mulG-objecGve	opGmizaGon
๏ Bayesian	inference ๏ stochasGc	opGmizaGon
Many-query problems
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
High-fidelity simulation: captive carry /38
Kevin	CarlbergBreaking	computa5onal	barriers 3
High-fidelity simulation: B61 captive carry
3
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
High-fidelity simulation: captive carry
๏ explore	flight	
envelope
๏ quanGfy	effects	of	
uncertainGes	on	store	load
๏ robust	design	of	
store	and	cavity
computa4onal	barrier
Many-query problems
+ Validated	and	predic(ve:	matches	wind-tunnel	experiments	to	within	5%	
- Extreme-scale:	100	million	cells,	200,000	Gme	steps	
- High	simula(on	costs:	6	weeks,	5000	cores
3
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Approach: exploit simulation data
4
Idea:	exploit	simula(on	data	collected	at	a	few	points
D
1. Training:	Solve	ODE	for																							and	collect	simulaGon	data	
2. Machine	learning:	IdenGfy	structure	in	data
3. Reduc(on:	Reduce	cost	of	ODE	solve	for
Many-query	problem:	solve	ODE	for	µ 2 Dquery
µ 2 Dtraining
µ 2 Dquery  Dtraining
ODE:
dx
dt
= f(x; t, µ), x(0, µ) = x0(µ), t 2 [0, Tfinal], µ 2 D
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Model reduction criteria
1. Accuracy:	achieves	less	than	1%	error
2. Low	cost:	achieves	at	least	100x	computaGonal	savings
3. Structure	preserva;on:	preserves	important	physical	properGes
4. Reliability:	guaranteed	saGsfacGon	of	any	error	tolerance	(fail	safe)
5. Cer;fica;on:	quanGfies	ROM-induced	epistemic	uncertainty
5
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Model reduction: previous state of the art
Linear	4me-invariant	systems:	mature	[Antoulas,	2005]	
‣ Balanced	truncaGon	[Moore,	1981;	Willcox	and	Peraire,	2002;	Rowley,	2005]	
‣ Transfer-funcGon	interpolaGon	[Bai,	2002;	Freund,	2003;	Gallivan	et	al,	2004;	Baur	et	al.,	2001]	
+ Accurate,	reliable,	cer(fied:	sharp	a	priori	error	bounds	
+ Inexpensive:	pre-assemble	operators	
+ Structure	preserva(on:	guaranteed	stability
Ellip4c/parabolic	PDEs:	mature	[Prud’Homme	et	al.,	2001;	Barrault	et	al.,	2004;	Rozza	et	al.,	2008]	
‣ Reduced-basis	method	
+ Accurate,	reliable,	cer(fied:	sharp	a	priori	error	bounds,	convergence	
+ Inexpensive:	pre-assemble	operators	
+ Structure	preserva(on:	preserve	operator	properGes
Nonlinear	dynamical	systems:	ineffecGve	
‣ Proper	orthogonal	decomposiGon	(POD)–Galerkin	[Sirovich,	1987]	
- Inaccurate,	unreliable:	ogen	unstable	
- Not	cer(fied:	error	bounds	grow	exponenGally	in	Gme		
- Expensive:	projecGon	insufficient	for	speedup	
- Structure	not	preserved:	dynamical-system	properGes	ignored
6
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.,	Choi,	Sargsyan,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
7
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
8
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011*;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.,	Choi,	Sargsyan,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
Collaborators:		
‣ Malhew	Barone	(Sandia)	
‣ Harbir	AnGl	(GMU)
			
‣ Charbel	Farhat	(Stanford	University)	
‣ Julien	CorGal	(Stanford	University)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Training:	Solve	ODE	for																							and	collect	simulaGon	data		
2. Machine	learning:	IdenGfy	structure	in	data	
3. Reduc(on:	Reduce	the	cost	of	solving	ODE	for	
µ 2 Dtraining
Training simulations: state tensor
9
dx
dt
= f(x; t, µ)ODE:
µ 2 Dquery  Dtraining
D
number	of	
Gme	steps	T
number	of	
state	variables	N
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Training:	Solve	ODE	for																							and	collect	simulaGon	data		
2. Machine	learning:	IdenGfy	structure	in	data	
3. Reduc(on:	Reduce	the	cost	of	solving	ODE	for	
µ 2 Dtraining
Training simulations: state tensor
9
dx
dt
= f(x; t, µ)ODE:
µ 2 Dquery  Dtraining
DX =
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Training:	Solve	ODE	for																							and	collect	simulaGon	data		
2. Machine	learning:	IdenGfy	structure	in	data	
3. Reduc(on:	Reduce	the	cost	of	solving	ODE	for	
Tensor decomposition
10
dx
dt
= f(x; t, µ)ODE:
Compute	dominant	leR	singular	vectors	of	mode-1	unfolding
µ 2 Dtraining
µ 2 Dquery  Dtraining
X(1) = = U ⌃ VT
X =
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Training:	Solve	ODE	for																							and	collect	simulaGon	data		
2. Machine	learning:	IdenGfy	structure	in	data	
3. Reduc(on:	Reduce	the	cost	of	solving	ODE	for	
Tensor decomposition
10
dx
dt
= f(x; t, µ)ODE:
Compute	dominant	leR	singular	vectors	of	mode-1	unfolding
columns	are	principal	components	of	the	spa(al	simula(on	data
How	to	integrate	these	data	with	the	computa;onal	model?
µ 2 Dtraining
µ 2 Dquery  Dtraining
X(1) = = U ⌃ VT
X =
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Previous state of the art: POD–Galerkin
1. Training:	Solve	ODE	for																							and	collect	simulaGon	data		
2. Machine	learning:	IdenGfy	structure	in	data	
3. Reduc(on:	Reduce	the	cost	of	solving	ODE	for	
µ 2 Dtraining
µ 2 Dquery  Dtraining
dx
dt
= f(x; t, µ)ODE:
1. Reduce	the	number	of	unknowns 2. Reduce	the	number	of	equaGons
D
DGalerkin	ODE:
dˆx
dt
= T
f( ˆx; t, µ)
dˆx
dt
) = 0
(
(T
(f( ˆx; t, µ)x(t) ⇡ ˜x(t) = ˆx(t)
11
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Captive carry
12
V1
‣ Unsteady	Navier–Stokes ‣ Re	=	6.3	x	106 ‣ M∞	=	0.6
Spa4al	discre4za4on	
‣ 2nd-order	finite	volume	
‣ DES	turbulence	model	
‣ 																	degrees	of	freedom1.2 ⇥ 106
Temporal	discre4za4on	
‣ 2nd-order	BDF	
‣ Verified	Gme	step		
‣ 																	Gme	instances
t = 1.5 ⇥ 10 3
8.3 ⇥ 103
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
High-fidelity model solution
13
vor(city	field
pressure	field
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
0
25
50
23
20
17
13
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Principal components
x(t) ⇡ ˆx(t)
1
401
21
101
14
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Galerkin performance
15
probe
Can	we	construct	a	beEer	projec;on?
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	204
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
high-fidelity:	
dim	1.2x106
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
pressure	at	probe
1.6
2.0
2.4
2.8
Gme
0 2 4 6 8 10 12
- Galerkin	projec(on	fails	regardless	of	basis	dimension
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Galerkin: time-continuous optimality
ODE Galerkin	ODE
dˆx
dt
= T
f( ˆx; t, µ)
+ Time-con(nuous	Galerkin	solu(on:	opGmal	in	the	minimum-residual	sense:
dˆx
dt
= T
f( ˆx; t, µ)
- Time-discrete	Galerkin	solu(on:	not	generally	opGmal	in	any	sense
dx
dt
= f(x; t) f( ˆx; t)
O∆E Galerkin	O∆E
rn
(x) := ↵0x t 0f(x; tn
) +
kX
j=1
↵j xn j
t
kX
j=1
j f(xn j
; tn j
)
rn
(xn
) = 0, n = 1, ... , Nn = 1, ... , T T
rn
( ˆxn
) = 0, n = 1, ... , Nn = 1, ... , T
16
r(v, x; t) := v f(x; t)
dˆx
dt
(x, t) = argmin
v2range( )
kr(v, x; t)k2
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Residual minimization and time discretization
ODE
residual	
minimiza(on
(me	
discre(za(on
dx
dt
= f(x; t)
Galerkin	ODE
dˆx
dt
(x, t) = argmin
v2range( )
kr(v, x; t)k2
, n
(ˆxn
)T
rn
( ˆxn
) = 0ˆxn
= argmin
v2range( )
kArn
(v)k2
n
(ˆxn
) := AT
A(↵0I t 0
@f
@x
( ˆxn
; t))
Least-squares	Petrov–Galerkin	(LSPG)	projec(on
residual	
minimiza(on
LSPG	O∆E
ˆxn
= argmin
v2range( )
kArn
(v)k2
[C.,	Bou-Mosleh,	Farhat,	2011]
n = 1, ... , T
(me	
discre(za(on
O∆E
rn
(xn
) = 0
n = 1, ... , T
Galerkin	O∆E
T
rn
( ˆxn
) = 0
n = 1, ... , T
17
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Discrete-time error bound
18
Theorem	[C.,	Barone,	AnGl,	2017]
If	the	following	condiGons	hold:	
1.													is	Lipschitz	conGnuous	with	Lipschitz	constant	
2.	The	Gme	step								is	small	enough	such	that																																											,	
3.	A	backward	differenGaGon	formula	(BDF)	Gme	integrator	is	used,	
4.	LSPG	employs												,	then	
f(·; t) 
0 < h := |↵0| | 0| tt
+ LSPG	sequen(ally	minimizes	the	error	bound
A = I
kxn
ˆxn
Gk2 
1
h
krn
G( ˆxn
G)k2+
1
h
kX
`=1
|↵`|kxn `
ˆxn `
G k2
kxn
ˆxn
LSPGk2 
1
h
min
ˆv
krn
LSPG( ˆv)k2+
1
h
kX
`=1
|↵`|kxn `
ˆxn `
LSPGk2
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
LSPG performance
19
probe
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	204
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
high-fidelity:	
dim	1.2x106
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG:	dim	368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG:	dim	204FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3LSPG:	dim	564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
pressure	at	probe
1.6
2.0
2.4
2.8
Gme
0 2 4 6 8 10 12
+ LSPG	is	far	more	accurate	than	Galerkin
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
20
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013*]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.,	Choi,	Sargsyan,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Wall-time problem
21
probe
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	204
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
high-fidelity:	
dim	1.2x106
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
Galerkin:	dim	564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG:	dim	368
FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3
LSPG:	dim	204FOM
Gal1
Gal2
Gal3
LSPG1
LSPG2
LSPG3LSPG:	dim	564
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
1.6
1.8
2
2.2
2.4
2.6
2.8
pressure	at	probe
1.6
2.0
2.4
2.8
Gme
0 2 4 6 8 10 12
‣ High-fidelity	simula(on:	1	hour,	48	cores
‣ Fastest	LSPG	simula(on:	1.3	hours,	48	cores
Why	does	this	occur?	
Can	we	fix	it?
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Training:	collect	residual	tensor									while	solving	ODE	for	
2. Machine	learning:	compute	residual	PCA						and	sampling	matrix
3. Reduc4on:	compute	regression	approximaGon
Cost reduction by gappy PCA [Everson and Sirovich, 1995]
minimize
ˆv
k A rn
( ˆv)k2
k2
Can	we	select						to	make	this	less	expensive?A
ˆv)k2
rn
⇡ ˜rn
= r(P r)+
Prn
r P
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
minimize
ˆv
k
k2
rn
(
rn
(
rn
(
rn
(
rn
(˜rn
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
index
value
r
˜rn
rn
Prn
Rijk
µ 2 Dtraining
22
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Training:	collect	residual	tensor									while	solving	ODE	for	
2. Machine	learning:	compute	residual	PCA						and	sampling	matrix
3. Reduc4on:	compute	regression	approximaGon
Cost reduction by gappy PCA [Everson and Sirovich, 1995]
minimize
ˆv
k A rn
( ˆv)k2
k2
Can	we	select						to	make	this	less	expensive?A
rn
( ˆv)k2 + Only	a	few	elements		
of	d		must	be	computedrn
rn
⇡ ˜rn
= r(P r)+
Prn
r P
(P r)+
P
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
minimize
ˆv
k
k2
rn
(
rn
(
rn
(
rn
(
rn
(
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
index
value
minimize| {z }
A
r
˜rn
rn
Prn
Rijk
µ 2 Dtraining
22
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Sample mesh [C., Farhat, Cortial, Amsallem, 2013]
vor(city	field pressure	field
LSPG	ROM	with	
32	min,	2	cores
+ 229x	savings	in	core–hours	
+ <	1%	error	in	(me-averaged	drag
+ HPC	on	a	laptop
sample	
mesh
minimize
ˆv
k(P r)+
Prn
( ˆv)k2
A = (P r)+
P
Prn
|{z}
A
high-fidelity	
5	hours,	48	cores
Implemented	in	three	computa;onal-mechanics	codes	at	Sandia
23
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Ahmed body [Ahmed, Ramm, Faitin, 1984]
24
V1
‣ Unsteady	Navier–Stokes ‣ Re	=	4.3	x	106 ‣ M∞	=	0.175
Spa4al	discre4za4on	
‣ 2nd-order	finite	volume	
‣ DES	turbulence	model	
‣ 																	degrees	of	freedom
Temporal	discre4za4on	
‣ 2nd-order	BDF	
‣ Time	step		
‣ 																	Gme	instances
t = 8 ⇥ 10 5
s
1.7 ⇥ 107
1.3 ⇥ 103
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Ahmed body results [C., Farhat, Cortial, Amsallem, 2013]
25
pressure		
field
+ 438x	savings	in	core–hours
+ HPC	on	a	laptop
sample	
mesh
high-fidelity	model	
13	hours,	512	cores
LSPG	ROM	with A = (P r)+
P
4	hours,	4	cores
+ Largest	nonlinear	dynamical	system	on	which	ROM	has	ever	had	success
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
26
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.,	Choi,	Sargsyan,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
27
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.,	Choi,	Sargsyan,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
28
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.,	Choi,	Sargsyan,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Our research
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.	and	Choi,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2018]
29
Collaborators:		
‣ MarGn	Drohmann	(formerly	Sandia)	
‣ Wayne	Uy	(Cornell	University)	
‣ Fei	Lu	(Johns	Hopkins	University)
			
‣ Malhias	Morzfeld	(U	of	Arizona)	
‣ Brian	Freno	(Sandia)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Surrogate modeling in UQ
30
outputsinputs µ surrogate	model qsurr
‣ surrogate	noise	model:		
‣ surrogate	likelihood:		
- inconsistent	with	HFM	noise	model
qmeas = qsurr(µ) + "
⇡surr(qmeas | µ) = ⇡"(qmeas qsurr(µ))
⇡"(·)
‣ high-fidelity-model	(HFM)	noise	model:	
‣ measurement	noise					has	probability	distribuGon	
‣ HFM	likelihood:
"
outputsinputs µ high-fidelity	model qHFM
qmeas = qHFM(µ) + "
⇡HFM(qmeas | µ) = ⇡"(qmeas qHFM(µ))
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Surrogate modeling in UQ
31
qHFM(µ) = qsurr(µ) + (µ)
‣ HFM	noise	model:	
‣ HFM	likelihood: ⇡HFM(qmeas | µ) = ⇡"(qmeas qHFM(µ))
= ⇡"(qmeas qsurr(µ) (µ))
qmeas = qHFM(µ) + "
= qsurr(µ) + (µ) + "
+ equivalent	to	HFM	formulaGon	
+ not	pracGcal:	the	(determinisGc)	error										is	generally	unknown(µ)
How	can	we	account	for	the	error										in	a	manner	that	is	
consistent	and	prac;cal?
(µ)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Surrogate modeling in UQ
32
qHFM(µ) = qsurr(µ) + (µ)
Approach:	sta(s(cal	model												for	the	error	that	models	its	uncertainty˜(µ)
˜qHFM(µ)
| {z }
stochastic
= qsurr(µ)
| {z }
deterministic
+ ˜(µ)
|{z}
stochastic
‣ staGsGcal	HFM	noise	model: qmeas = ˜qHFM(µ) + "
= qsurr(µ) + ˜(µ) + "
+ consistent	with	HFM	noise	model	
+ pracGcal	if	the	staGsGcal	error	model					is	computable
⇡]HFM
(qmeas | µ) = ⇡"+˜(qmeas qsurr(µ))‣ stochasGc	HFM	likelihood:
˜
Desired	proper4es	in	sta4s4cal	error	model	
1.	cheaply	computable:	similar	cost	to	evaluaGng	the	surrogate	
2.	low	variance:	introduces	lille	epistemic	uncertainty	
3. 	generalizable:	correctly	models	the	error
˜(µ)
How	can	we	construct	a	sta;s;cal	error	model	for	reduced-order	models?
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Approximate-solution surrogate models
33
r(x(µ); µ) = 0<latexit 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MEw+7Cx12QPr1p1/Qt/3WvtqyIYi4cOypeStCHFReNuytFFPmG3a84UUTxT44xXwoGusCPvMJlOD1rRV3lejREiA7aNJFUyu7HIsMLyugU0X2nzNt2rghCe7NHhClbntbIftJLQ+jCvR6cCyV53+GzMMVqQOG1K7yTcpynhnuEUGOmP3rKiiYYRdmGdwK0QZXZReeueQc3iaaxSMZ20pKjG86T+RM6313jWjTD3Ki4K+2Q52dWqb/JIuxTo7NTT1p5/jFpxkzQ6txCsmRmQvjMuCNBZRG+ASRjdnjOHdOKtk5u7iOCVzB8dmRZwy9oXtLcJpQ1fWjuXcktX2z/Wg4v2NEdkkCrBWYnY1UXHtPlBjgiAFOwcmosZ4CCSbTYKwPpc0Ap/gIGPbLDH8WD3moSdZMZWt52opzoEliw6qw/QlIG9DhKOAhKOvBJmMf6g200fdDL5gLsmeODtkFmlSOM5ysjFo0i/SfKQ7nJOk7Hf5TVunrTGTdrscumsaDG6LvOU3gNp3lzyWyLOlZ0Debq3MmXK424ZdKfmdDma0NOcFgtrEkyhMcCVy21YgivJWbbrZuhGWbjnkxkCLEWASB6ktZAzAFmOAGwGY1l8AcZXrftuigKRk693eCBF5RU9zVvOc+PoAIjn5490bP+eWAja5weTMdZuQOR0OB7y6LsAxJEsOyjBMaw/SAK8gODJrGVQqaItKE0hiqFESOmgClAIKF4tAdDvWOHGFu0S/A8g/ihLUBI8SzCWDGRIaQpRDCXhEuS388jwQu+BQalWSC2CsXGazOjbNXjA8pltogTw/SBsI40y99whWEKd2KbSEqYoHcJqTZfJvCrzDI2vUCngeGaXt4fc1oLHb3UkiFSOfYTfAakguTPGO1j3RTagUiim5ODe8dvX1Ndg98IAWZ1EkVvoNK/D3S5QptIjpbPCQLmRqPDKHscwvDMIAtwNEnGdkUi0W7ZMFHcNJBbcbh5z97ttbMdbhNMb+wInTtA2STkMKicam6dF4lxwZ+tFtjoFk9Q+D/tOS+IxUBgs2X/0Rajy02xVajN7hT7Z3/nGTkd9+mEJS6t2XuIx3WlxgtsYrLa76YoMLHe1g8MJnjtIoi6ZzEdWfUIOp2AY9psMOPS/QDFxQ4x3xZBnUJDuY6nHfsmXj7YdCXtxllHjidO8SzEhWK8B8ewVSbjZwJ0DIPZrHpyNHOhNAzrMuAoTwvpPEzWc6GMazcR/hMfSBLA6cI3mpgT5G50uGSB3F+G2XquyPG0YOq3jY2/BZzNod9fmq1ABubeZ3KYY5EINMngixUTZTY7bEGxAlAi0B++BIRMBVKuUuKpWVZaBqjWzHaxf6CBOSI6qaXuVzV+3dATvNSZxxqYzMb2/upScAEEZZTxmsz58liGtihkz5ZVXMl9pabZIVoiiqtp8DlW1iYDyISSGzzdR//Wc77ywImQBRLEb47+tYVNVe7NT1ELIaSag38C8T3hUdjtFqDu4HaFDFzB9C747tdImYPjWnofsA0ZhS6LH1iKfUBcPfNcLTpyh7BIeY3AiOyoRpuoHdwAR4nRGSIh5aMMUURVxnZBdC7wSq2iQa+IsluWpJPvCwVScSFIuuxhL3AFpap0+8p08kqeOWB3y79DwTkXwt5mEBTKUIZM98ontYiXbS/Dnfyi67YtzP13PqqmRjchdZFAT2AVJbEYkRWjkXtH6a8XZDZY2BNRfx2VC3rp8uly8xR2FXOV+I+5Bxh3AMqLsSvHEWIhRgDK4wm1FDtsTGSJbLXGq2qlz1UsCZLXJJ1BjkolyhBGnAJjno/R1YHtkuLLCwsRrlcT3IRtlgng3t+RpUI3smpm7HziKS3MJTXZWwN5IB30TyzYNQTQb8dTe9zxx3+4r9v20Ka8kUReDAjJI0YqeOijm4EEhgdU8LtL+OBp4wQ23xX57lzgg4hna5Skg+DQMmjBHAjjd4serifM+53UwjDtbwHEYvzJ2FsJV1DsIafGw+n6Wm6lQTnwnvLoduhIHq9pGEaDLcJmqXOcVZuTnDyoUl+M52wFgL9arY/BKMvlVAz5QOa+kTCaIBShZ+rHHfUWvyU8QFGmCSv1NJ3xXrQNJ4NN4MZp0IbCWCSTw269OE4VlOK66lR5w2S3kaZ5e0/eokMOJE6/JEyeEHRCnXMkACJ7oAALvYP4yc3qaRwhVGyy5FasdHbpaGNho8QipziGHnV+JczcCUcUuicm+dOmNaGVCjMsIH4CmbrNLCpyBJm6GS7COQduToW6fXPPofL4laxnrLVUpQ8xFXhGwGYzvZsNjhut9qffJUl/aGinFYXdj3UTfrSmTPEsmme62H4fvbToSZK6+HAGhhZfE2csvR0rodmmeLnnf+YVtt1N/wn4b+FgeKuCxM+jdkxqI/DoFldloQh2pchffCgqLIwsesCZcz7AubIU+IBTdUTlnShME29X+i/PuStqKxa15dpn0+JLFA8kNHfKX9xonYq9u2zfmbdQD109MsuswgdzK98lnr/01o5SX5it0vVwn9h42VGO6+65jzx6vo86hKhHtahn8qK1g8hiPsLfdBHtLxV37CK9HI/jfJshTtjbXyrjVPKdghR1tF55bQte+kjF43uqU2lezvbVDrir4tvFneAaeidNO+rW2545KDeuqQh00hNos1beK6s8h6pJDJuQgj8sKneXsbxkw54t7igqa94XvcIM2YTT2tjBLo5EP72vqKnWjUryAnpHAWy00jcV/LLGB0I/OchQJYavUaUEr3cphYwStRCDDHUW04P5rYJCRjYucBaR4H0aF3zgSslzBxSitSV45SFudmh3mF2w7c9jzRSVeIIhZDRNyYMbnMDXxy3zYAqD8cIz+Omm23gvDbB0yAf6JitDbsgHD/eaSgksR+TWze5D0zB0BmXNzZWfHdGypJPuoicEPUc03qNk/ZsARhiTyJjOoWKW/o35PBXDAwl6UE0M0/cFQHTjqIe0e5W6Rcx8uWPKW7T4hrjAwH1IaD4OSyKNzmVDHsJoj/VXUYxjslsuLQG2S2y8N+Y/5mEYc5FFUM41FA+h/3YQxXrFNWOrDF/YVgKM3T8+wWPJfN4EzyUeYSQEidxzGf6PM7PdkK0m8upUdYDXSl8YK0jmOR4aW2R0XQnuc1RDLvKriEsWM/tR2wvHKkRyyYamFXVn8zPcc0Yva0KDH9BeoXMQVPcAM/rKIEDSLK3V+7MQFkfflVuRiJbNnbE8P8XjOUuwWUKMF3ayQWbHrjLWtVyib1XN3K1x7L/FZyYZh7RGfsugpdCcOwJ42QVCBbskP4/QvG/rBCQPUuPqMn295bxdNFhfVPK9F/PC9E/inEt5l0zydqP2x97hP0zYHotlpO6HSrg2JsYWeBIVq75T1vQDTBNriXwTF8+1GIfqlF2SLfRkUKyITUvBpAGaLPlWi3Zv4pBWxJkXx7K0y7PJ8oMFcLeJbGhC94ijynRyKED8NRgQAr1UgV30qvFzE2RA/28Vfumfi6FUhAOh6rxhzHacWTLz5D1Z+jcYUnKgiWvqO9Io4jDHf/9kbURGTi80NEOTr9iw5ZKCkv7+I9G2tppzhwWPnm8e9ncff9Hb29v+w35o4PfbK0Xvf7Pd2vvnDt1/3dnb3LDjdEeE4nE7+ZyH4G+Y5Zv9rrHb3j/vf9nb29/+IoSUyXZ9FkuLhfhovU/pTW1Px+fKzjT3715ndD+ePdvZ2d/Z+2N/407/zX27+7dq/rP3b2tba3tq3a39ae7nWX3u3Nl77z7X/Wvvvtf/59O2ni0+TT1MG/fWvOOf3a8a/T/n/AZHNsGc=</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
High-fidelity	model	
‣ governing	equaGons:	
‣ quanGty	of	interest:		
Types	of	approximate	solu4ons	
‣ Reduced-order	model:	
‣ Low-fidelity	model:			
‣ Inexact	solu(on:	compute																																	such	that
rLF(xLF; µ) = 0, ˜x = p(xLF)<latexit 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</latexit><latexit sha1_base64="hcF8sa3hvaquhsoiV7TN+x2EoNA=">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
T
r( ˆx; µ) = 0, ˜x = ˆx
x(k)
, k = 1, ... , K
kr(x(K)
; µ) = 0k2  ✏, ˜x = x(K)
Approximate-solu4on	surrogate	model	
‣ approximate	soluGon:	
‣ quanGty	of	interest:		
˜x(µ) ⇡ x(µ)
qsurr(µ) := q(˜x(µ))
qHFM(µ) := q(x(µ))
What	methods	exist	for	quan;fying	the	error																																																?(µ) := qHFM(µ) qsurr(µ)
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1) Error indicators: residual norm
34
‣ Applica(ons:	terminaGon	criterion,	greedy	methods,	trust	regions	
[Bui-Thanh	et	al.,	2008;	Hine	and	Kunkel,	2012;	Wu	and	Hetmaniuk,	2015;	Zahr,	2016]	
+ Informa(ve:	zero	for	high-fidelity	model	
- Determinis(c:	not	a	staGsGcal	error	model	
- Low	quality:	relaGonship	to	error	depends	on	condiGoning
kr(˜x; µ)k2<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
‣ SubsGtute	(2)	into	the	residual	of	(1)	and	take	the	norm:
r(x(µ); µ) = 0<latexit 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</latexit><latexit sha1_base64="t1vvd/XPojOY0yVN3uB39k9YovM=">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
˜x(µ) ⇡ x(µ)
‣ HFM	governing	equaGons:
‣ Approximate	soluGon:
(1)
(2)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1) Error indicators: dual-weighted residual
35
‣ Solve	for	the	error
(2)x ˜x = [
@r
@x
(˜x)] 1
r(˜x) + O(kx ˜xk2
)
<latexit 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</latexit><latexit 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</latexit><latexit 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2QPLlt18wp/3W3tWxoYi4cOypeStCHFReNuytFFPmEXW04VUTxT44xXwr6usCPvIJlO91vRV3lejREiA7aNJFUyu7HIsMLyugU0X2nzNt2rghCe7NHhClbntbIftJLQ+jCvR6cCyV53+GzMMVqQOG1KrwPcpSnhnuEUGOmP3rKiiYYRdmEdw60QZXZReeueQc3iaaxSMZ20pKjG86T+RM6313jWjTD3Mi4K+2Q52dWqr/NIu4/n7NTT1p5/jFpxiTM6sxCsmRmQvjMuCNBpRC9fSRjdnjOHdOKtk0uziOCVzJ8dmRZwy9oXtLcJpQ1fWzuXcktX2z/Wg4v2NEdkkCrBWYnYrUHHtPl+jgiAFOwcmosZYD+SbTYKwPpc0Ap/gIEPbbDH8WBXioSdZMZWF42opzoEliw6qw/QlIG9DhIOAxIOvRJmMf6g200fdDJ5n7smeODtkFmlSOM5ysidn0i/SfKA7nJOk7Hf5TVunrTGJdbsYumsaDG6LvOU3gNp3lzwWyLOlZ19ebq3MmXK424ZdKfmZDma0NOcFgtrEkyhMcCVy21YgivJWbbrZuhGWbjnkxkCLEWASB6ktZAzAFmOAGwGY1l8AcZXrftaiAKRk6+3eCBF5SU9zVvOc+PoAIjn5490bP+OWAja5weTMdZuQOR0OB7y6DV8cSTLDkpwDOsPkgAvIHgyaxlUqmgLSlOIYigRUjqoAhQCildLAPQ7VrixRbsE/wOIP8oSlATPEowlAxlSmkIUQ0m4BPntPDK80HtgUKoVUotgbJwmM/piCx6wfGabKAF8PwjbSKPMPXcIllAntqm0hClKh7Ba02UyL8s8Q+NLVAo4ntnl7SG3teDxWx0JIpVjH+F3QCpI7ozxDtZ9hwyoFIopObh3/OIz9TXYvTBAVidR5AI4zetwpwuUqfRI6awwUG4kKryyxzEM7wyCAHeDRFxnJBLtli0TxV0DiQW3m8fc/W4b2/EW4fTGvsCJE7RNUg6Dyom9WI4UiXO3nK0X2eoUTFL7POw7LYnHQGGwZP/RF6HKT7NVqc3sFfpkf+cbOx316YclLK3aeYnHdKfFCW5jsNrupisysNzVDg4neO4gibpkMh9Z9Qk5nIJh2G8y4ND/AsXEDTFe00KeQUG6j6Ue+yVfPtp2JOydVUaNJ07zLsWEYL2Bw7NXJOFmA3cOgNhvWHA2cqBL/jrMuAoTwvpPEzWc6GMazcR/hMfSBLA6cI3mpgT5G50uGSB3F+G2XquyPG0YOq3jY2/BZzNod9fmq1ABubeZ3KYY5EINMngixUTZTY7bEGxAlAi0B++BIRMBVKuUuKpWVZaBqjWzHaxf6CBOSI6qaXuVzd90dAjvNSZxxqYzMb2/upCcAEEZZTxmsz58liGtihkz5ZVXMl9pabZIVoiiqtp8DlW1iYDyISSGzzdR//WM77ywImQBRLEb47+tYVNVe7NT1ELIaSag38C8T3hUdjtFqDu4HaFDFzB9C747tdImYPjWnofsA0ZhS6LH1iKfUBdPlN40x19r8ubcrCWJM5RdwGMMTmRHJcJU/eAOIEKczggJMQ9tmCKqIiavPD7AOcwqGuSaOItleSrJvnAwFSeSlMsuxhJ3QJpap498J4/kqSNWh/w7NLxTEfxtJmGBDGXIZI98YrtYyfYS/Pkfim774sxP17NqamQjchcZ1AR2QRKbEUkRGrlXtP5acXaDpQ0B9VdxmZAXHp8sF29xRyFXud+Ie5BxB7CMKLtSPDEWYhSgDK5wW5HD9kSGyFZLnKp26lz1kgBZbfIJ1JhkohxhxCkA5vkofR3YHhmurLAw8Vol8X3IRpkg3s0teRpUI7tm5u4HjuLSXEKTnRWwN9JB38SyTUMQzUY8tfc9T9wX64p9P23KK0nUxaCADFK0oqcOijl4UEhgNY+LtD+OBl5ww22x394lDoh4hnZ5Cgg+DYMmzJEATrf48WrivMt5HQzjzhZwHMavjJ2FcBX19kNaPKy+n+VmKpQT3wmvboeuxMGqtlEE6DJcpirXeYUZ+eGDCsXleM52ANg77eoYvJL8ibzQlrV1522QyQSxACVLP/K4r+g1eft/kSao1N90wnfVOpAEPo0Xo0kXAmuZQAK//eo0UViG46pb6QGX3UKe5OkVfY8KOZw48Zo8cULYAXHKlQyA4IkOIPAO5i8zp6d5hFC1wZJbsdrRoauFgY0Wj5DqHHLY+ZU4dyMQVeySmOxLF96IVibEuIzwAWjqNrukwBlo4ma4BOsYtD0Z6vbJNY/O51uylrHeUpUyxFzkFQGbwfhuNjxmuN6XepUr9aWtkVIcdjfWTSS+qZI8SyaZ7rYfh+9tOhJkrr4cAaGFl8TZyy9HSuh2aZ4ued/5zLbbqT9hv4h7LA8V8NgZ9O5JDUR+GILKbDShjlS5i28FhcWRBQ9YE65nWBe2Qh8Qiu6onDOlCYLtav/svLuStmJxa55dJj26YPFAckOH/OW9xonYq9v2jXkbdd/1E5PsKkwgt/J98tkbd80o5YX59lov14m9hw3VmO6+69izx+uoc6hKRLtaBj9qK5g8xiPsbTfB3lJx1z7C69EI/rcJ8pStzZUybjXPKVhhR9uF55bQla9kDJ63OqX21Wxv7ZCrCr5t/BmegWfitJN+re25o1LDuqpQBw2hNkv1raL6c4i65JAJ2c/jskKnOftbBsz56o6iguZ95TvcoE0Yjb0tzNJo5MP7mrpK3agU735nJPBWC01j8R9LbCD0o7McRULYKnVa0Eq3ctgYQSsRyHBHES24/xoYZGTjImcBKd6HUeE3joQsV3AxSmuSVw7SVqdmh/kF284c9nxRiRcIYlbDhByY8TlMTfwyH7YAKD8co79Omq33wjBbh0yAfx0i9LZswHC/uaTgUkR+SOwOJD1zh0Dm3FzZ2TEdWyrJPmpi8ENU8w1q9u8IcIQhibzJDCpW6e+onzIBHLCwF+XEEE1/MFQHjnpIu0e5W+TchwuWvGW7T4grDMyHlMbDoCTy6Fwm1DGs5kh/FfUYBrvl8iJQm+T2S0P+Yz6mEQd5FNVMY9EA+l83YYxXbBOW+uCFfQXg8M3TUyyW/NdN4EzyEWZSgMgdh/GfBzP7PdlKEq9uZQdYjfSlsYJ0juORoWV2y4XQHmc1xDKvqrhEMaM/tZ1wvHIkh2xYamFXFj/zPUf0ojY06DH9BSoXcUUPMIM/6iEEzeJKnR87dkHkfbkVuVjJ7Bnb00M8nrMUuwXUaEE3K2RW7DpjbesVymb13N0K195LfFqyYVh7xKcsegrdiQOwp00QVKAb8ps0/YuGfnDCADWuPuOXU97bRZPFRTXPaxE/fO8E/qmEd9k0Tydqf+w97tO0zYFodtpOqLRrQ2JsocdBodo7ZX0vwDSBtvgXQfF8u1GIfukF2WJfBsWKyIQUfBKA2aJPlGj3Jj5pRaxJUTx7qwy7PB9oMJeL+IYGRC85inwnhyLEr3IRAcBrFchVnwovF3E2xC9m8ZfumTh6VQgAut4rxlzFqQUTb/6DlV+hMQUnqoiWviO9Ig5jzPd/9kZUxORicwME+bo9Sw4ZKOlPHyJ9W6spZ3jwePjd497Dx9/1Hu7u/WGvNXF67JWjd7/b6z387g/ff9t7uLNrwemOCMfhdPI/C8HfMM8xe99itTt/3Pu+93Bv748YWiLT9VkkKR7up/Eypb9yNRWfL77Y2LV/GNn9cPbo4e7Ow92f9jb+9O/8R5N/t/ava/+2trW2u/b92p/WXq71196tje/99l7v3rf3vvu0++nnT8NP/8mgv7nHOf+yZvz7lPwvunGo8g==</latexit><latexit 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</latexit>
‣ Applica(ons:	adapGve	mesh	refinement		
[Babuska	and	Miller,	1984;	Becker	and	Rannacher,	1996;	Rannacher,	1999;	Vendit	and	Darmofal,	2000;	Fidkowski,	2007]	
+ Accurate:	second-order-accurate	approximaGon	
- Determinis(c:	not	a	staGsGcal	error	model
‣ Approximate	HFM	residual	to	first	order
0 = r(x) = r(˜x) +
@r
@x
(˜x)(x ˜x) + O(kx ˜xk2
)
<latexit 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</latexit><latexit 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
‣ Approximate	HFM	quanGty	of	interest	to	first	order
(1)q(x) = q(˜x) +
@q
@x
(˜x)(x ˜x) + O(kx ˜xk2
)
‣ SubsGtute	(2)	in	(1): q(x) q(˜x) = yT
r(˜x) + O(kx ˜xk2
)
@r
@x
(˜x)T
y =
@q
@x
(˜x)T
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
2) Rigorous a posteriori error bound
36
Proposi4on
If	the	following	condiGons	hold:	
1.													is	inf–sup	stable,	i.e.,	for	all												,	there	exists																			s.t.	
				
2.									is	Lipschitz	conGnuous,	i.e.,	there	exits													such	that																					
	then	the	quanGty-of-interest	error	can	be	bounded	as
r(·; µ)<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
µ 2 D<latexit 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</latexit><latexit sha1_base64="8ngsVhJumVyxt51asezAd56Tyv0=">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
↵(µ) > 0
q(·) > 0
|q(x) q(˜x)| 
↵
kr(˜x; µ)k2
‣ Applica(ons:	reduced-order	models	
[Rathinam	and	Petzold,	2003;	Grepl	and	Patera,	2005;	Antoulas,	2005;	Hinze	and	Volkwein,	2005;	C.	et	al.,	2017]	
+ Cer(fica(on:	guaranteed	bound	
- Lack	sharpness:	orders-of-magnitude	overesGmaGon	
- Difficult	to	implement:	require	bounds	for	inf–sup/Lipschitz	constants	
- Determinis(c:	not	a	staGsGcal	error	model	
-
kr(z1; µ) r(z2; µ)k2 ↵(µ)kz1 z2k2, 8z1, z2 2 RN
|q(z1) q(z2)|  kz1 z2k2, 8z1, z2 2 RN
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
3) Model-discrepancy approach
‣ Applica(ons:	
‣ Model	calibraGon	[Kennedy,	O’Hagan,	2001;	Higdon	et	al.,	2003;	Higdon	et	al.,	2004]	
‣ MulGfidelity	opGmizaGon	[Gano	et	al.,	2005;	Huang	et	al.,	2006;	March,	Willcox,	2012;	Ng,	Eldred,	2012]	
+ General:	applicable	to	any	surrogate	model	
+ Sta(s(cal:	interpretable	as	a	staGsGcal	error	model	
+ Epistemic	uncertainty	quan(fied:	through	variance	
- Poorly	informa(ve	inputs:	parameters						weakly	related	to	the	error	
- Poor	scalability:	difficult	in	high-dimensional	parameter	spaces	
- Thus,	can	introduce	large	epistemic	uncertainty:	large	variance
µ<latexit 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37
quan(ty	of	interest	
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+VocWx6eKqI5lFq0qW9EvZ1hR15z5PZbL8VfZXn1RghMmDbSFIlsxuLDCssr/eB5ittfkD3qiCEJ3t0uILVea0cBK0ktAHM69OpQLI3HT4bc4wWJE6b0ksBd2lKuGc4BUb6o7esaKJhhHMcH26FKLOLylv3DGoWT2OViumkJUU1WST1T+R8e41n3QhzL+OisE+Wk12t+jqPtFt5zk49be35x6gVVzmjMwvBmpkBGTjjggCdRvQKloTR7TlzSCfeOrk6iwheyfzekWkB71v7gvY2obThc2vnUm7pavvHenDRnuaIDFIlOCsRuzvomLbYzxEBkIJdQHMxA+xHss1GAdiAC1rjDzDwoQ32OB7sYpGwk8zY6roR9VSHwJJFZw0AmjKw30HCYUDCoVfCPMYfdLvpg04m73PXBA+8HTKrFGk8Rxm5+RPpN0ke0l3OWTLxu7zGzZPWuMqajVbOihaj6zJP6T2Q5s2I3xJxruzsy9O9lSlTHnfLoDs1J6vxlJ7mtFhYk2AKjQGuXG7DElxJzrJdN0M3ysK9mM4RYCkCRPIgrYWcA8hyDGAzGMviCzC+at2XQxSInHy9xQMpKi/pad5ykRtHB0A8P3+kYwd3xELQAT+YjLF2AyKnw/GQRy/jiyNZdlCCY1h/kAR4AcGTWcugUkVbUJpCFEOJkNJBFaAQULxeAqDfscKNLdol+M9A/FGWoCR4lmAsGciQ0hSiGErCJchv55Hhhd4Dg1KtkFoEY+M0mdPXW/CA5TPbRAng+0HYRhpl7rtDsIQ6sU2lJUxROoTVmi6TeVnmGZpcolLA8cwubw+5rQWP3+pIEKkc+wi/A1JBcmeMd7Dum2RApVBMycG949efqa/B7oUBsjqJItfAaV6HO12gTKVHSmeFgXIjUeG1PY5heGcQBLgbJOI6I5Fot2yZKO4aSCy43Tzh7nfb2I63CKc39gVOnKBtknIYVE40Nk+LxLlhztaLbHUKJql9HvadlsRjoDBYsv/oi1Dlp9mq1Gb2Gn2yv/ONnY769MMSllbtvMRjutPiBLcxWG130xUZWO5qB4cTPHeQRF0ymY+s+oQcTsEw7DcZcOh/iWLihhgvayHPoCDdx1KP/ZIvH207EvbmKqPGE6d5l2JCsN7D4dkrknCzgTsHQOz3LDgbOdBVfx1mXIUJYf2niRpO9DGNZuI/wmNpAlgduEZzU4L8jU6XDJC7i3Bbr1VZnjYMndbxse/DZzNod9fmq1ABubeZ3KYY5EINMngixUTZTY7bEGxAlAi0B++BIRMBVKuUuK5WVZaBqjWzHaxf6CBOSI6qaXuVzd93dAjvNSZxxqYzMb2/GklOgKCMMh6zWR8+y5BWxZyZ8sorma+0NFskK0RRVW0+h6raRED5EBLD55uo/3rGd15YEbIAotiN8d/WsKmqvdkpaiHkNBPQb2DeJzwqu50i1B3cjtChC5i+Bd+dWmsTMHxrz0P2AaOwJdFja5FPqYsnSm+W4681eX9u1pLEOcpG8BiDE9lRiTBVP7gDiBCnM0JCzEMbpoiqiOuE7FrglVhFg1xTZ7EsTyXZFw5m4kSSctnFWOIOSDPr9JHv5JE8dcTqkH+Hhncqgr/NJCyQoQyZ7JFPbBcr2V6CP/9D0W0Pzvx0PaumRjYid5FBTWAXJLEZkRShkftF668VZzdY2hBQfxWXCXnt8clq+RZ3FHKV+424Bxl3AMuIsivFE2MhRgHK4Aq3FTlsT2SIbLXEqWqnzlUvCZDVJp9AjUkmyhFGnAJgno/S14HtkeHKCgsTr1US34dslAni3dySp0E1smtm7n7gOC7NJTTZWQF7Ix30TSzbNATRbMRTe9+LxH29rtj306a8kkRdDArIIEUreup5sQAPCgms5nGR9sfRwAtuuC3227vkO8LgoV2eAoJPw6ApcySA0y1+vJo473JeB8O4swUch/ErY2chXEX9/ZAWD2vgZ7mZCuXEd8Kr26ErcbCqbRQBugyXqcp1XmFGfnWgQnE5WbAdAPZmuzoGryT/RF5ry9q6807IZIpYgJKlH3ncV/Sa/AZAkSao1N90wnfVOpAEPo2X42kXAmuZQAK//eo0UViG46pb6QGX3UKe5OkVfY8KOZw49Zo8dULYAXHKlQyA4IkOIPAO5i8zp6d5hFC1wZJbs9rRoeuFgY0Wj5DqHHLY+ZU4dyMQVeySmOxLI29EKxNiXEb4ADR1m11S4Aw0cTNcgnUM2p4Mdfvkmkfn8y1Zy1hvqUoZYi7yioDNYHw3Gx4zXO9LvdCV+tLWSCkOuxvrJhLfVEmeJZNMd9uPw/c2HQkyV1+OgNDCS+Ls5ZcjJXS7NE9XvO/8zLbbqT9hv457Ig8V8NgZ9O5JDUR+HoLKbDShjlS5i28FhcWRBQ9YE65nWBe2Rh8Qiu6onDOlCYLtav/ZeXclbcXi1jy7THo0YvFAckOH/OW9xonYq9v2jXkbdd/1E5PsKkwgt/J98tl7d80o5ch8h62X68Tew4ZqTHffdeLZ43XUOVQlol0vgx+1FUwe4xH2tltgb6m4ax/h9WgE/9sCecrW5koZt57nFKywo+3Cc0voylcyBs9bnVL7era3dshVBd82/hzPwHNx2km/1vbCUalhXVWog4ZQm6X61lH9OURdcsiE7OdxWaHTnP0tA+Z8dkdRQfM+8x1u0CaMxt4WZmk08uF9TV2lblSKN8AzEnirhaax+I8lNhD60VmOIiFsnTotaKVbOWyMoJUIZLijiBbcfw0MMrJxkbOAFO/DqPAbR0KWK7gYpTXJawdpq1Ozw/yCbWcOe76oxAsEMathQg7M+BymJn6ZD1sAlB+O0V8nzdZ7YZitQybAvxERels2YLjfXFJwKSI/J3YHkp65QyBzbq7s7JiOLZVkHzUx+CGq+QY1+9cEOMKQRN5kBhWr9HfUD5oADljYi3JiiKY/GKoDRz2k3aPcLXLuwwVL3rLdJ8QVBuZDSuNhUBJ5dC4T6hhWc6S/inoMg91yOQjUJrn90pD/mI9pxEEeRTXTWDSA/tdNmOAV25SlPjywrwAcvnl6isWS/7oJnEk+wkwKELnjMP4jYWa/J1tJ4tWt7ACrkb4yVpDOcTwytMxvuRDa46yGWOZVFZcoZvSnthOOV47kkA1LLezK4me+F4he1IYGPaa/QOUyrugBZvCnPYSgeVyp82PHLoi8L7ciFyuZPRN7eognC5Zit4AaLelmhcyKXWesbb1C2bxeuFvh2nuJT0s2DGuP+JRFT6E7cQD2tAmCCnRDfplmMGroBycMUOPqM34/5b1dNFlcVIu8FvHD907gn0p4l83ydKr2x97jPk3bHIhmp+2ESrs2JMYWehwUqr1T1vcCTBNoiz8IiufbjUL0Sy/IFvsyKFZEJqTgkwDMFn2iRLs38UkrYk2K4tlbZdjl+UCDuVzGNzQgeslR5Ds5FCF+m4sIAF6rQK76VHi5iLMhfjeLv3TPxNGrQgDQ9V4x5ipOLZh48x+s/ApNKDhRRbTyHekVcRhjvv+DN6IiJhebGyDI1+1ZcshASX8AEenbWk05x4PH9leP+9uPv+pv7+79dq81cXrslaN3v9rrb3/126+/7G/v7FpwuiPCcTid/M9C8DfMc8zel1jtzu/2vu5v7+39DkNLZLo+yyTFw/0sXqX0t65m4vPok96u/fPI7oezR9u7O9u73+31fv9P/KeTf7XxDxv/uHF/Y3fj643fb7zcGGy825hstBv/svGvG/92+++3/3H7X7f/zaC//AXn/N2G8e/2f/4Prcx7dg==</latexit><latexit sha1_base64="+dEK4DwuTqbUh1zw5Peof32SZXo=">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
qsurr
-5
0
5
10
parameter µ
= qHFM qsurr
˜”(µ) ≥ N(µ(µ); ‡2
(µ))
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Objective
38
Error	modeling:	staGsGcal	model	for	the	error	
‣ strength	of	#3	
+ Sta(s(cal:	interpretable	as	a	staGsGcal	error	model		
+ Epistemic	uncertainty	quan(fied:	through	variance
A	posteriori:	use	residual-based	quanGGes	computed	by	the	surrogate	
‣ strength	of	#1	and	#2		
+ Informa(ve	inputs:	quanGGes	are	strongly	related	to	the	error	
+ Thus,	can	lead	to	lower	epistemic	uncertainty:	lower	variance
Goal:	combine	the	strengths	of	
1. error	indicators,	
2. rigorous	a	posteriori	error	bounds,	and	
3. the	model-discrepancy	approach
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Main idea
39
Key observation
10 5
10 4
10 5
10 4
Residual r/error bound
error(energy
norm)|||uhured|||
10 5
10 4
10 5
10 4
Residual r
error(energy
norm)|||uhured|||
(r; ||| u|||)
( µ
u ; ||| u|||)
ROMs generate error indicators that
correlate with the error
ROMES Kevin Carlberg, Martin Dr
+ Can	produce	lower-variance	models	than	the	model-discrepancy	approach
Idea:	Apply	machine	learning	regression	to	generate	a	mapping	from		
residual-based	quan((es	to	a	random	variable	for	the	error
Machine-learning	error	models
‣ Observa4on:	residual-based	quanGGes	are	informaGve	of	the	error
‣ So,	these	are	informaGve	features:	can	predict	the	error	with	low	variance
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Machine-learning error models: formulation
40
‣ features:		
‣ regression	funcGon:		
‣ noise:			
‣ Note:	model-discrepancy	approach	uses														
fl(µ) œ RNfl
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‘(fl)<latexit 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”(µ) = f(fl(µ))
¸ ˚˙ ˝
deterministic
+ ‘(fl(µ))
¸ ˚˙ ˝
stochastic<latexit 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sha1_base64="QeA/IRV43eQBeSyQq50pSBRI7Xs=">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</latexit>
f(fl) = E[” | fl]<latexit sha1_base64="0bLzeycmdaBFc1feZw7k+ZGI2EA=">AAArDnicnVrdcty2Fd6kf4n657SXveFU4xlbXWu0imwn6aiNbdmJXVuSV7KcjijvYLlYLiISpEFwJZlmn6FPksvedXrbB+hNL9pn6QEIgiAIrp1qkjEBfOcc4BycHwA7TSOS8a2tf3/w4Q9++KMf/+Sjj9d++rOf/+KX1z751UmW5CzAL4IkStg3U5ThiFD8ghMe4W9ShlE8jfDL6fkDMf5yiVlGEnrMr1J8
fl = µ
‣ Desired	properGes	in	error	model	
1.	cheaply	computable:	features											are	inexpensive	to	compute	
2.	low	variance:	noise	model										has	low	variance	
3.	generalizable:	empirical	distribuGons	of					and						‘close’	on	test	data
fl(µ)
˜‘(fl)
” ˜”
˜”
‣ regression-funcGon	model:		
‣ noise	model:
˜f(¥ f)
˜”(µ) = ˜f(fl(µ))
¸ ˚˙ ˝
deterministic
+ ˜‘(fl(µ))
¸ ˚˙ ˝
stochastic
˜‘(¥ ‘)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Training and machine learning
1. Training:	Solve	high-fidelity	and	mulGple	surrogates	for																							µ 2 Dtraining
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for µ 2 Dquery  Dtraining
D
41
= qHFM qsurr<latexit 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8keXrbq/hX+utfadzUwFg8dlC8laUOKi8bdlKPFsenhmSKaR6lJl/ZKONAVduQ9T2azg1b0VZ5XY4TIgG0jSZXMbiwyrLC83gear7T5Ad2rghCe7NHhClbntbIftJLQ+jCvR6cCyd50+GzMMVqQOG1KLwXcpSnhnuEUGOmP3rKiiYYRznF8uBWizC4qb90zqFk8jVUqppOWFNVkkdSfyPn2Gs+6EeZexkVhnywnu1r1dR5pt/KcnXra2vOPUSuuckbnFoI1MwPSd8YFATqL6BUsCaPbc+aQTrx1cnUWEbyS+YMj0wLet/YF7W1CacMX1s6l3NLV9o/14KI9zREZpEpwViJ2d9AxbXGQIwIgBbuA5mIGOIhkm40CsD4XtMYfYOAjG+xxPNjFImEnmbHVdSPqqQ6BJYvO6gM0ZWCvg4SjgIQjr4R5jD/odtMHnUw+4K4JHng7ZFYp0niOMnLzJ9Jvkjyku5yzZOJ3eY2bJ61xlTUbrZwVLUbXZZ7SeyDNmxG/JeJc2TmQp3srU6Y87pZBd2pOV+MpPc1psbAmwRQaA1y53IYluJKcZbtuhm6UhXsxnSPAUgSI5EFaCzkHkOUYwGYwlsUXYHzVui+HKBA5+XqLB1JUXtLTvOUiN44OgHh+/kjH9u+IhaB9fjAZY+0GRE6H4yGPXsYXR7LsoATHsP4gCfACgiezlkGliragNIUohhIhpYMqQCGgeL0EQL9jhRtbtEvw34D4oyxBSfAswVgykCGlKUQxlIRLkN/OI8MLvQcGpVohtQjGxmkyp6+34AHLZ7aJEsD3g7CNNMrcc4dgCXVim0pLmKJ0CKs1XSbzsswzNLlEpYDjmV3eHnJbCx6/1ZEgUjn2EX4HpILkzhjvYN03yYBKoZiSg3vHrz9TX4PdCwNkdRJFroHTvA53u0CZSo+UzgoD5Uaiwmt7HMPwziAIcDdIxHVGItFu2TJR3DWQWHC7ecLd77axHW8RTm/sC5w4Qdsk5TConGhsnhaJc8OcrRfZ6hRMUvs87DsticdAYbBk/9EXocpPs1Wpzew1+mR/5xs7HfXphyUsrdp5icd0p8UJbmOw2u6mKzKw3NUODid47iCJumQyH1n1CTmcgmHYbzLg0P8SxcQNMV7WQp5BQbqPpR77JV8+2nYk7M1VRo0nTvMuxYRgvYfDs1ck4WYDdw6A2O9ZcDZyoKv+Osy4ChPC+k8TNZzoYxrNxH+Ex9IEsDpwjeamBPkbnS4ZIHcX4bZeq7I8bRg6reNj34fPZtDurs1XoQJybzO5TTHIhRpk8ESKibKbHLch2IAoEWgP3gNDJgKoVilxXa2qLANVa2Y7WL/QQZyQHFXT9iqbv+/oCN5rTOKMTWdien81kpwAQRllPGazPnyWIa2KOTPllVcyX2lptkhWiKKq2nwOVbWJgPIhJIbPN1H/9ZzvvLAiZAFEsRvjv61hU1V7s1PUQshpJqDfwLxPeFR2O0WoO7gdoUMXMH0Lvju11iZg+Naeh+wDRmFLosfWIp9SFw98WzBOnKNsBI8xOJEdlQhT9YM7gAhxOiMkxDy0YYqoirhOyK4FXolVNMg1dRbL8lSSfeFgJk4kKZddjCXugDSzTh/5Th7JU0esDvl3aHinIvjbTMICGcqQyR75xHaxku0l+PM/FN328NxP17NqamQjchcZ1AR2QRKbEUkRGrlXtP5acXaDpQ0B9VdxmZDXHp+ulm9xRyFXud+Ie5BxB7CMKLtSPDEWYhSgDK5wW5HD9kSGyFZLnKp26lz1kgBZbfIJ1JhkohxhxCkA5vkofR3YHhmurLAw8Vol8X3IRpkg3s0teRpUI7tm5u4HjuPSXEKTnRWwN9JB38SyTUMQzUY8tfe9SNzX64p9P23KK0nUxaCADFK0oqcOigV4UEhgNY+LtD+OBl5ww22x394l3xEGD+3yFBB8GgZNmSMBnG7x49XEeZfzOhjGnS3gOIxfGTsL4SrqHYS0eFh9P8vNVCgnvhNe3Q5diYNVbaMI0GW4TFWu8woz8nsFFYrLyYLtALA329UxeCX5E3mtLWvrzjshkyliAUqWfuxxX9Fr8hsARZqgUn/TCd9V60AS+DRejqddCKxlAgn89qvTRGEZjqtupQdcdgt5mqdX9D0q5HDi1Gvy1AlhB8QpVzIAgic6gMA7mL/MnJ7mEULVBktuzWpHh64XBjZaPEKqc8hh51fi3I1AVLFLYrIvjbwRrUyIcRnhA9DUbXZJgTPQxM1wCdYxaHsy1O2Tax6dz7dkLWO9pSpliLnIKwI2g/HdbHjMcL0v9UJX6ktbI6U47G6sm0h8UyV5lkwy3W0/Dt/bdCTIXH05AkILL4mzl1+OlNDt0jxd8b7zM9tup/6E/TruiTxUwGNn0LsnNRD5eQgqs9GEOlLlLr4VFBZHFjxgTbieYV3YGn1AKLqjcs6UJgi2q/1n592VtBWLW/PsMunxiMUDyQ0d8pf3Gidir27bN+Zt1APXT0yyqzCB3Mr3yWfv3TWjlCPzHbZerhN7DxuqMd1914lnj9dR51CViHa9DH7UVjB5jEfY226DvaXirn2E16MR/G8b5Clbmytl3HqeU7DCjrYLzy2hK1/JGDxvdUrt69ne2iFXFXzb+HM8A8/FaSf9WtsLR6WGdVWhDhpCbZbqW0f15xB1ySETcpDHZYXOcva3DJjz2zuKCpr3W9/hBm3CaOxtYZZGIx/e19RV6kaleAM8I4G3Wmgai/9YYgOhH53lKBLC1qnTgla6lcPGCFqJQIY7imjB/dfAICMbFzkLSPE+jAq/cSRkuYKLUVqTvHaQtjo1O8wv2HbmsOeLSrxAELMaJuTAjM9hauKX+bAFQPnhGP110my9F4bZOmQC/BsRobdlA4b7zSUFlyLyc2J3IOmZOwIy5+bKzo7p2FJJ9lETgx+imm9Qs39NgCMMSeRNZlCxSn9H/aAJ4ICFvSgnhmj6g6E6cNRD2j3K3SLnPlyw5C3bfUJcYWA+pDQeBiWRR+cyoY5hNUf6q6jHMNgtl8NAbZLbLw35j/mYRhzkUVQzjUUD6H/dhAlesU1Z6sND+wrA0ZunZ1gs+a+bwJnkI8ykAJE7DuM/Emb2e7KVJF7dyg6wGukrYwXpHMcjQ8v8lguhPc5qiGVeVXGJYkZ/ajvheOVIDtmw1MKuLH7me4HoRW1o0GP6C1Qu44oeYAZ/2kMImseVOj924oLI+3IrcrGS2TOxp4d4smApdguo0ZJuVsis2HXG2tYrlM3rhbsVrr2X+Kxkw7D2iE9Z9BS6EwdgT5sgqEA35Jdp+qOGfnDCADWuPuP3U97bRZPFRbXIaxE/fO8E/qmEd9ksT6dqf+w97tO0zYFodtpOqLRrQ2JsoSdBodo7ZX0vwDSBtvjDoHi+3ShEv/SCbLEvg2JFZEIKPg3AbNGnSrR7E5+0ItakKJ69VYZdng80mMtlfEMDopccRb6TQxHit7mIAOC1CuSqT4WXizgb4nez+Ev3TBy9KgQAXe8VY67i1IKJN//Byq/QhIITVUQr35FeEYcx5vs/eiMqYnKxuQGCfN2eJYcMlPQHEJG+rdWUczx47Hz9uLfz+Ovezt7+7/ZbE6fHXjl67+v93s7Xv/vmq97O7p4FpzsiHIfTyf8sBH/DPMfsf4XV7v5+/5vezv7+7zG0RKbrs0xSPNzP4lVKf+tqJj6PPtvas38e2f1w/mhnb3dn7/v9rT/8K//p5F9t/PPGv2zc39jb+GbjDxsvN/ob7zYmG/++8Z8b/7Xx37f/8+nXn/7x0z8x6C9/wTn/sGH8+xT9HwKpkaU=</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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model
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Training and machine learning
1. Training:	Solve	high-fidelity	and	mulGple	surrogates	for																							µ 2 Dtraining
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for µ 2 Dquery  Dtraining
D
41
= qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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model
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Training and machine learning
1. Training:	Solve	high-fidelity	and	mulGple	surrogates	for																							µ 2 Dtraining
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for µ 2 Dquery  Dtraining
D
41
= qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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model
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Training and machine learning
1. Training:	Solve	high-fidelity	and	mulGple	surrogates	for																							µ 2 Dtraining
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for µ 2 Dquery  Dtraining
D
41
= qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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model
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Training and machine learning
1. Training:	Solve	high-fidelity	and	mulGple	surrogates	for																							µ 2 Dtraining
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for µ 2 Dquery  Dtraining
D
41
= qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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model
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Training and machine learning
1. Training:	Solve	high-fidelity	and	mulGple	surrogates	for																							µ 2 Dtraining
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for µ 2 Dquery  Dtraining
D
41
= qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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high-fidelity	
model
surrogate	
models
‣ randomly	divide	data	into	(1)	training	data	and	(2)	tesGng	data	
‣ construct	regression-funcGon	model					via	cross	validaGon	on	training	data	
‣ construct	noise	model					from	sample	variance	on	test	data	
˜f
˜‘
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Reduction
42
features ⇢
1. Training:	Solve	high-fidelity	and	reduced-order	models	for																							
2. Machine	learning:	Construct	regression	model
3. Reduc(on:	predict	surrogate-model	error	for
µ 2 Dtraining
µ 2 Dquery  Dtraining
Doutputsinputs µ surrogate	model qsurr<latexit 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sha1_base64="eTUIsuHu8WCf8qj9Di3PqrhEvOQ=">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</latexit>
˜qHFM(µ)
¸ ˚˙ ˝
stochastic
= qsurr(µ)
¸ ˚˙ ˝
deterministic
+ ˜”(µ)
¸˚˙˝
stochastic<latexit 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regression	model machine	learning	
error	model ˜”<latexit 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sha1_base64="VnF8wDq0xTx1sO0lxALydqRvJfs=">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</latexit>
˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit sha1_base64="qpFvcXRrO0Mec8Yp6uVtTwOURus=">AAArQHicnVpbb9y4FZ7d3nbTW7Z9LFAINQIk3onh8TrJ7hZuc3GyzTaxnbHjbGE5A46Go+FaohSKGttR1P+xb/0tBfrQn9B/UKAPRYH2pU89pCiKoqhJtsYuIpLfOYc8h+dCcqZpRDK+ufm3997/zne/9/0ffPDhlR/+6Mc/+enVj352nCU5C/DzIIkS9tU
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Regression	model:	construct	regression	model					to	trade	off:	
‣ High	capacity:	low	variance,	more	data	to	generalize	
‣ Low	capacity:	high	variance,	less	data	to	generalize
˜f<latexit 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Error-model construction
43
Feature	engineering:	select	features					to	trade	off:	
1.	Number	of	features	
‣ Large	number:	costly,	low	variance,	high-capacity	regression	
‣ Small	number:	cheap,	high	variance,	low-capacity	regression	
2.	Quality	of	features	
‣ High	quality:	expensive,	low	variance	
‣ Low	quality:	cheap,	high	variance
⇢
Method	2:	Large	number	of	features	and	high-dimensional	regression	
[Trehan,	C.,	Durlofsky,	2017;	Freno,	C.,	2018]	
Method	1:	Dual-weighted	residual	and	Gaussian	process	regression	
[Drohmann,	C.,	2015;	C.,	Uy,	Lu,	Morzfeld,	2018]
˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit 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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Regression	model:	construct	regression	model					to	trade	off:	
‣ High	capacity:	low	variance,	more	data	to	generalize	
‣ Low	capacity:	high	variance,	less	data	to	generalize
˜f<latexit 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Error-model construction
43
Feature	engineering:	select	features					to	trade	off:	
1.	Number	of	features	
‣ Large	number:	costly,	low	variance,	high-capacity	regression	
‣ Small	number:	cheap,	high	variance,	low-capacity	regression	
2.	Quality	of	features	
‣ High	quality:	expensive,	low	variance	
‣ Low	quality:	cheap,	high	variance
⇢
Method	2:	Large	number	of	features	and	high-dimensional	regression	
[Trehan,	C.,	Durlofsky,	2017;	Freno,	C.,	2018]	
Method	1:	Dual-weighted	residual	and	Gaussian	process	regression	
[Drohmann,	C.,	2015;	C.,	Uy,	Lu,	Morzfeld,	2018]
˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit 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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Feature: dual-weighted residual [Drohmann, C., 2015]
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q(x) q(˜x) = yT
r(˜x) + O(kx ˜xk2
)<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
@r
@x
(˜x)T
y =
@q
@x
(˜x)T
<latexit 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</latexit><latexit sha1_base64="o3vxFqLJSv8I6puYLSOT7u9l9Hc=">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
‣ Want	to	avoid	HFM-scale	solves,	so	approximate	dual	as
and	construct	a	ROM	for	the	dual
y
T @r
@x
(˜x)T
yˆy = y
T @q
@x
(˜x)T
y ¥ ˜y = yˆy
‣ One	feature:	
‣ can	control	feature	quality	via	dimension	of	
‣ Regression	model:		Gaussian	process	[Rasmussen,	Williams,	2006]
q(x) q(˜x) ⇡ ˆyT
y
T
r(˜x)
y
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Thermal block
1 2 3
4 5 6
7 8 9
D
N1
N0
4c(x; µ)u(x; µ) = 0 in ⌦ x(µ) = 0 on D
rc(µ)x(µ) · n = 0 on N0 rc(µ)x(µ) · n = 1 on N1
Inputs µ 2 [0.1, 10]9 define di↵usivity c in subdomains
Output z(µ) =
R
N1
x(µ)dx is compliant
ROM constructed via RB–Greedy [?]
ROMES Kevin Carlberg, Martin Drohmann 15 / 22
Application: Bayesian inference
45
1 2 3
4 5 6
7 8 9
D
N1
N0
4c(x; µ)u(x; µ) = 0 in ⌦ x(µ) = 0 on D
rc(µ)x(µ) · n = 0 on N0 rc(µ)x(µ) · n = 1 on N1
Inputs µ 2 [0.1, 10]9 define di↵usivity c in subdomains
Output z(µ) =
R
N1
x(µ)dx is compliant
ROM constructed via RB–Greedy [?]
ROMES Kevin Carlberg, Martin Drohmann 15 / 22
‣ Inputs																									define	diffusivity	in	c	in	subdomains	
‣ Outputs					are	24	measured	temperatures				
‣ ROM	constructed	via	RB-Greedy	[Patera	and	Rozza,	2006]	
‣ 															:	Gaussian	with	variance	0.1	
‣ 			
‣ Posterior	sampling:															samples	w/	implicit	sampling	[Tu	et	al.,	2013]	
µ 2 [0.1, 10]9
q
⇡prior(µ)
" ⇠ N(0, 1 ⇥ 10 3
)
1 ⇥ 105
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
2.5
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0
0.15
0.15
0 0.04 0.08 0.12
q24(x)q24(ˆx)
-0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0
2.5
0 1 2
q1(x)q1(ˆx)
low	
variance
costly
⇢1 ⇢24
rank( 1) = 22 rank( 24) = 7
high	
quality
Machine learning error models
46
˜i (µ) ⇠ N( ⇢i (µ), ↵1 + ↵2|⇢i (µ)|↵3
)
-0.2 -0.1 0 0.1 0.2 0.3 0.4
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.52.5
0
2.5
0.2 0 0.2 0.4
q1(x)q1(ˆx)
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
q24(x)q24(ˆx)
0
0.15
0.15
0.1 0 0.1 0.2
high	
variance
cheap
rank( 1) = 1 rank( 24) = 1
⇢1 ⇢24
low	
quality
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Wall-time performance
47
‣ ROM:	
+ cheapest	
- inconsistent	formulaGon
simulaGon	Gme
HFM ROM ROM+	
high-var
ROM+	
low-var
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Wall-time performance
47
‣ ROM:	
+ cheapest	
- inconsistent	formulaGon
‣ ROM	+	error	models:	
+ cheaper	than	HFM	
- more	expensive	than	ROM	
+ consistent	formulaGon
simulaGon	Gme
HFM ROM ROM+	
high-var
ROM+	
low-var
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Posteriors: ROM
48
⇡HFM
post(µ|qmeas)
⇡surr
post(µ | qmeas)
true
prior
⇡surr
post(µ | qmeas)
⇡HFM
post (µ | qmeas)
+ HFM	posterior:	close	to	true	parameters	
- ROM	posterior:	far	from	prior	and	true	parameters
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Posteriors: ROM + high-variance error model
49
⇡HFM
post(µ|qmeas)
true
prior
⇡HFM
post (µ | qmeas)
+ ROM	+	high-variance	error	model	posterior:	close	to	prior
⇡
]HFM
post (µ | qmeas)
⇡
]HFM
post (µ | qmeas)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
⇡HFM
post(µ|qmeas)
true
prior
⇡HFM
post (µ | qmeas)
⇡
]HFM
post (µ | qmeas)
Posteriors: ROM + low-variance error model
50
+ ROM	+	low-variance	error	model	posterior:	close	to	HFM	posterior
⇡
]HFM
post (µ | qmeas)
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Regression	model:	construct	regression	model					to	trade	off:	
‣ High	capacity:	low	variance,	more	data	to	generalize	
‣ Low	capacity:	high	variance,	less	data	to	generalize
˜f<latexit 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Error-model construction
51
Feature	engineering:	select	features					to	trade	off:	
1.	Number	of	features	
‣ Large	number:	costly,	low	variance,	high-capacity	regression	
‣ Small	number:	cheap,	high	variance,	low-capacity	regression	
2.	Quality	of	features	
‣ High	quality:	expensive,	low	variance	
‣ Low	quality:	cheap,	high	variance
⇢
Method	2:	Large	number	of	features	and	high-dimensional	regression	
[Trehan,	C.,	Durlofsky,	2017;	Freno,	C.,	2018]	
Method	1:	Dual-weighted	residual	and	Gaussian	process	regression	
[Drohmann,	C.,	2015;	C.,	Uy,	Lu,	Morzfeld,	2018]
˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit 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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
1. Error	indicators:	
‣ residual	norm:	
‣ dual-weighted	residual:
Feature engineering [Freno, C., 2018]
52
Proposed	features:	
‣ parameters	
‣ low	quality,	cheap	
‣ used	by	model	discrepancy	
‣ residual	norm		
- small	number,	low	quality,	costly	
‣ residual	
- large	number,		low	quality,	costly
µ
kr( ˆx; µ)k2
r( ˆx; µ)
‣ residual	samples	
+ moderate	number,	cheap	
- low	quality	
‣ residual	PCA	
+ moderate	number,	high-quality	
- costly	
‣ gappy	PCA	
+ moderate	number,	high-quality	
+ cheap
ˆr := T
r r( ˆx; µ)
Pr( ˆx; µ)
ˆrg := (P r)+
Pr( ˆx; µ)
kr(˜x; µ)k2<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
2. Rigorous	a	posteriori	error	bound: |q(x) q(˜x)| 
↵
kr(˜x; µ)k2
<latexit 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</latexit><latexit 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
q(x) q(˜x) = yT
r(˜x) + O(kx ˜xk2
)
3. Model	discrepancy:
Idea:	Use	tradi(onal	error	quan(fica(on	as	inspira(on	for	features
˜”(µ) ≥ N(µ(µ); ‡2
(µ))
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Application: Predictive capability assessment project
53
x
y
z
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
0.011
Deformation
Magnitude [m]
AB
‣ high-fidelity	model	dimension:	
‣ reduced-order	model	dimensions:	
‣ inputs				:	elasGc	modulus,	Poisson	raGo,	applied	pressure	
‣ quan((es	of	interest:	y-displacement	at	A,	radial	displacement	at	B	
‣ training	data:	150	training	examples,	150	tesGng	examples
µ
2.8 ⇥ 105
1, ... , 5<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit sha1_base64="EzhCuj0Dot+YWL6K4S/CVctu3A4=">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</latexit>
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
krk2
[µ;krk2]
[µ;Pr](nr=10)
[µ;ˆrg](nr=10)
[µ;Pr](nr=100)
[µ;ˆrg](nr=100)
[µ;Pr](nr=1000)
[µ;ˆrg](nr=1000)
[µ;ˆr]
µ
ANN
k-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
ˆuy : log10 FVU
Application: Predictive capability assessment project
54
Introduction Parameterized Nonlinear Equations Approach Experiments Summary
PCAP: FVU for QoI Error Prediction
ˆ r: log10 FVU ˆ y: log10 FVU
RegressionMethods
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
Features Features
Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination)
• SVR: RBF and MLP perform the best
• [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301)
Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44
log10(1 R2
) log10(1 R2
)
y-displacement	at	A radial	displacement	at	B
regression	methods
features features
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
krk2
[µ;krk2]
[µ;Pr](nr=10)
[µ;ˆrg](nr=10)
[µ;Pr](nr=100)
[µ;ˆrg](nr=100)
[µ;Pr](nr=1000)
[µ;ˆrg](nr=1000)
[µ;ˆr]
µ
ANN
k-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
ˆuy : log10 FVU
Application: Predictive capability assessment project
54
Introduction Parameterized Nonlinear Equations Approach Experiments Summary
PCAP: FVU for QoI Error Prediction
ˆ r: log10 FVU ˆ y: log10 FVU
RegressionMethods
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
Features Features
Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination)
• SVR: RBF and MLP perform the best
• [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301)
Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44
log10(1 R2
) log10(1 R2
)
- parameters	(model-discrepancy	approach):	large	variance
y-displacement	at	A radial	displacement	at	B
regression	methods
features features
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
krk2
[µ;krk2]
[µ;Pr](nr=10)
[µ;ˆrg](nr=10)
[µ;Pr](nr=100)
[µ;ˆrg](nr=100)
[µ;Pr](nr=1000)
[µ;ˆrg](nr=1000)
[µ;ˆr]
µ
ANN
k-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
ˆuy : log10 FVU
Application: Predictive capability assessment project
54
Introduction Parameterized Nonlinear Equations Approach Experiments Summary
PCAP: FVU for QoI Error Prediction
ˆ r: log10 FVU ˆ y: log10 FVU
RegressionMethods
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
Features Features
Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination)
• SVR: RBF and MLP perform the best
• [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301)
Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44
log10(1 R2
) log10(1 R2
)
- parameters	(model-discrepancy	approach):	large	variance
- small	number	of	low-quality	features:	large	variance
y-displacement	at	A radial	displacement	at	B
regression	methods
features features
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
krk2
[µ;krk2]
[µ;Pr](nr=10)
[µ;ˆrg](nr=10)
[µ;Pr](nr=100)
[µ;ˆrg](nr=100)
[µ;Pr](nr=1000)
[µ;ˆrg](nr=1000)
[µ;ˆr]
µ
ANN
k-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
ˆuy : log10 FVU
Application: Predictive capability assessment project
54
Introduction Parameterized Nonlinear Equations Approach Experiments Summary
PCAP: FVU for QoI Error Prediction
ˆ r: log10 FVU ˆ y: log10 FVU
RegressionMethods
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
Features Features
Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination)
• SVR: RBF and MLP perform the best
• [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301)
Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44
log10(1 R2
) log10(1 R2
)
- parameters	(model-discrepancy	approach):	large	variance
- small	number	of	low-quality	features:	large	variance
‣ PCA	of	the	residual:	lowest	variance	overall	but	costly
y-displacement	at	A radial	displacement	at	B
regression	methods
features features
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
krk2
[µ;krk2]
[µ;Pr](nr=10)
[µ;ˆrg](nr=10)
[µ;Pr](nr=100)
[µ;ˆrg](nr=100)
[µ;Pr](nr=1000)
[µ;ˆrg](nr=1000)
[µ;ˆr]
µ
ANN
k-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
ˆuy : log10 FVU
Application: Predictive capability assessment project
54
Introduction Parameterized Nonlinear Equations Approach Experiments Summary
PCAP: FVU for QoI Error Prediction
ˆ r: log10 FVU ˆ y: log10 FVU
RegressionMethods
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
Features Features
Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination)
• SVR: RBF and MLP perform the best
• [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301)
Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44
log10(1 R2
) log10(1 R2
)
- parameters	(model-discrepancy	approach):	large	variance
- small	number	of	low-quality	features:	large	variance
‣ PCA	of	the	residual:	lowest	variance	overall	but	costly
+ gappy	PCA	of	the	residual:	nearly	as	low	variance,	but	much	cheaper
y-displacement	at	A radial	displacement	at	B
regression	methods
features features
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
krk2
[µ;krk2]
[µ;Pr](nr=10)
[µ;ˆrg](nr=10)
[µ;Pr](nr=100)
[µ;ˆrg](nr=100)
[µ;Pr](nr=1000)
[µ;ˆrg](nr=1000)
[µ;ˆr]
µ
ANN
k-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
ˆuy : log10 FVU
Application: Predictive capability assessment project
54
Introduction Parameterized Nonlinear Equations Approach Experiments Summary
PCAP: FVU for QoI Error Prediction
ˆ r: log10 FVU ˆ y: log10 FVU
RegressionMethods
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
krk2
[µ;krk2]
[µ;Pr](q=10)
[µ;ˆrg](q=10)
[µ;Pr](q=100)
[µ;ˆrg](q=100)
[µ;Pr](q=1000)
[µ;ˆrg](q=1000)
[µ;ˆr]
µ
MLP
-NN
RF
SVR: RBF
SVR: Linear
OLS: Quadratic
OLS: Linear
5
4
3
2
1
0
Features Features
Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination)
• SVR: RBF and MLP perform the best
• [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301)
Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44
log10(1 R2
) log10(1 R2
)
- parameters	(model-discrepancy	approach):	large	variance
- small	number	of	low-quality	features:	large	variance
‣ PCA	of	the	residual:	lowest	variance	overall	but	costly
+ gappy	PCA	of	the	residual:	nearly	as	low	variance,	but	much	cheaper
+ neural	networks	and	SVR:	RBF	yield	lowest-variance	models
y-displacement	at	A radial	displacement	at	B
regression	methods
features features
/38
Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
10 8 6 4 2 0 2 4
Predicted error, ˆuy [⇥10 3
]
10
8
6
4
2
0
2
4
Exacterror,uy[⇥103
]
Exact
krk2
SVR: RBF
r2
=0.94712, MSE=2.424⇥10 7
µ
ANN
r2
=0.96851, MSE=1.444⇥10 7
[µ; ˆrg] (nr=10)
ANN
r2
=0.99944, MSE=2.554⇥10 9
Application: Predictive capability assessment project
55
kr( ˆx; µ)k2
‣ TradiGonal	features						and																						:	
- high	noise	variance	
- expensive	for																						:	compute	enGre	residual
kr( ˆx; µ)k2<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
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</latexit><latexit 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
kr( ˆx; µ)k2
[µ;ˆrg]‣ Proposed	features											:	
+ low	noise	variance	
+ extremely	cheap:	only	compute	10	elements	of	the	residual
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Summary
56
Accurate,	low-cost,	structure-preserving,	
reliable,	cer;fied	nonlinear	model	reduc;on
‣ accuracy:	LSPG	projecGon	[C.,	Bou-Mosleh,	Farhat,	2011;	C.,	Barone,	AnGl,	2017]	
‣ low	cost:	sample	mesh	[C.,	Farhat,	CorGal,	Amsallem,	2013]	
‣ low	cost:	reduce	temporal	complexity	
[C.,	Ray,	van	Bloemen	Waanders,	2015;	C.,	Brencher,	Haasdonk,	Barth,	2017;	Choi	and	C.,	2017]	
‣ structure	preserva(on	[C.,	Tuminaro,	Boggs,	2015;	Peng	and	C.,	2017;	C.	and	Choi,	2017]	
‣ reliability:	adapGvity	[C.,	2015]	
‣ cer(fica(on:	machine	learning	error	models	
[Drohmann	and	C.,	2015;	Trehan,	C.,	Durlofsky,	2017;	Freno	and	C.,	2017]
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Kevin	CarlbergAdvances	in	nonlinear	model	reduc4on
Questions?
57
Machine-learning	error	models:	
‣ Freno,	C.	“Machine-learning	error	models	for	approximate	solu4ons	to	
parameterized	systems	of	nonlinear	equa4ons,”	arXiv	e-Print,	1808.02097	(2018).	
‣ Trehan,	C,	and	Durlofsky.	“Error	modeling	for	surrogates	of	dynamical	systems	using	
machine	learning,”	InternaGonal	Journal	for	Numerical	Methods	in	Engineering,	Vol.	
112,	No.	12,	p.	1801–1827	(2017).		
‣ Drohmann	and	C.	“The	ROMES	method	for	staGsGcal	modeling	of	reduced-order-
model	error,”	SIAM/ASA	Journal	on	Uncertainty	QuanGficaGon,	Vol.	3,	No.	1,	p.116–
145	(2015).
LSPG	reduced-order	model:	
‣ C,	Barone,	and	An4l.	“Galerkin	v.	least-squares	Petrov–Galerkin	projec4on	in	
nonlinear	model	reduc4on,”	Journal	of	Computa4onal	Physics,	Vol.	330,	p.	693–
734	(2017).	
‣ C,	Farhat,	CorGal,	and	Amsallem.	“The	GNAT	method	for	nonlinear	model	
reducGon:	EffecGve	implementaGon	and	applicaGon	to	computaGonal	fluid	
dynamics	and	turbulent	flows,”	Journal	of	ComputaGonal	Physics,	Vol.	242,	p.	623–
647	(2013).	
‣ C,	Bou-Mosleh,	and	Farhat.	“Efficient	non-linear	model	reducGon	via	a	least-
squares	Petrov–Galerkin	projecGon	and	compressive	tensor	approximaGons,”	
InternaGonal	Journal	for	Numerical	Methods	in	Engineering,	Vol.	86,	No.	2,	p.	155–
181	(2011).

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MUMS Opening Workshop - Machine-Learning Error Models for Quantifying the Epistemic Uncertainty in Low-Fidelity Models - Kevin Carlberg, August 21, 2018

  • 1. Kevin Carlberg Sandia Na(onal Laboratories SAMSI MUMS Opening Workshop Duke University August 21, 2018 Advances in nonlinear model reduction:
 least-squares Petrov–Galerkin projection and machine-learning error models Kevin Carlberg iCME *Talk Stanford University May 22, 2017 Breaking computaDonal barriers Using data to enable extreme-scale simulaDons for many-query problems -8 -6 -4 -2 0 2 -8 -6 -4 -2 0 2 support vector machine error predicDon R2 = 0.990 error ROM h-refinement e), b) prolongation (fine) reduced-order model high-fidelity model Sandia Na(onal Laboratories is a mul(mission laboratory managed and operated by Na(onal Technology and Engineering Solu(ons of Sandia, LLC., a wholly owned subsidiary of Honeywell Interna(onal, Inc., for the U.S. Department of Energy’s Na(onal Nuclear Security Administra(on under contract DE-NA-0003525. ⇡HFM post(µ|qmeas) true prior ⇡HFM post (µ | qmeas) ⇡ ]HFM post (µ | qmeas) ⇡ ]HFM post (µ | qmeas)
  • 2. /38 Kevin CarlbergAdvances in nonlinear model reduc4on High-fidelity simulation 2 computa4onal barrier +Indispensable across science and engineering - High fidelity: extreme-scale nonlinear dynamical system models /38 Kevin CarlbergBreaking computa5onal barriers High-fidelity simulation computaDonal barrier ๏ uncertainty propagaDon ๏ mulD-objecDve opDmizaDon +Indispensable across science and engineering - High fidelity: extreme-scale nonlinear dynamical system models Many-query problems ๏ Bayesian inference ๏ stochasDc opDmizaDon Magnetohydrodynamics courtesy J. Shadid, Sandia Turbulent reac5ng flows courtesy J. Chen, Sandia Antarc5c ice sheet modeling courtesy R. Tuminaro, Sandia 2 /38High-fidelity simulation computaDonal barrier ๏ uncertainty propagaDon ๏ mulD-objecDve opDmizaDon +Indispensable across science and engineering - High fidelity: extreme-scale nonlinear dynamical system models Many-query problems ๏ Bayesian inference ๏ stochasDc opDmizaDon Magnetohydrodynamics courtesy J. Shadid, Sandia Turbulent reac5ng flows courtesy J. Chen, Sandia Antarc5c ice sheet modeling courtesy R. Tuminaro, Sandia /38High-fidelity simulation computaDonal barrier ๏ uncertainty propagaDon ๏ mulD-objecDve opDmizaDon +Indispensable across science and engineering - High fidelity: extreme-scale nonlinear dynamical system models Many-query problems ๏ Bayesian inference ๏ stochasDc opDmizaDon Magnetohydrodynamics courtesy J. Shadid, Sandia Turbulent reac5ng flows courtesy J. Chen, Sandia Antarc5c ice sheet modeling courtesy R. Tuminaro, Sandia ๏ uncertainty propagaGon ๏ mulG-objecGve opGmizaGon ๏ Bayesian inference ๏ stochasGc opGmizaGon Many-query problems
  • 3. /38 Kevin CarlbergAdvances in nonlinear model reduc4on High-fidelity simulation: captive carry /38 Kevin CarlbergBreaking computa5onal barriers 3 High-fidelity simulation: B61 captive carry 3
  • 4. /38 Kevin CarlbergAdvances in nonlinear model reduc4on High-fidelity simulation: captive carry ๏ explore flight envelope ๏ quanGfy effects of uncertainGes on store load ๏ robust design of store and cavity computa4onal barrier Many-query problems + Validated and predic(ve: matches wind-tunnel experiments to within 5% - Extreme-scale: 100 million cells, 200,000 Gme steps - High simula(on costs: 6 weeks, 5000 cores 3
  • 5. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Approach: exploit simulation data 4 Idea: exploit simula(on data collected at a few points D 1. Training: Solve ODE for and collect simulaGon data 2. Machine learning: IdenGfy structure in data 3. Reduc(on: Reduce cost of ODE solve for Many-query problem: solve ODE for µ 2 Dquery µ 2 Dtraining µ 2 Dquery Dtraining ODE: dx dt = f(x; t, µ), x(0, µ) = x0(µ), t 2 [0, Tfinal], µ 2 D
  • 6. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Model reduction criteria 1. Accuracy: achieves less than 1% error 2. Low cost: achieves at least 100x computaGonal savings 3. Structure preserva;on: preserves important physical properGes 4. Reliability: guaranteed saGsfacGon of any error tolerance (fail safe) 5. Cer;fica;on: quanGfies ROM-induced epistemic uncertainty 5
  • 7. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Model reduction: previous state of the art Linear 4me-invariant systems: mature [Antoulas, 2005] ‣ Balanced truncaGon [Moore, 1981; Willcox and Peraire, 2002; Rowley, 2005] ‣ Transfer-funcGon interpolaGon [Bai, 2002; Freund, 2003; Gallivan et al, 2004; Baur et al., 2001] + Accurate, reliable, cer(fied: sharp a priori error bounds + Inexpensive: pre-assemble operators + Structure preserva(on: guaranteed stability Ellip4c/parabolic PDEs: mature [Prud’Homme et al., 2001; Barrault et al., 2004; Rozza et al., 2008] ‣ Reduced-basis method + Accurate, reliable, cer(fied: sharp a priori error bounds, convergence + Inexpensive: pre-assemble operators + Structure preserva(on: preserve operator properGes Nonlinear dynamical systems: ineffecGve ‣ Proper orthogonal decomposiGon (POD)–Galerkin [Sirovich, 1987] - Inaccurate, unreliable: ogen unstable - Not cer(fied: error bounds grow exponenGally in Gme - Expensive: projecGon insufficient for speedup - Structure not preserved: dynamical-system properGes ignored 6
  • 8. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017] 7
  • 9. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research 8 Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011*; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017] Collaborators: ‣ Malhew Barone (Sandia) ‣ Harbir AnGl (GMU) ‣ Charbel Farhat (Stanford University) ‣ Julien CorGal (Stanford University)
  • 10. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Training: Solve ODE for and collect simulaGon data 2. Machine learning: IdenGfy structure in data 3. Reduc(on: Reduce the cost of solving ODE for µ 2 Dtraining Training simulations: state tensor 9 dx dt = f(x; t, µ)ODE: µ 2 Dquery Dtraining D number of Gme steps T number of state variables N
  • 11. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Training: Solve ODE for and collect simulaGon data 2. Machine learning: IdenGfy structure in data 3. Reduc(on: Reduce the cost of solving ODE for µ 2 Dtraining Training simulations: state tensor 9 dx dt = f(x; t, µ)ODE: µ 2 Dquery Dtraining DX =
  • 12. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Training: Solve ODE for and collect simulaGon data 2. Machine learning: IdenGfy structure in data 3. Reduc(on: Reduce the cost of solving ODE for Tensor decomposition 10 dx dt = f(x; t, µ)ODE: Compute dominant leR singular vectors of mode-1 unfolding µ 2 Dtraining µ 2 Dquery Dtraining X(1) = = U ⌃ VT X =
  • 13. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Training: Solve ODE for and collect simulaGon data 2. Machine learning: IdenGfy structure in data 3. Reduc(on: Reduce the cost of solving ODE for Tensor decomposition 10 dx dt = f(x; t, µ)ODE: Compute dominant leR singular vectors of mode-1 unfolding columns are principal components of the spa(al simula(on data How to integrate these data with the computa;onal model? µ 2 Dtraining µ 2 Dquery Dtraining X(1) = = U ⌃ VT X =
  • 14. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Previous state of the art: POD–Galerkin 1. Training: Solve ODE for and collect simulaGon data 2. Machine learning: IdenGfy structure in data 3. Reduc(on: Reduce the cost of solving ODE for µ 2 Dtraining µ 2 Dquery Dtraining dx dt = f(x; t, µ)ODE: 1. Reduce the number of unknowns 2. Reduce the number of equaGons D DGalerkin ODE: dˆx dt = T f( ˆx; t, µ) dˆx dt ) = 0 ( (T (f( ˆx; t, µ)x(t) ⇡ ˜x(t) = ˆx(t) 11
  • 15. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Captive carry 12 V1 ‣ Unsteady Navier–Stokes ‣ Re = 6.3 x 106 ‣ M∞ = 0.6 Spa4al discre4za4on ‣ 2nd-order finite volume ‣ DES turbulence model ‣ degrees of freedom1.2 ⇥ 106 Temporal discre4za4on ‣ 2nd-order BDF ‣ Verified Gme step ‣ Gme instances t = 1.5 ⇥ 10 3 8.3 ⇥ 103
  • 18. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Galerkin performance 15 probe Can we construct a beEer projec;on? FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 368 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 204 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 high-fidelity: dim 1.2x106 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 564 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 pressure at probe 1.6 2.0 2.4 2.8 Gme 0 2 4 6 8 10 12 - Galerkin projec(on fails regardless of basis dimension
  • 19. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Galerkin: time-continuous optimality ODE Galerkin ODE dˆx dt = T f( ˆx; t, µ) + Time-con(nuous Galerkin solu(on: opGmal in the minimum-residual sense: dˆx dt = T f( ˆx; t, µ) - Time-discrete Galerkin solu(on: not generally opGmal in any sense dx dt = f(x; t) f( ˆx; t) O∆E Galerkin O∆E rn (x) := ↵0x t 0f(x; tn ) + kX j=1 ↵j xn j t kX j=1 j f(xn j ; tn j ) rn (xn ) = 0, n = 1, ... , Nn = 1, ... , T T rn ( ˆxn ) = 0, n = 1, ... , Nn = 1, ... , T 16 r(v, x; t) := v f(x; t) dˆx dt (x, t) = argmin v2range( ) kr(v, x; t)k2
  • 20. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Residual minimization and time discretization ODE residual minimiza(on (me discre(za(on dx dt = f(x; t) Galerkin ODE dˆx dt (x, t) = argmin v2range( ) kr(v, x; t)k2 , n (ˆxn )T rn ( ˆxn ) = 0ˆxn = argmin v2range( ) kArn (v)k2 n (ˆxn ) := AT A(↵0I t 0 @f @x ( ˆxn ; t)) Least-squares Petrov–Galerkin (LSPG) projec(on residual minimiza(on LSPG O∆E ˆxn = argmin v2range( ) kArn (v)k2 [C., Bou-Mosleh, Farhat, 2011] n = 1, ... , T (me discre(za(on O∆E rn (xn ) = 0 n = 1, ... , T Galerkin O∆E T rn ( ˆxn ) = 0 n = 1, ... , T 17
  • 21. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Discrete-time error bound 18 Theorem [C., Barone, AnGl, 2017] If the following condiGons hold: 1. is Lipschitz conGnuous with Lipschitz constant 2. The Gme step is small enough such that , 3. A backward differenGaGon formula (BDF) Gme integrator is used, 4. LSPG employs , then f(·; t)  0 < h := |↵0| | 0| tt + LSPG sequen(ally minimizes the error bound A = I kxn ˆxn Gk2  1 h krn G( ˆxn G)k2+ 1 h kX `=1 |↵`|kxn ` ˆxn ` G k2 kxn ˆxn LSPGk2  1 h min ˆv krn LSPG( ˆv)k2+ 1 h kX `=1 |↵`|kxn ` ˆxn ` LSPGk2
  • 22. /38 Kevin CarlbergAdvances in nonlinear model reduc4on LSPG performance 19 probe FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 368 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 204 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 high-fidelity: dim 1.2x106 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 564 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 LSPG: dim 368 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 LSPG: dim 204FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3LSPG: dim 564 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 pressure at probe 1.6 2.0 2.4 2.8 Gme 0 2 4 6 8 10 12 + LSPG is far more accurate than Galerkin
  • 23. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research 20 Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013*] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
  • 24. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Wall-time problem 21 probe FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 368 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 204 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 high-fidelity: dim 1.2x106 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 Galerkin: dim 564 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 LSPG: dim 368 FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3 LSPG: dim 204FOM Gal1 Gal2 Gal3 LSPG1 LSPG2 LSPG3LSPG: dim 564 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 1.6 1.8 2 2.2 2.4 2.6 2.8 pressure at probe 1.6 2.0 2.4 2.8 Gme 0 2 4 6 8 10 12 ‣ High-fidelity simula(on: 1 hour, 48 cores ‣ Fastest LSPG simula(on: 1.3 hours, 48 cores Why does this occur? Can we fix it?
  • 25. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Training: collect residual tensor while solving ODE for 2. Machine learning: compute residual PCA and sampling matrix 3. Reduc4on: compute regression approximaGon Cost reduction by gappy PCA [Everson and Sirovich, 1995] minimize ˆv k A rn ( ˆv)k2 k2 Can we select to make this less expensive?A ˆv)k2 rn ⇡ ˜rn = r(P r)+ Prn r P 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 minimize ˆv k k2 rn ( rn ( rn ( rn ( rn (˜rn 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 index value r ˜rn rn Prn Rijk µ 2 Dtraining 22
  • 26. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Training: collect residual tensor while solving ODE for 2. Machine learning: compute residual PCA and sampling matrix 3. Reduc4on: compute regression approximaGon Cost reduction by gappy PCA [Everson and Sirovich, 1995] minimize ˆv k A rn ( ˆv)k2 k2 Can we select to make this less expensive?A rn ( ˆv)k2 + Only a few elements of d must be computedrn rn ⇡ ˜rn = r(P r)+ Prn r P (P r)+ P 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 minimize ˆv k k2 rn ( rn ( rn ( rn ( rn ( 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 9 10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 index value minimize| {z } A r ˜rn rn Prn Rijk µ 2 Dtraining 22
  • 27. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Sample mesh [C., Farhat, Cortial, Amsallem, 2013] vor(city field pressure field LSPG ROM with 32 min, 2 cores + 229x savings in core–hours + < 1% error in (me-averaged drag + HPC on a laptop sample mesh minimize ˆv k(P r)+ Prn ( ˆv)k2 A = (P r)+ P Prn |{z} A high-fidelity 5 hours, 48 cores Implemented in three computa;onal-mechanics codes at Sandia 23
  • 28. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Ahmed body [Ahmed, Ramm, Faitin, 1984] 24 V1 ‣ Unsteady Navier–Stokes ‣ Re = 4.3 x 106 ‣ M∞ = 0.175 Spa4al discre4za4on ‣ 2nd-order finite volume ‣ DES turbulence model ‣ degrees of freedom Temporal discre4za4on ‣ 2nd-order BDF ‣ Time step ‣ Gme instances t = 8 ⇥ 10 5 s 1.7 ⇥ 107 1.3 ⇥ 103
  • 29. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Ahmed body results [C., Farhat, Cortial, Amsallem, 2013] 25 pressure field + 438x savings in core–hours + HPC on a laptop sample mesh high-fidelity model 13 hours, 512 cores LSPG ROM with A = (P r)+ P 4 hours, 4 cores + Largest nonlinear dynamical system on which ROM has ever had success
  • 30. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research 26 Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
  • 31. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research 27 Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
  • 32. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research 28 Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C., Choi, Sargsyan, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
  • 33. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Our research Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C. and Choi, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2018] 29 Collaborators: ‣ MarGn Drohmann (formerly Sandia) ‣ Wayne Uy (Cornell University) ‣ Fei Lu (Johns Hopkins University) ‣ Malhias Morzfeld (U of Arizona) ‣ Brian Freno (Sandia)
  • 34. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Surrogate modeling in UQ 30 outputsinputs µ surrogate model qsurr ‣ surrogate noise model: ‣ surrogate likelihood: - inconsistent with HFM noise model qmeas = qsurr(µ) + " ⇡surr(qmeas | µ) = ⇡"(qmeas qsurr(µ)) ⇡"(·) ‣ high-fidelity-model (HFM) noise model: ‣ measurement noise has probability distribuGon ‣ HFM likelihood: " outputsinputs µ high-fidelity model qHFM qmeas = qHFM(µ) + " ⇡HFM(qmeas | µ) = ⇡"(qmeas qHFM(µ))
  • 35. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Surrogate modeling in UQ 31 qHFM(µ) = qsurr(µ) + (µ) ‣ HFM noise model: ‣ HFM likelihood: ⇡HFM(qmeas | µ) = ⇡"(qmeas qHFM(µ)) = ⇡"(qmeas qsurr(µ) (µ)) qmeas = qHFM(µ) + " = qsurr(µ) + (µ) + " + equivalent to HFM formulaGon + not pracGcal: the (determinisGc) error is generally unknown(µ) How can we account for the error in a manner that is consistent and prac;cal? (µ)
  • 36. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Surrogate modeling in UQ 32 qHFM(µ) = qsurr(µ) + (µ) Approach: sta(s(cal model for the error that models its uncertainty˜(µ) ˜qHFM(µ) | {z } stochastic = qsurr(µ) | {z } deterministic + ˜(µ) |{z} stochastic ‣ staGsGcal HFM noise model: qmeas = ˜qHFM(µ) + " = qsurr(µ) + ˜(µ) + " + consistent with HFM noise model + pracGcal if the staGsGcal error model is computable ⇡]HFM (qmeas | µ) = ⇡"+˜(qmeas qsurr(µ))‣ stochasGc HFM likelihood: ˜ Desired proper4es in sta4s4cal error model 1. cheaply computable: similar cost to evaluaGng the surrogate 2. low variance: introduces lille epistemic uncertainty 3. generalizable: correctly models the error ˜(µ) How can we construct a sta;s;cal error model for reduced-order models?
  • 37. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Approximate-solution surrogate models 33 r(x(µ); µ) = 0<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit> High-fidelity model ‣ governing equaGons: ‣ quanGty of interest: Types of approximate solu4ons ‣ Reduced-order model: ‣ Low-fidelity model: ‣ Inexact solu(on: compute such that rLF(xLF; µ) = 0, ˜x = p(xLF)<latexit 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</latexit><latexit 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MDsEtJ522y5tG0nSg5oszUtV7SyEjzMryofQ+8r+iD3lFWIce+Cul8xrV5I2LfoluhaXl8hD/rZ+429Jnt41aqrZPjrXmtfO8FYPHRQvpSkDSkuGndTji7yMbupc66I4pkaZ7wSDnSFK/IOk8nkoBV9lefVGCEyYNtIUiVzNRYZVlhet4DmK23epntVEMKTPTpcweq8VnaDVhJaF+Z16FQg2esOn405RgsSp03p/Yb7NCXcM5wCI/3RW1Y00TDCLqxTuBWizC4qb90zqFk8jVUqppOWFNVoltSfyPn2Gs+6EeZexUVhnywnu1r1TR5pFwydnXra2vMPUStupUYXFoI1MwPSdcYFATqP6G0yCaPbc+aQTrx1cgsYEbyS+aMj0wJuWfuC9jahtOFLa+dSbulq+8d6cNGe5ogMUiU4KxG7BumYNjvIEQGQgp1BczEDHESyzUYBWJcLWuIPMPCxDfY4HuyOlLCTzNjq5hT1VPvAkkVndQGaMrCzgoTjgIRjr4RpjD/odtMHK5l8wF0TPPCukFmlSOM5ysglpki/SfKQ7nJOkpHf5TVunrTGrdxssHBWtBhdl3lK74E0rwf8lohzZedAnu6tTJnyuFsG3ak5WwzH9DSnxcKaBFNoDHDlchuW4Epylu26GbpRFu75eIoASxEgkgdpLeQUQJZDAJvBWBZfgPFV677nokDk5OsdHkhReUVP85az3Dg6AOL5+SMd270nFoJ2+cFkjLUbEDkdjoc8+l4BcSTLDkpwDOsPkgAvIHgyaxlUqmgLSlOIYigRUlZQBSgEFC+XA T r( ˆx; µ) = 0, ˜x = ˆx x(k) , k = 1, ... , K kr(x(K) ; µ) = 0k2  ✏, ˜x = x(K) Approximate-solu4on surrogate model ‣ approximate soluGon: ‣ quanGty of interest: ˜x(µ) ⇡ x(µ) qsurr(µ) := q(˜x(µ)) qHFM(µ) := q(x(µ)) What methods exist for quan;fying the error ?(µ) := qHFM(µ) qsurr(µ)
  • 38. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1) Error indicators: residual norm 34 ‣ Applica(ons: terminaGon criterion, greedy methods, trust regions [Bui-Thanh et al., 2008; Hine and Kunkel, 2012; Wu and Hetmaniuk, 2015; Zahr, 2016] + Informa(ve: zero for high-fidelity model - Determinis(c: not a staGsGcal error model - Low quality: relaGonship to error depends on condiGoning kr(˜x; µ)k2<latexit 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yp6P/eUVYhx44K6WTF1LyTsW/RLdCEvrJAHw+z91m6T3T1v1eUx/HW3tS+aYCweOihfStKGFBeNuylHF/mU3c05UUTxTI0zXgn7usKOvKfJbLbfir7K82qMEBmwbSSpktmNRYYVltfbQPOVNt+he1UQwpM9OlzB6rxW9oNWElof5vXoVCDZmw6fjTlGCxLnS+mNhps0JdwznAIj/dFbVjTRMMIurCO4FaLMLipv3TOoWTyNVSqmk5YU1WSR1J/IifYaz7oR5p7HRWGfJSe7WvVlHmlXCp2detra8w9RK+6hRqcWgjUzA9J3xgUBOono/TEJo9tz5pBOvHVy7xcRvJL5kyPTAt629gXtbUJpw9fWzqXc0tX2j/Xgoj3NERmkSnBWInbx0TFtsZ8jAiAFu4DmYgbYj2SbjQKwPhe0xh9g4AMb7HE82K0oYSeZsdVdKeqpDoEli87qAzRlYK+DhIOAhAOvhHmMP+h20wedTN7nrgkeeDtkVinSeI4ycm0p0u+O3KW7nLNk4nd5jbsmrXEPNxutnBUtRtdlntKbH83rEb8X4lzS2ZfneStTpjzulkG3aI5X4yk9zWmxsCbBFBoDXLnchiW4kpxlu26GbpSFezadI8BSBIjkQVoLOQeQ5RjAZjCWxRdgfNW6b7YoEDn5eo0HUlSe09O85SI3jg6AeH7+SMf2b4iFoH1+MBlj7QZEzoPjIY++SUAcybKDEhzD+oMkwAsInsxaBpUq2oLSFKIYSoSUDqoAhYDi9RIA/Y4VbmzRLsF/BeKPsgQlwbMEY8lAhpSmEMVQEi5Bfh+PDC/05heUaoXUIhgbp8mcvpuDByyf2CZKAN8PwjbSKHPPHYIl1IltKi1hitIhrNZ0mczzMs/Q5ByVAo5ndnlfyG0tePxWR4JI5dhH+B2QCpI7Y7yDdV+DAyqFYkoO7i2/u019DXYTDJDVSRS5w07zOtzpAmUqPVI6KwyUG4kKr+1xDMM7gyDA3SARFxiJRLtly0Rx10Biwe3mCXe/28Z2vEU4vbGvbOIEbZOUw6ByorF5WiTO9Xi2XmSrUzBJ7fOw77QkHgKFwZL9R1+EKj/NVqU2s9fok/2db+x01KcflrC0auclHtKdFie4jcFqu5uuyMByVzs4nOC5gyTqksl8YNUn5HAKhmG/yYBD/0sUEzfEeNMMeQYF6T6UeuyXfPlg25Gw124ZNZ44zbsUE4L1EhHPXpGEmw3cOQBivyTC2ciB3lOgw4yrMCGs/zRRw4k+ptFM/Ed4LE0AqwPXaG5KkL/R6ZIBcncRbuu1KsvThqHTOj72bfhsBu3u2nwVKiD3NpPbFINcqEEGT6SYKLvJcRuCDYgSgfbgPTBkIoBqlRLX1arKMlC1ZraD9QsdxAnJUTVtr7L5y5oO4L3GJM7YdCam95cjyQkQlFHGYzbrw2cZ0qqYM1NeeiXzlZZmi2SFKKqqzedQVZsIKB9CYvh8E/VfT/nOCytCFkAUuzH+2xo2VbU3O0UthJxmAvoNzPuER2W3U4S6g9sROnQB07fgu1NrbQKGb+15yD5gFLYkemwt8il18cC3u+DEOcpG8BiDE9lRiTBVP7gDiBCnM0JCzEMbpoiqiOuE7FrglVhFg1xTZ7EsTyXZFw5m4kSSctnFWOIOSDPr9JHv5JE8dcTqkH+Hhncqgr+/JCyQoQyZ7JFPbBcr2V6CP/9D0W2fn/rpelZNjWxE7iKDmsAuSGIzIilCI/eK1l8rzm6wtCGg/iIuE/LO5uPV8g3uKOQq92txDzLuAJYRZVeKJ8ZCjAKUwRVuK3LYnsgQ2WqJU9VOnateEiCrTT6BGpNMlCOMOAXAPB+lrwPbI8OVFRYmXqQkvg/ZKBPEu7klT4NqZNfM3P3AcVyaS2iyswL2Rjrom1i2aQii2Yin9r4XiftuYLHvp0958eS8NSgggxSt6KmDYgEeFBJYzeMi7Y+jgVfacFvs93WJAyKOlfL8D3wOBk2ZCwGca/Hj1ZR5k5M6GMbdLOAgjF8ZOwXhKurth7R4WH0/y81UKCe+s13djluJI1VtowjQNbhMVavzujLyowkVisvJgsX+2Qv56hi8jPyJvI2XtXLnVZbJFLHQJEs/9Diu6BX56YIiTVCpv+OE76d1IAl8Gi/H0y4E1jKBBH7v1WmisAzHSbfSA866hTzO0wv6BhVyLHHqNXnqBK8D4pQTGQDBUxxA4B3MX2ZOT/MIoWqDJbdmnaND1wsDGy0eG9UJ5LDbK3HuFiCq2PUw2ZdG3lhWJsS4jPDRZ+owu6TA6WfiYLgE6wC0PQ3q9snVjs7nm7GWsd5SlTLELOQVAZvB+G42PGa4fpd6Dy31oq2RUhxzN1ZM9D2aMsmzWJLpbvtx+N6mI0HmussREFpySZy98HKkhO6V5umK953PbKOdehL2W8Qn8jgBj5pB75nUQORXLajMRhPqSJX791Y4WBxW8IA14XqGdWFr9AFB6I7KOVOaINiu9s/OeyppKxb35dk10sMRiwSSuznkL+81Tqxe3bNvzHuo+66HmGQXYQK5j++Tz14XbMYnR+ard71cJ+oeNlRjujuuE8/urqPOoSoR7XoZ/JCtYPLojrC33QZ7i3hlXYRXohH8bxvkKVubC2Xcep5TsMKOtgvPLaELX8kYPG91Su3r2d7aIZcUfBv4czwDz8U5J/1C2zNHpYZ1VaEOGkJtlupbR/XnEHXJIROyn8dlhU5y9rcMmPPVDUUFzfvKd6xBmzAae0OYpdGYh/cFdZW6SyleXM9I4H0WmsYiP5bYQNBHZzmKhLB16rRwlW7lsDHCVSKE4Y4iWlj/FTDIyMZFTgFSvA+jAm8cCVmu4GKU1iSvHaStTs2O8Qu2nTns+aISLxDErIYJOTDjc5ia+GU+bAFQfjhGf3U0W++FYbYOmQD/tEXozdiA4X5zScGliPwK2g1IeuYOgMy5ubKzYzq2VJJ9yMTgh6jmu9PsH0HgCEMSeYcZVKzS31G/wwI4YGEvyokemv5gqA4c9ZB2j3K3yLkPFyx5y3afEFcYmA8pjQdASczRuUaoY1jNkf4q6jEMdsvleaA2yb2XhvzHfEwjDvIQqpnGogH0v27CBK/Ypiz17nP78P/B68cnWCz5r5vAmeQjzKQAkTsO479tZvZ7sokkXtrKjq4a6StjBekcxCNDy/yaC6E9zmqIZV5VcYliRn9sO+F45UiO17DUwq4sftp7gegVbWjQY/oLVC7jih5dBn+RRAiax5U6OXbkgsibcitypZLZM7Gnh3iyYCl2C6jRkm5TyKzYdcba1kuUzeuFuwmuvZH4pGTDsPaIT1n0/LkTB2BPmyCoQFfkB3X6o4Z+cMIANa4+42df3tlFk8VFtchrET9854T8qYS32SxPp2pn7B3u07TNgWh2zk6otGtDYmyhR0Gh2ttkfa++NIG2+OdB8XyjUYh+4QXZYl8ExYrIhBR8HIDZoo+VaPcOPmlFrElRPHufDLs2H2gw58v4igZEzzmKfCfHIcRPihEBwAsVyCWfCi8XcTbEz33x1+2ZOHpJCAC63ivGXMSpBRPv/IOVX6AJBSeqiFa+w7wiDmPM93/wRlTE5GJzAwT5oj1LDhko6e82Ir5VVNKfmyrnePC49+3D3r2H3/bu7e79dq81cXrslaN3v93r3fv2t99907u3s2vB6Y4Ix+F08j8LUaxKchWZY/a+wWp3frf3Xe/e3t7vMLREpuuzTFI83M/iVUp/omsmPo++2Nq1f9XZ/XD64N7uzr3dH/e2fv8v/Beff7Xxjxv/vHF7Y3fju43fb7zY6G+83Zhs/NvGf2/8aeN/Pi0//fun//j0nwz6y19wzj9sGP8+/df/ASgZxeg=</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit> ‣ SubsGtute (2) into the residual of (1) and take the norm: r(x(µ); µ) = 0<latexit 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</latexit><latexit 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MEw+7Cx12QPr1p1/Qt/3WvtqyIYi4cOypeStCHFReNuytFFPmG3a84UUTxT44xXwoGusCPvMJlOD1rRV3lejREiA7aNJFUyu7HIsMLyugU0X2nzNt2rghCe7NHhClbntbIftJLQ+jCvR6cCyV53+GzMMVqQOG1K7yTcpynhnuEUGOmP3rKiiYYRdmGdwK0QZXZReeueQc3iaaxSMZ20pKjG86T+RM6313jWjTD3Ki4K+2Q52dWqb/JIuxTo7NTT1p5/jFpxkzQ6txCsmRmQvjMuCNBZRG+ASRjdnjOHdOKtk5u7iOCVzB8dmRZwy9oXtLcJpQ1fWjuXcktX2z/Wg4v2NEdkkCrBWYnY1UXHtPlBjgiAFOwcmosZ4CCSbTYKwPpc0Ap/gIGPbLDH8WD3moSdZMZWt52opzoEliw6qw/QlIG9DhKOAhKOvBJmMf6g200fdDL5gLsmeODtkFmlSOM5ysjFo0i/SfKQ7nJOk7Hf5TVunrTGTdrscumsaDG6LvOU3gNp3lzyWyLOlZ0Debq3MmXK424ZdKfmdDma0NOcFgtrEkyhMcCVy21YgivJWbbrZuhGWbjnkxkCLEWASB6ktZAzAFmOAGwGY1l8AcZXrftuigKRk693eCBF5RU9zVvOc+PoAIjn5490bP+eWAja5weTMdZuQOR0OB7y6LsAxJEsOyjBMaw/SAK8gODJr ˜x(µ) ⇡ x(µ) ‣ HFM governing equaGons: ‣ Approximate soluGon: (1) (2)
  • 39. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1) Error indicators: dual-weighted residual 35 ‣ Solve for the error (2)x ˜x = [ @r @x (˜x)] 1 r(˜x) + O(kx ˜xk2 ) <latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit> ‣ Applica(ons: adapGve mesh refinement [Babuska and Miller, 1984; Becker and Rannacher, 1996; Rannacher, 1999; Vendit and Darmofal, 2000; Fidkowski, 2007] + Accurate: second-order-accurate approximaGon - Determinis(c: not a staGsGcal error model ‣ Approximate HFM residual to first order 0 = r(x) = r(˜x) + @r @x (˜x)(x ˜x) + O(kx ˜xk2 ) <latexit 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</latexit><latexit 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 ‣ Approximate HFM quanGty of interest to first order (1)q(x) = q(˜x) + @q @x (˜x)(x ˜x) + O(kx ˜xk2 ) ‣ SubsGtute (2) in (1): q(x) q(˜x) = yT r(˜x) + O(kx ˜xk2 ) @r @x (˜x)T y = @q @x (˜x)T
  • 40. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 2) Rigorous a posteriori error bound 36 Proposi4on If the following condiGons hold: 1. is inf–sup stable, i.e., for all , there exists s.t. 2. is Lipschitz conGnuous, i.e., there exits such that then the quanGty-of-interest error can be bounded as r(·; µ)<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit> µ 2 D<latexit 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</latexit><latexit 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 ↵(µ) > 0 q(·) > 0 |q(x) q(˜x)|  ↵ kr(˜x; µ)k2 ‣ Applica(ons: reduced-order models [Rathinam and Petzold, 2003; Grepl and Patera, 2005; Antoulas, 2005; Hinze and Volkwein, 2005; C. et al., 2017] + Cer(fica(on: guaranteed bound - Lack sharpness: orders-of-magnitude overesGmaGon - Difficult to implement: require bounds for inf–sup/Lipschitz constants - Determinis(c: not a staGsGcal error model - kr(z1; µ) r(z2; µ)k2 ↵(µ)kz1 z2k2, 8z1, z2 2 RN |q(z1) q(z2)|  kz1 z2k2, 8z1, z2 2 RN
  • 41. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 3) Model-discrepancy approach ‣ Applica(ons: ‣ Model calibraGon [Kennedy, O’Hagan, 2001; Higdon et al., 2003; Higdon et al., 2004] ‣ MulGfidelity opGmizaGon [Gano et al., 2005; Huang et al., 2006; March, Willcox, 2012; Ng, Eldred, 2012] + General: applicable to any surrogate model + Sta(s(cal: interpretable as a staGsGcal error model + Epistemic uncertainty quan(fied: through variance - Poorly informa(ve inputs: parameters weakly related to the error - Poor scalability: difficult in high-dimensional parameter spaces - Thus, can introduce large epistemic uncertainty: large variance µ<latexit 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37 quan(ty of interest qHFM<latexit 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</latexit><latexit sha1_base64="+dEK4DwuTqbUh1zw5Peof32SZXo=">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 qsurr -5 0 5 10 parameter µ = qHFM qsurr ˜”(µ) ≥ N(µ(µ); ‡2 (µ))
  • 42. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Objective 38 Error modeling: staGsGcal model for the error ‣ strength of #3 + Sta(s(cal: interpretable as a staGsGcal error model + Epistemic uncertainty quan(fied: through variance A posteriori: use residual-based quanGGes computed by the surrogate ‣ strength of #1 and #2 + Informa(ve inputs: quanGGes are strongly related to the error + Thus, can lead to lower epistemic uncertainty: lower variance Goal: combine the strengths of 1. error indicators, 2. rigorous a posteriori error bounds, and 3. the model-discrepancy approach
  • 43. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Main idea 39 Key observation 10 5 10 4 10 5 10 4 Residual r/error bound error(energy norm)|||uhured||| 10 5 10 4 10 5 10 4 Residual r error(energy norm)|||uhured||| (r; ||| u|||) ( µ u ; ||| u|||) ROMs generate error indicators that correlate with the error ROMES Kevin Carlberg, Martin Dr + Can produce lower-variance models than the model-discrepancy approach Idea: Apply machine learning regression to generate a mapping from residual-based quan((es to a random variable for the error Machine-learning error models ‣ Observa4on: residual-based quanGGes are informaGve of the error ‣ So, these are informaGve features: can predict the error with low variance
  • 44. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Machine-learning error models: formulation 40 ‣ features: ‣ regression funcGon: ‣ noise: ‣ Note: model-discrepancy approach uses fl(µ) œ RNfl <latexit 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”(µ) = f(fl(µ)) ¸ ˚˙ ˝ deterministic + ‘(fl(µ)) ¸ ˚˙ ˝ stochastic<latexit 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f(fl) = E[” | fl]<latexit sha1_base64="0bLzeycmdaBFc1feZw7k+ZGI2EA=">AAArDnicnVrdcty2Fd6kf4n657SXveFU4xlbXWu0imwn6aiNbdmJXVuSV7KcjijvYLlYLiISpEFwJZlmn6FPksvedXrbB+hNL9pn6QEIgiAIrp1qkjEBfOcc4BycHwA7TSOS8a2tf3/w4Q9++KMf/+Sjj9d++rOf/+KX1z751UmW5CzAL4IkStg3U5ThiFD8ghMe4W9ShlE8jfDL6fkDMf5yiVlGEnrMr1J8 fl = µ ‣ Desired properGes in error model 1. cheaply computable: features are inexpensive to compute 2. low variance: noise model has low variance 3. generalizable: empirical distribuGons of and ‘close’ on test data fl(µ) ˜‘(fl) ” ˜” ˜” ‣ regression-funcGon model: ‣ noise model: ˜f(¥ f) ˜”(µ) = ˜f(fl(µ)) ¸ ˚˙ ˝ deterministic + ˜‘(fl(µ)) ¸ ˚˙ ˝ stochastic ˜‘(¥ ‘)
  • 45. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Training and machine learning 1. Training: Solve high-fidelity and mulGple surrogates for µ 2 Dtraining 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dquery Dtraining D 41 = qHFM qsurr<latexit 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</latexit><latexit 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8keXrbq/hX+utfadzUwFg8dlC8laUOKi8bdlKPFsenhmSKaR6lJl/ZKONAVduQ9T2azg1b0VZ5XY4TIgG0jSZXMbiwyrLC83gear7T5Ad2rghCe7NHhClbntbIftJLQ+jCvR6cCyd50+GzMMVqQOG1KLwXcpSnhnuEUGOmP3rKiiYYRznF8uBWizC4qb90zqFk8jVUqppOWFNVkkdSfyPn2Gs+6EeZexkVhnywnu1r1dR5pt/KcnXra2vOPUSuuckbnFoI1MwPSd8YFATqL6BUsCaPbc+aQTrx1cnUWEbyS+YMj0wLet/YF7W1CacMX1s6l3NLV9o/14KI9zREZpEpwViJ2d9AxbXGQIwIgBbuA5mIGOIhkm40CsD4XtMYfYOAjG+xxPNjFImEnmbHVdSPqqQ6BJYvO6gM0ZWCvg4SjgIQjr4R5jD/odtMHnUw+4K4JHng7ZFYp0niOMnLzJ9Jvkjyku5yzZOJ3eY2bJ61xlTUbrZwVLUbXZZ7SeyDNmxG/JeJc2TmQp3srU6Y87pZBd2pOV+MpPc1psbAmwRQaA1y53IYluJKcZbtuhm6UhXsxnSPAUgSI5EFaCzkHkOUYwGYwlsUXYHzVui+HKBA5+XqLB1JUXtLTvOUiN44OgHh+/kjH9u+IhaB9fjAZY+0GRE6H4yGPXsYXR7LsoATHsP4gCfACgiezlkGliragNIUohhIhpYMqQCGgeL0EQL9jhRtbtEvw34D4oyxBSfAswVgykCGlKUQxlIRLkN/OI8MLvQcGpVohtQjGxmkyp6+34AHLZ7aJEsD3g7CNNMrcc4dgCXVim0pLmKJ0CKs1XSbzsswzNLlEpYDjmV3eHnJbCx6/1ZEgUjn2EX4HpILkzhjvYN03yYBKoZiSg3vHrz9TX4PdCwNkdRJFroHTvA53u0CZSo+UzgoD5Uaiwmt7HMPwziAIcDdIxHVGItFu2TJR3DWQWHC7ecLd77axHW8RTm/sC5w4Qdsk5TConGhsnhaJc8OcrRfZ6hRMUvs87DsticdAYbBk/9EXocpPs1Wpzew1+mR/5xs7HfXphyUsrdp5icd0p8UJbmOw2u6mKzKw3NUODid47iCJumQyH1n1CTmcgmHYbzLg0P8SxcQNMV7WQp5BQbqPpR77JV8+2nYk7M1VRo0nTvMuxYRgvYfDs1ck4WYDdw6A2O9ZcDZyoKv+Osy4ChPC+k8TNZzoYxrNxH+Ex9IEsDpwjeamBPkbnS4ZIHcX4bZeq7I8bRg6reNj34fPZtDurs1XoQJybzO5TTHIhRpk8ESKibKbHLch2IAoEWgP3gNDJgKoVilxXa2qLANVa2Y7WL/QQZyQHFXT9iqbv+/oCN5rTOKMTWdien81kpwAQRllPGazPnyWIa2KOTPllVcyX2lptkhWiKKq2nwOVbWJgPIhJIbPN1H/9ZzvvLAiZAFEsRvjv61hU1V7s1PUQshpJqDfwLxPeFR2O0WoO7gdoUMXMH0Lvju11iZg+Naeh+wDRmFLosfWIp9SFw98WzBOnKNsBI8xOJEdlQhT9YM7gAhxOiMkxDy0YYqoirhOyK4FXolVNMg1dRbL8lSSfeFgJk4kKZddjCXugDSzTh/5Th7JU0esDvl3aHinIvjbTMICGcqQyR75xHaxku0l+PM/FN328NxP17NqamQjchcZ1AR2QRKbEUkRGrlXtP5acXaDpQ0B9VdxmZDXHp+ulm9xRyFXud+Ie5BxB7CMKLtSPDEWYhSgDK5wW5HD9kSGyFZLnKp26lz1kgBZbfIJ1JhkohxhxCkA5vkofR3YHhmurLAw8Vol8X3IRpkg3s0teRpUI7tm5u4HjuPSXEKTnRWwN9JB38SyTUMQzUY8tfe9SNzX64p9P23KK0nUxaCADFK0oqcOigV4UEhgNY+LtD+OBl5ww22x394l3xEGD+3yFBB8GgZNmSMBnG7x49XEeZfzOhjGnS3gOIxfGTsL4SrqHYS0eFh9P8vNVCgnvhNe3Q5diYNVbaMI0GW4TFWu8woz8nsFFYrLyYLtALA329UxeCX5E3mtLWvrzjshkyliAUqWfuxxX9Fr8hsARZqgUn/TCd9V60AS+DRejqddCKxlAgn89qvTRGEZjqtupQdcdgt5mqdX9D0q5HDi1Gvy1AlhB8QpVzIAgic6gMA7mL/MnJ7mEULVBktuzWpHh64XBjZaPEKqc8hh51fi3I1AVLFLYrIvjbwRrUyIcRnhA9DUbXZJgTPQxM1wCdYxaHsy1O2Tax6dz7dkLWO9pSpliLnIKwI2g/HdbHjMcL0v9UJX6ktbI6U47G6sm0h8UyV5lkwy3W0/Dt/bdCTIXH05AkILL4mzl1+OlNDt0jxd8b7zM9tup/6E/TruiTxUwGNn0LsnNRD5eQgqs9GEOlLlLr4VFBZHFjxgTbieYV3YGn1AKLqjcs6UJgi2q/1n592VtBWLW/PsMunxiMUDyQ0d8pf3Gidir27bN+Zt1APXT0yyqzCB3Mr3yWfv3TWjlCPzHbZerhN7DxuqMd1914lnj9dR51CViHa9DH7UVjB5jEfY226DvaXirn2E16MR/G8b5Clbmytl3HqeU7DCjrYLzy2hK1/JGDxvdUrt69ne2iFXFXzb+HM8A8/FaSf9WtsLR6WGdVWhDhpCbZbqW0f15xB1ySETcpDHZYXOcva3DJjz2zuKCpr3W9/hBm3CaOxtYZZGIx/e19RV6kaleAM8I4G3Wmgai/9YYgOhH53lKBLC1qnTgla6lcPGCFqJQIY7imjB/dfAICMbFzkLSPE+jAq/cSRkuYKLUVqTvHaQtjo1O8wv2HbmsOeLSrxAELMaJuTAjM9hauKX+bAFQPnhGP110my9F4bZOmQC/BsRobdlA4b7zSUFlyLyc2J3IOmZOwIy5+bKzo7p2FJJ9lETgx+imm9Qs39NgCMMSeRNZlCxSn9H/aAJ4ICFvSgnhmj6g6E6cNRD2j3K3SLnPlyw5C3bfUJcYWA+pDQeBiWRR+cyoY5hNUf6q6jHMNgtl8NAbZLbLw35j/mYRhzkUVQzjUUD6H/dhAlesU1Z6sND+wrA0ZunZ1gs+a+bwJnkI8ykAJE7DuM/Emb2e7KVJF7dyg6wGukrYwXpHMcjQ8v8lguhPc5qiGVeVXGJYkZ/ajvheOVIDtmw1MKuLH7me4HoRW1o0GP6C1Qu44oeYAZ/2kMImseVOj924oLI+3IrcrGS2TOxp4d4smApdguo0ZJuVsis2HXG2tYrlM3rhbsVrr2X+Kxkw7D2iE9Z9BS6EwdgT5sgqEA35Jdp+qOGfnDCADWuPuP3U97bRZPFRbXIaxE/fO8E/qmEd9ksT6dqf+w97tO0zYFodtpOqLRrQ2JsoSdBodo7ZX0vwDSBtvjDoHi+3ShEv/SCbLEvg2JFZEIKPg3AbNGnSrR7E5+0ItakKJ69VYZdng80mMtlfEMDopccRb6TQxHit7mIAOC1CuSqT4WXizgb4nez+Ev3TBy9KgQAXe8VY67i1IKJN//Byq/QhIITVUQr35FeEYcx5vs/eiMqYnKxuQGCfN2eJYcMlPQHEJG+rdWUczx47Hz9uLfz+Ovezt7+7/ZbE6fHXjl67+v93s7Xv/vmq97O7p4FpzsiHIfTyf8sBH/DPMfsf4XV7v5+/5vezv7+7zG0RKbrs0xSPNzP4lVKf+tqJj6PPtvas38e2f1w/mhnb3dn7/v9rT/8K//p5F9t/PPGv2zc39jb+GbjDxsvN/ob7zYmG/++8Z8b/7Xx37f/8+nXn/7x0z8x6C9/wTn/sGH8+xT9HwKpkaU=</latexit><latexit 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8keXrbq/hX+utfadzUwFg8dlC8laUOKi8bdlKPFsenhmSKaR6lJl/ZKONAVduQ9T2azg1b0VZ5XY4TIgG0jSZXMbiwyrLC83gear7T5Ad2rghCe7NHhClbntbIftJLQ+jCvR6cCyd50+GzMMVqQOG1KLwXcpSnhnuEUGOmP3rKiiYYRznF8uBWizC4qb90zqFk8jVUqppOWFNVkkdSfyPn2Gs+6EeZexkVhnywnu1r1dR5pt/KcnXra2vOPUSuuckbnFoI1MwPSd8YFATqL6BUsCaPbc+aQTrx1cnUWEbyS+YMj0wLet/YF7W1CacMX1s6l3NLV9o/14KI9zREZpEpwViJ2d9AxbXGQIwIgBbuA5mIGOIhkm40CsD4XtMYfYOAjG+xxPNjFImEnmbHVdSPqqQ6BJYvO6gM0ZWCvg4SjgIQjr4R5jD/odtMHnUw+4K4JHng7ZFYp0niOMnLzJ9Jvkjyku5yzZOJ3eY2bJ61xlTUbrZwVLUbXZZ7SeyDNmxG/JeJc2TmQp3srU6Y87pZBd2pOV+MpPc1psbAmwRQaA1y53IYluJKcZbtuhm6UhXsxnSPAUgSI5EFaCzkHkOUYwGYwlsUXYHzVui+HKBA5+XqLB1JUXtLTvOUiN44OgHh+/kjH9u+IhaB9fjAZY+0GRE6H4yGPXsYXR7LsoATHsP4gCfACgiezlkGliragNIUohhIhpYMqQCGgeL0EQL9jhRtbtEvw34D4oyxBSfAswVgykCGlKUQxlIRLkN/OI8MLvQcGpVohtQjGxmkyp6+34AHLZ7aJEsD3g7CNNMrcc4dgCXVim0pLmKJ0CKs1XSbzsswzNLlEpYDjmV3eHnJbCx6/1ZEgUjn2EX4HpILkzhjvYN03yYBKoZiSg3vHrz9TX4PdCwNkdRJFroHTvA53u0CZSo+UzgoD5Uaiwmt7HMPwziAIcDdIxHVGItFu2TJR3DWQWHC7ecLd77axHW8RTm/sC5w4Qdsk5TConGhsnhaJc8OcrRfZ6hRMUvs87DsticdAYbBk/9EXocpPs1Wpzew1+mR/5xs7HfXphyUsrdp5icd0p8UJbmOw2u6mKzKw3NUODid47iCJumQyH1n1CTmcgmHYbzLg0P8SxcQNMV7WQp5BQbqPpR77JV8+2nYk7M1VRo0nTvMuxYRgvYfDs1ck4WYDdw6A2O9ZcDZyoKv+Osy4ChPC+k8TNZzoYxrNxH+Ex9IEsDpwjeamBPkbnS4ZIHcX4bZeq7I8bRg6reNj34fPZtDurs1XoQJybzO5TTHIhRpk8ESKibKbHLch2IAoEWgP3gNDJgKoVilxXa2qLANVa2Y7WL/QQZyQHFXT9iqbv+/oCN5rTOKMTWdien81kpwAQRllPGazPnyWIa2KOTPllVcyX2lptkhWiKKq2nwOVbWJgPIhJIbPN1H/9ZzvvLAiZAFEsRvjv61hU1V7s1PUQshpJqDfwLxPeFR2O0WoO7gdoUMXMH0Lvju11iZg+Naeh+wDRmFLosfWIp9SFw98WzBOnKNsBI8xOJEdlQhT9YM7gAhxOiMkxDy0YYqoirhOyK4FXolVNMg1dRbL8lSSfeFgJk4kKZddjCXugDSzTh/5Th7JU0esDvl3aHinIvjbTMICGcqQyR75xHaxku0l+PM/FN328NxP17NqamQjchcZ1AR2QRKbEUkRGrlXtP5acXaDpQ0B9VdxmZDXHp+ulm9xRyFXud+Ie5BxB7CMKLtSPDEWYhSgDK5wW5HD9kSGyFZLnKp26lz1kgBZbfIJ1JhkohxhxCkA5vkofR3YHhmurLAw8Vol8X3IRpkg3s0teRpUI7tm5u4HjuPSXEKTnRWwN9JB38SyTUMQzUY8tfe9SNzX64p9P23KK0nUxaCADFK0oqcOigV4UEhgNY+LtD+OBl5ww22x394l3xEGD+3yFBB8GgZNmSMBnG7x49XEeZfzOhjGnS3gOIxfGTsL4SrqHYS0eFh9P8vNVCgnvhNe3Q5diYNVbaMI0GW4TFWu8woz8nsFFYrLyYLtALA329UxeCX5E3mtLWvrzjshkyliAUqWfuxxX9Fr8hsARZqgUn/TCd9V60AS+DRejqddCKxlAgn89qvTRGEZjqtupQdcdgt5mqdX9D0q5HDi1Gvy1AlhB8QpVzIAgic6gMA7mL/MnJ7mEULVBktuzWpHh64XBjZaPEKqc8hh51fi3I1AVLFLYrIvjbwRrUyIcRnhA9DUbXZJgTPQxM1wCdYxaHsy1O2Tax6dz7dkLWO9pSpliLnIKwI2g/HdbHjMcL0v9UJX6ktbI6U47G6sm0h8UyV5lkwy3W0/Dt/bdCTIXH05AkILL4mzl1+OlNDt0jxd8b7zM9tup/6E/TruiTxUwGNn0LsnNRD5eQgqs9GEOlLlLr4VFBZHFjxgTbieYV3YGn1AKLqjcs6UJgi2q/1n592VtBWLW/PsMunxiMUDyQ0d8pf3Gidir27bN+Zt1APXT0yyqzCB3Mr3yWfv3TWjlCPzHbZerhN7DxuqMd1914lnj9dR51CViHa9DH7UVjB5jEfY226DvaXirn2E16MR/G8b5Clbmytl3HqeU7DCjrYLzy2hK1/JGDxvdUrt69ne2iFXFXzb+HM8A8/FaSf9WtsLR6WGdVWhDhpCbZbqW0f15xB1ySETcpDHZYXOcva3DJjz2zuKCpr3W9/hBm3CaOxtYZZGIx/e19RV6kaleAM8I4G3Wmgai/9YYgOhH53lKBLC1qnTgla6lcPGCFqJQIY7imjB/dfAICMbFzkLSPE+jAq/cSRkuYKLUVqTvHaQtjo1O8wv2HbmsOeLSrxAELMaJuTAjM9hauKX+bAFQPnhGP110my9F4bZOmQC/BsRobdlA4b7zSUFlyLyc2J3IOmZOwIy5+bKzo7p2FJJ9lETgx+imm9Qs39NgCMMSeRNZlCxSn9H/aAJ4ICFvSgnhmj6g6E6cNRD2j3K3SLnPlyw5C3bfUJcYWA+pDQeBiWRR+cyoY5hNUf6q6jHMNgtl8NAbZLbLw35j/mYRhzkUVQzjUUD6H/dhAlesU1Z6sND+wrA0ZunZ1gs+a+bwJnkI8ykAJE7DuM/Emb2e7KVJF7dyg6wGukrYwXpHMcjQ8v8lguhPc5qiGVeVXGJYkZ/ajvheOVIDtmw1MKuLH7me4HoRW1o0GP6C1Qu44oeYAZ/2kMImseVOj924oLI+3IrcrGS2TOxp4d4smApdguo0ZJuVsis2HXG2tYrlM3rhbsVrr2X+Kxkw7D2iE9Z9BS6EwdgT5sgqEA35Jdp+qOGfnDCADWuPuP3U97bRZPFRbXIaxE/fO8E/qmEd9ksT6dqf+w97tO0zYFodtpOqLRrQ2JsoSdBodo7ZX0vwDSBtvjDoHi+3ShEv/SCbLEvg2JFZEIKPg3AbNGnSrR7E5+0ItakKJ69VYZdng80mMtlfEMDopccRb6TQxHit7mIAOC1CuSqT4WXizgb4nez+Ev3TBy9KgQAXe8VY67i1IKJN//Byq/QhIITVUQr35FeEYcx5vs/eiMqYnKxuQGCfN2eJYcMlPQHEJG+rdWUczx47Hz9uLfz+Ovezt7+7/ZbE6fHXjl67+v93s7Xv/vmq97O7p4FpzsiHIfTyf8sBH/DPMfsf4XV7v5+/5vezv7+7zG0RKbrs0xSPNzP4lVKf+tqJj6PPtvas38e2f1w/mhnb3dn7/v9rT/8K//p5F9t/PPGv2zc39jb+GbjDxsvN/ob7zYmG/++8Z8b/7Xx37f/8+nXn/7x0z8x6C9/wTn/sGH8+xT9HwKpkaU=</latexit><latexit 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high-fidelity model surrogate models
  • 46. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Training and machine learning 1. Training: Solve high-fidelity and mulGple surrogates for µ 2 Dtraining 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dquery Dtraining D 41 = qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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high-fidelity model surrogate models
  • 47. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Training and machine learning 1. Training: Solve high-fidelity and mulGple surrogates for µ 2 Dtraining 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dquery Dtraining D 41 = qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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high-fidelity model surrogate models
  • 48. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Training and machine learning 1. Training: Solve high-fidelity and mulGple surrogates for µ 2 Dtraining 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dquery Dtraining D 41 = qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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high-fidelity model surrogate models
  • 49. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Training and machine learning 1. Training: Solve high-fidelity and mulGple surrogates for µ 2 Dtraining 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dquery Dtraining D 41 = qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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8keXrbq/hX+utfadzUwFg8dlC8laUOKi8bdlKPFsenhmSKaR6lJl/ZKONAVduQ9T2azg1b0VZ5XY4TIgG0jSZXMbiwyrLC83gear7T5Ad2rghCe7NHhClbntbIftJLQ+jCvR6cCyd50+GzMMVqQOG1KLwXcpSnhnuEUGOmP3rKiiYYRznF8uBWizC4qb90zqFk8jVUqppOWFNVkkdSfyPn2Gs+6EeZexkVhnywnu1r1dR5pt/KcnXra2vOPUSuuckbnFoI1MwPSd8YFATqL6BUsCaPbc+aQTrx1cnUWEbyS+YMj0wLet/YF7W1CacMX1s6l3NLV9o/14KI9zREZpEpwViJ2d9AxbXGQIwIgBbuA5mIGOIhkm40CsD4XtMYfYOAjG+xxPNjFImEnmbHVdSPqqQ6BJYvO6gM0ZWCvg4SjgIQjr4R5jD/odtMHnUw+4K4JHng7ZFYp0niOMnLzJ9Jvkjyku5yzZOJ3eY2bJ61xlTUbrZwVLUbXZZ7SeyDNmxG/JeJc2TmQp3srU6Y87pZBd2pOV+MpPc1psbAmwRQaA1y53IYluJKcZbtuhm6UhXsxnSPAUgSI5EFaCzkHkOUYwGYwlsUXYHzVui+HKBA5+XqLB1JUXtLTvOUiN44OgHh+/kjH9u+IhaB9fjAZY+0GRE6H4yGPXsYXR7LsoATHsP4gCfACgiezlkGliragNIUohhIhpYMqQCGgeL0EQL9jhRtbtEvw34D4oyxBSfAswVgykCGlKUQxlIRLkN/OI8MLvQcGpVohtQjGxmkyp6+34AHLZ7aJEsD3g7CNNMrcc4dgCXVim0pLmKJ0CKs1XSbzsswzNLlEpYDjmV3eHnJbCx6/1ZEgUjn2EX4HpILkzhjvYN03yYBKoZiSg3vHrz9TX4PdCwNkdRJFroHTvA53u0CZSo+UzgoD5Uaiwmt7HMPwziAIcDdIxHVGItFu2TJR3DWQWHC7ecLd77axHW8RTm/sC5w4Qdsk5TConGhsnhaJc8OcrRfZ6hRMUvs87DsticdAYbBk/9EXocpPs1Wpzew1+mR/5xs7HfXphyUsrdp5icd0p8UJbmOw2u6mKzKw3NUODid47iCJumQyH1n1CTmcgmHYbzLg0P8SxcQNMV7WQp5BQbqPpR77JV8+2nYk7M1VRo0nTvMuxYRgvYfDs1ck4WYDdw6A2O9ZcDZyoKv+Osy4ChPC+k8TNZzoYxrNxH+Ex9IEsDpwjeamBPkbnS4ZIHcX4bZeq7I8bRg6reNj34fPZtDurs1XoQJybzO5TTHIhRpk8ESKibKbHLch2IAoEWgP3gNDJgKoVilxXa2qLANVa2Y7WL/QQZyQHFXT9iqbv+/oCN5rTOKMTWdien81kpwAQRllPGazPnyWIa2KOTPllVcyX2lptkhWiKKq2nwOVbWJgPIhJIbPN1H/9ZzvvLAiZAFEsRvjv61hU1V7s1PUQshpJqDfwLxPeFR2O0WoO7gdoUMXMH0Lvju11iZg+Naeh+wDRmFLosfWIp9SFw98WzBOnKNsBI8xOJEdlQhT9YM7gAhxOiMkxDy0YYqoirhOyK4FXolVNMg1dRbL8lSSfeFgJk4kKZddjCXugDSzTh/5Th7JU0esDvl3aHinIvjbTMICGcqQyR75xHaxku0l+PM/FN328NxP17NqamQjchcZ1AR2QRKbEUkRGrlXtP5acXaDpQ0B9VdxmZDXHp+ulm9xRyFXud+Ie5BxB7CMKLtSPDEWYhSgDK5wW5HD9kSGyFZLnKp26lz1kgBZbfIJ1JhkohxhxCkA5vkofR3YHhmurLAw8Vol8X3IRpkg3s0teRpUI7tm5u4HjuPSXEKTnRWwN9JB38SyTUMQzUY8tfe9SNzX64p9P23KK0nUxaCADFK0oqcOigV4UEhgNY+LtD+OBl5ww22x394l3xEGD+3yFBB8GgZNmSMBnG7x49XEeZfzOhjGnS3gOIxfGTsL4SrqHYS0eFh9P8vNVCgnvhNe3Q5diYNVbaMI0GW4TFWu8woz8nsFFYrLyYLtALA329UxeCX5E3mtLWvrzjshkyliAUqWfuxxX9Fr8hsARZqgUn/TCd9V60AS+DRejqddCKxlAgn89qvTRGEZjqtupQdcdgt5mqdX9D0q5HDi1Gvy1AlhB8QpVzIAgic6gMA7mL/MnJ7mEULVBktuzWpHh64XBjZaPEKqc8hh51fi3I1AVLFLYrIvjbwRrUyIcRnhA9DUbXZJgTPQxM1wCdYxaHsy1O2Tax6dz7dkLWO9pSpliLnIKwI2g/HdbHjMcL0v9UJX6ktbI6U47G6sm0h8UyV5lkwy3W0/Dt/bdCTIXH05AkILL4mzl1+OlNDt0jxd8b7zM9tup/6E/TruiTxUwGNn0LsnNRD5eQgqs9GEOlLlLr4VFBZHFjxgTbieYV3YGn1AKLqjcs6UJgi2q/1n592VtBWLW/PsMunxiMUDyQ0d8pf3Gidir27bN+Zt1APXT0yyqzCB3Mr3yWfv3TWjlCPzHbZerhN7DxuqMd1914lnj9dR51CViHa9DH7UVjB5jEfY226DvaXirn2E16MR/G8b5Clbmytl3HqeU7DCjrYLzy2hK1/JGDxvdUrt69ne2iFXFXzb+HM8A8/FaSf9WtsLR6WGdVWhDhpCbZbqW0f15xB1ySETcpDHZYXOcva3DJjz2zuKCpr3W9/hBm3CaOxtYZZGIx/e19RV6kaleAM8I4G3Wmgai/9YYgOhH53lKBLC1qnTgla6lcPGCFqJQIY7imjB/dfAICMbFzkLSPE+jAq/cSRkuYKLUVqTvHaQtjo1O8wv2HbmsOeLSrxAELMaJuTAjM9hauKX+bAFQPnhGP110my9F4bZOmQC/BsRobdlA4b7zSUFlyLyc2J3IOmZOwIy5+bKzo7p2FJJ9lETgx+imm9Qs39NgCMMSeRNZlCxSn9H/aAJ4ICFvSgnhmj6g6E6cNRD2j3K3SLnPlyw5C3bfUJcYWA+pDQeBiWRR+cyoY5hNUf6q6jHMNgtl8NAbZLbLw35j/mYRhzkUVQzjUUD6H/dhAlesU1Z6sND+wrA0ZunZ1gs+a+bwJnkI8ykAJE7DuM/Emb2e7KVJF7dyg6wGukrYwXpHMcjQ8v8lguhPc5qiGVeVXGJYkZ/ajvheOVIDtmw1MKuLH7me4HoRW1o0GP6C1Qu44oeYAZ/2kMImseVOj924oLI+3IrcrGS2TOxp4d4smApdguo0ZJuVsis2HXG2tYrlM3rhbsVrr2X+Kxkw7D2iE9Z9BS6EwdgT5sgqEA35Jdp+qOGfnDCADWuPuP3U97bRZPFRbXIaxE/fO8E/qmEd9ksT6dqf+w97tO0zYFodtpOqLRrQ2JsoSdBodo7ZX0vwDSBtvjDoHi+3ShEv/SCbLEvg2JFZEIKPg3AbNGnSrR7E5+0ItakKJ69VYZdng80mMtlfEMDopccRb6TQxHit7mIAOC1CuSqT4WXizgb4nez+Ev3TBy9KgQAXe8VY67i1IKJN//Byq/QhIITVUQr35FeEYcx5vs/eiMqYnKxuQGCfN2eJYcMlPQHEJG+rdWUczx47Hz9uLfz+Ovezt7+7/ZbE6fHXjl67+v93s7Xv/vmq97O7p4FpzsiHIfTyf8sBH/DPMfsf4XV7v5+/5vezv7+7zG0RKbrs0xSPNzP4lVKf+tqJj6PPtvas38e2f1w/mhnb3dn7/v9rT/8K//p5F9t/PPGv2zc39jb+GbjDxsvN/ob7zYmG/++8Z8b/7Xx37f/8+nXn/7x0z8x6C9/wTn/sGH8+xT9HwKpkaU=</latexit><latexit 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high-fidelity model surrogate models
  • 50. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Training and machine learning 1. Training: Solve high-fidelity and mulGple surrogates for µ 2 Dtraining 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dquery Dtraining D 41 = qHFM qsurr<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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high-fidelity model surrogate models ‣ randomly divide data into (1) training data and (2) tesGng data ‣ construct regression-funcGon model via cross validaGon on training data ‣ construct noise model from sample variance on test data ˜f ˜‘
  • 51. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Reduction 42 features ⇢ 1. Training: Solve high-fidelity and reduced-order models for 2. Machine learning: Construct regression model 3. Reduc(on: predict surrogate-model error for µ 2 Dtraining µ 2 Dquery Dtraining Doutputsinputs µ surrogate model qsurr<latexit 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˜qHFM(µ) ¸ ˚˙ ˝ stochastic = qsurr(µ) ¸ ˚˙ ˝ deterministic + ˜”(µ) ¸˚˙˝ stochastic<latexit 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regression model machine learning error model ˜”<latexit 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˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit sha1_base64="qpFvcXRrO0Mec8Yp6uVtTwOURus=">AAArQHicnVpbb9y4FZ7d3nbTW7Z9LFAINQIk3onh8TrJ7hZuc3GyzTaxnbHjbGE5A46Go+FaohSKGttR1P+xb/0tBfrQn9B/UKAPRYH2pU89pCiKoqhJtsYuIpLfOYc8h+dCcqZpRDK+ufm3997/zne/9/0ffPDhlR/+6Mc/+enVj352nCU5C/DzIIkS9tU
  • 52. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Regression model: construct regression model to trade off: ‣ High capacity: low variance, more data to generalize ‣ Low capacity: high variance, less data to generalize ˜f<latexit 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Error-model construction 43 Feature engineering: select features to trade off: 1. Number of features ‣ Large number: costly, low variance, high-capacity regression ‣ Small number: cheap, high variance, low-capacity regression 2. Quality of features ‣ High quality: expensive, low variance ‣ Low quality: cheap, high variance ⇢ Method 2: Large number of features and high-dimensional regression [Trehan, C., Durlofsky, 2017; Freno, C., 2018] Method 1: Dual-weighted residual and Gaussian process regression [Drohmann, C., 2015; C., Uy, Lu, Morzfeld, 2018] ˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit 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  • 53. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Regression model: construct regression model to trade off: ‣ High capacity: low variance, more data to generalize ‣ Low capacity: high variance, less data to generalize ˜f<latexit 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Error-model construction 43 Feature engineering: select features to trade off: 1. Number of features ‣ Large number: costly, low variance, high-capacity regression ‣ Small number: cheap, high variance, low-capacity regression 2. Quality of features ‣ High quality: expensive, low variance ‣ Low quality: cheap, high variance ⇢ Method 2: Large number of features and high-dimensional regression [Trehan, C., Durlofsky, 2017; Freno, C., 2018] Method 1: Dual-weighted residual and Gaussian process regression [Drohmann, C., 2015; C., Uy, Lu, Morzfeld, 2018] ˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit 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  • 54. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Feature: dual-weighted residual [Drohmann, C., 2015] 44 q(x) q(˜x) = yT r(˜x) + O(kx ˜xk2 )<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit> @r @x (˜x)T y = @q @x (˜x)T <latexit 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</latexit><latexit 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 ‣ Want to avoid HFM-scale solves, so approximate dual as and construct a ROM for the dual y T @r @x (˜x)T yˆy = y T @q @x (˜x)T y ¥ ˜y = yˆy ‣ One feature: ‣ can control feature quality via dimension of ‣ Regression model: Gaussian process [Rasmussen, Williams, 2006] q(x) q(˜x) ⇡ ˆyT y T r(˜x) y
  • 55. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Thermal block 1 2 3 4 5 6 7 8 9 D N1 N0 4c(x; µ)u(x; µ) = 0 in ⌦ x(µ) = 0 on D rc(µ)x(µ) · n = 0 on N0 rc(µ)x(µ) · n = 1 on N1 Inputs µ 2 [0.1, 10]9 define di↵usivity c in subdomains Output z(µ) = R N1 x(µ)dx is compliant ROM constructed via RB–Greedy [?] ROMES Kevin Carlberg, Martin Drohmann 15 / 22 Application: Bayesian inference 45 1 2 3 4 5 6 7 8 9 D N1 N0 4c(x; µ)u(x; µ) = 0 in ⌦ x(µ) = 0 on D rc(µ)x(µ) · n = 0 on N0 rc(µ)x(µ) · n = 1 on N1 Inputs µ 2 [0.1, 10]9 define di↵usivity c in subdomains Output z(µ) = R N1 x(µ)dx is compliant ROM constructed via RB–Greedy [?] ROMES Kevin Carlberg, Martin Drohmann 15 / 22 ‣ Inputs define diffusivity in c in subdomains ‣ Outputs are 24 measured temperatures ‣ ROM constructed via RB-Greedy [Patera and Rozza, 2006] ‣ : Gaussian with variance 0.1 ‣ ‣ Posterior sampling: samples w/ implicit sampling [Tu et al., 2013] µ 2 [0.1, 10]9 q ⇡prior(µ) " ⇠ N(0, 1 ⇥ 10 3 ) 1 ⇥ 105
  • 56. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 2.5 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0 0.15 0.15 0 0.04 0.08 0.12 q24(x)q24(ˆx) -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0 2.5 0 1 2 q1(x)q1(ˆx) low variance costly ⇢1 ⇢24 rank( 1) = 22 rank( 24) = 7 high quality Machine learning error models 46 ˜i (µ) ⇠ N( ⇢i (µ), ↵1 + ↵2|⇢i (µ)|↵3 ) -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.52.5 0 2.5 0.2 0 0.2 0.4 q1(x)q1(ˆx) -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 q24(x)q24(ˆx) 0 0.15 0.15 0.1 0 0.1 0.2 high variance cheap rank( 1) = 1 rank( 24) = 1 ⇢1 ⇢24 low quality
  • 57. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Wall-time performance 47 ‣ ROM: + cheapest - inconsistent formulaGon simulaGon Gme HFM ROM ROM+ high-var ROM+ low-var
  • 58. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Wall-time performance 47 ‣ ROM: + cheapest - inconsistent formulaGon ‣ ROM + error models: + cheaper than HFM - more expensive than ROM + consistent formulaGon simulaGon Gme HFM ROM ROM+ high-var ROM+ low-var
  • 59. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Posteriors: ROM 48 ⇡HFM post(µ|qmeas) ⇡surr post(µ | qmeas) true prior ⇡surr post(µ | qmeas) ⇡HFM post (µ | qmeas) + HFM posterior: close to true parameters - ROM posterior: far from prior and true parameters
  • 60. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Posteriors: ROM + high-variance error model 49 ⇡HFM post(µ|qmeas) true prior ⇡HFM post (µ | qmeas) + ROM + high-variance error model posterior: close to prior ⇡ ]HFM post (µ | qmeas) ⇡ ]HFM post (µ | qmeas)
  • 61. /38 Kevin CarlbergAdvances in nonlinear model reduc4on ⇡HFM post(µ|qmeas) true prior ⇡HFM post (µ | qmeas) ⇡ ]HFM post (µ | qmeas) Posteriors: ROM + low-variance error model 50 + ROM + low-variance error model posterior: close to HFM posterior ⇡ ]HFM post (µ | qmeas)
  • 62. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Regression model: construct regression model to trade off: ‣ High capacity: low variance, more data to generalize ‣ Low capacity: high variance, less data to generalize ˜f<latexit 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Error-model construction 51 Feature engineering: select features to trade off: 1. Number of features ‣ Large number: costly, low variance, high-capacity regression ‣ Small number: cheap, high variance, low-capacity regression 2. Quality of features ‣ High quality: expensive, low variance ‣ Low quality: cheap, high variance ⇢ Method 2: Large number of features and high-dimensional regression [Trehan, C., Durlofsky, 2017; Freno, C., 2018] Method 1: Dual-weighted residual and Gaussian process regression [Drohmann, C., 2015; C., Uy, Lu, Morzfeld, 2018] ˜”(µ) = ˜f(fl(µ)) + ˜‘(fl(µ))<latexit 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  • 63. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 1. Error indicators: ‣ residual norm: ‣ dual-weighted residual: Feature engineering [Freno, C., 2018] 52 Proposed features: ‣ parameters ‣ low quality, cheap ‣ used by model discrepancy ‣ residual norm - small number, low quality, costly ‣ residual - large number, low quality, costly µ kr( ˆx; µ)k2 r( ˆx; µ) ‣ residual samples + moderate number, cheap - low quality ‣ residual PCA + moderate number, high-quality - costly ‣ gappy PCA + moderate number, high-quality + cheap ˆr := T r r( ˆx; µ) Pr( ˆx; µ) ˆrg := (P r)+ Pr( ˆx; µ) kr(˜x; µ)k2<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit> 2. Rigorous a posteriori error bound: |q(x) q(˜x)|  ↵ kr(˜x; µ)k2 <latexit 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</latexit><latexit 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 q(x) q(˜x) = yT r(˜x) + O(kx ˜xk2 ) 3. Model discrepancy: Idea: Use tradi(onal error quan(fica(on as inspira(on for features ˜”(µ) ≥ N(µ(µ); ‡2 (µ))
  • 64. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Application: Predictive capability assessment project 53 x y z 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 Deformation Magnitude [m] AB ‣ high-fidelity model dimension: ‣ reduced-order model dimensions: ‣ inputs : elasGc modulus, Poisson raGo, applied pressure ‣ quan((es of interest: y-displacement at A, radial displacement at B ‣ training data: 150 training examples, 150 tesGng examples µ 2.8 ⇥ 105 1, ... , 5<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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</latexit>
  • 65. /38 Kevin CarlbergAdvances in nonlinear model reduc4on krk2 [µ;krk2] [µ;Pr](nr=10) [µ;ˆrg](nr=10) [µ;Pr](nr=100) [µ;ˆrg](nr=100) [µ;Pr](nr=1000) [µ;ˆrg](nr=1000) [µ;ˆr] µ ANN k-NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 ˆuy : log10 FVU Application: Predictive capability assessment project 54 Introduction Parameterized Nonlinear Equations Approach Experiments Summary PCAP: FVU for QoI Error Prediction ˆ r: log10 FVU ˆ y: log10 FVU RegressionMethods krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 Features Features Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination) • SVR: RBF and MLP perform the best • [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301) Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44 log10(1 R2 ) log10(1 R2 ) y-displacement at A radial displacement at B regression methods features features
  • 66. /38 Kevin CarlbergAdvances in nonlinear model reduc4on krk2 [µ;krk2] [µ;Pr](nr=10) [µ;ˆrg](nr=10) [µ;Pr](nr=100) [µ;ˆrg](nr=100) [µ;Pr](nr=1000) [µ;ˆrg](nr=1000) [µ;ˆr] µ ANN k-NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 ˆuy : log10 FVU Application: Predictive capability assessment project 54 Introduction Parameterized Nonlinear Equations Approach Experiments Summary PCAP: FVU for QoI Error Prediction ˆ r: log10 FVU ˆ y: log10 FVU RegressionMethods krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 Features Features Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination) • SVR: RBF and MLP perform the best • [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301) Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44 log10(1 R2 ) log10(1 R2 ) - parameters (model-discrepancy approach): large variance y-displacement at A radial displacement at B regression methods features features
  • 67. /38 Kevin CarlbergAdvances in nonlinear model reduc4on krk2 [µ;krk2] [µ;Pr](nr=10) [µ;ˆrg](nr=10) [µ;Pr](nr=100) [µ;ˆrg](nr=100) [µ;Pr](nr=1000) [µ;ˆrg](nr=1000) [µ;ˆr] µ ANN k-NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 ˆuy : log10 FVU Application: Predictive capability assessment project 54 Introduction Parameterized Nonlinear Equations Approach Experiments Summary PCAP: FVU for QoI Error Prediction ˆ r: log10 FVU ˆ y: log10 FVU RegressionMethods krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 Features Features Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination) • SVR: RBF and MLP perform the best • [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301) Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44 log10(1 R2 ) log10(1 R2 ) - parameters (model-discrepancy approach): large variance - small number of low-quality features: large variance y-displacement at A radial displacement at B regression methods features features
  • 68. /38 Kevin CarlbergAdvances in nonlinear model reduc4on krk2 [µ;krk2] [µ;Pr](nr=10) [µ;ˆrg](nr=10) [µ;Pr](nr=100) [µ;ˆrg](nr=100) [µ;Pr](nr=1000) [µ;ˆrg](nr=1000) [µ;ˆr] µ ANN k-NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 ˆuy : log10 FVU Application: Predictive capability assessment project 54 Introduction Parameterized Nonlinear Equations Approach Experiments Summary PCAP: FVU for QoI Error Prediction ˆ r: log10 FVU ˆ y: log10 FVU RegressionMethods krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 Features Features Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination) • SVR: RBF and MLP perform the best • [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301) Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44 log10(1 R2 ) log10(1 R2 ) - parameters (model-discrepancy approach): large variance - small number of low-quality features: large variance ‣ PCA of the residual: lowest variance overall but costly y-displacement at A radial displacement at B regression methods features features
  • 69. /38 Kevin CarlbergAdvances in nonlinear model reduc4on krk2 [µ;krk2] [µ;Pr](nr=10) [µ;ˆrg](nr=10) [µ;Pr](nr=100) [µ;ˆrg](nr=100) [µ;Pr](nr=1000) [µ;ˆrg](nr=1000) [µ;ˆr] µ ANN k-NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 ˆuy : log10 FVU Application: Predictive capability assessment project 54 Introduction Parameterized Nonlinear Equations Approach Experiments Summary PCAP: FVU for QoI Error Prediction ˆ r: log10 FVU ˆ y: log10 FVU RegressionMethods krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 Features Features Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination) • SVR: RBF and MLP perform the best • [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301) Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44 log10(1 R2 ) log10(1 R2 ) - parameters (model-discrepancy approach): large variance - small number of low-quality features: large variance ‣ PCA of the residual: lowest variance overall but costly + gappy PCA of the residual: nearly as low variance, but much cheaper y-displacement at A radial displacement at B regression methods features features
  • 70. /38 Kevin CarlbergAdvances in nonlinear model reduc4on krk2 [µ;krk2] [µ;Pr](nr=10) [µ;ˆrg](nr=10) [µ;Pr](nr=100) [µ;ˆrg](nr=100) [µ;Pr](nr=1000) [µ;ˆrg](nr=1000) [µ;ˆr] µ ANN k-NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 ˆuy : log10 FVU Application: Predictive capability assessment project 54 Introduction Parameterized Nonlinear Equations Approach Experiments Summary PCAP: FVU for QoI Error Prediction ˆ r: log10 FVU ˆ y: log10 FVU RegressionMethods krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear krk2 [µ;krk2] [µ;Pr](q=10) [µ;ˆrg](q=10) [µ;Pr](q=100) [µ;ˆrg](q=100) [µ;Pr](q=1000) [µ;ˆrg](q=1000) [µ;ˆr] µ MLP -NN RF SVR: RBF SVR: Linear OLS: Quadratic OLS: Linear 5 4 3 2 1 0 Features Features Fraction of variance unexplained (FVU) is 1 r2 (r2 is coe cient of determination) • SVR: RBF and MLP perform the best • [µ; ˆrg] and [µ; Pr] well with only q = 100 samples (compared to Nu = 278, 301) Freno & Carlberg Machine-Learning Error Models for Approximate Solutions 44 log10(1 R2 ) log10(1 R2 ) - parameters (model-discrepancy approach): large variance - small number of low-quality features: large variance ‣ PCA of the residual: lowest variance overall but costly + gappy PCA of the residual: nearly as low variance, but much cheaper + neural networks and SVR: RBF yield lowest-variance models y-displacement at A radial displacement at B regression methods features features
  • 71. /38 Kevin CarlbergAdvances in nonlinear model reduc4on 10 8 6 4 2 0 2 4 Predicted error, ˆuy [⇥10 3 ] 10 8 6 4 2 0 2 4 Exacterror,uy[⇥103 ] Exact krk2 SVR: RBF r2 =0.94712, MSE=2.424⇥10 7 µ ANN r2 =0.96851, MSE=1.444⇥10 7 [µ; ˆrg] (nr=10) ANN r2 =0.99944, MSE=2.554⇥10 9 Application: Predictive capability assessment project 55 kr( ˆx; µ)k2 ‣ TradiGonal features and : - high noise variance - expensive for : compute enGre residual kr( ˆx; µ)k2<latexit 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</latexit><latexit 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</latexit><latexit 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</latexit><latexit 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BeNmylHi2PTw1NFNI9SkybtlXCgK+zIe5pMpwetaKs8r0YPkQHbRpIqmd1YpFthed0Cqq+0eZvuVUEIT/ZodwWr81rZD1pJaH2Y16NDgWSvO3zW5xg1SJw2pZcC7lKVcMtwHEbao9dXNNEwwjmOD9dClNmu8pY9g5ruaSyvmEFaUlTjeVJ/JOfbazzqRph7GReFfbKc7GrV13mk3cpzduppbc8/RK24yhmdWQhWzQxI3+kXBOg0olewJIxuz5ldOonWydVZRPBK5o+OTAu4Ze0L2tuE0oYvrJ1LuaWr7R/ri4v2MEdkkCLBWYnY3UHHtPlBjgiAOHYOjcUMcBDJOhsFYH0uaEU8wMCHNtgTeLCLRcJOMmKr60Y0Uh0CUxad1QdoysBeBwmHAQmHXgmzGH/Q7aYPOpl8wEMT3PF2yKxSpPEcZeTmT6TfJLlPdzmnydgf8ho3T1rjKmt2sXRmtBhdl3lK74E0ry/4LRHnys6BPN1bmTLlcbcMulNzshxN6GlOi4U1CabQGODK6TYswZXkTNt1M3SjLNyzyQwBliJAJF+ktZAzAFmOAGwGY9n6AoyvWvflEAUiJ19vcUeKykt6mrec58bRARDPzx/p2P4dsRC0zw8mY6xdgcjpcNzl0cv44kiWvSjBMaw9SAI8geDJrGZQqaIuKE0hiqFESOmgClAIKF4tAdDvWOGuLdoe/A9g/VF6UBI8UzCWDGRIaQpRDCVhD/LbeaR7offAoFRrSS2CsXGazOjrLfiC5RPbRAng+0HYRrrK3HO7YAl11jaVljBF6RBWa7pM5mWZZ2h8iUoBxyO7vD3k1hbcf6sjQaRw7CP8Dkgtkjt9vIN13yQDKoXWlBzcW379mcYa7F4YIKuTKHINnOZ1uNsFylR6pHRWGPAbWRVe2eIYhjcGQYCbQSKuMxKJds2WieKugcSC281jHn63jR14i+X0xr7AiRO0TVIOg/xE1+apS5wb5my+yGanYJLa52HfqSceAs5gyf6jL0KVn2arUpvZK/TJ9s43djrq0w9LWFq18xIP6U6Ls7iNwWq7m87IQL+rHRxO8NxBEmXJZD6wyhMKOAXDsN9kwEv/CxSTMMR4WQt5Bi3SfSj1tV/y5YNtR8LeXGWUeOJU71IMCNZ7ODx7RRJuVnDnAIj9ngVnIwe66q/DjKswIaz/NFHDiT6mUU38R3gsTQCrA9eobkqQv9LpkgFydxFu7bUKy1OHodM6PvYWfDaDNndtvAo5yL3N5FbFIBeqkMETKSbKrnLchmAFokSgPngPDJkIoFilxFWlqrIMFK2Z7WD5QgdxQnJUSduzbP6+o0N4rzGJMzacieH95YXkBAjKKOMxG/XhswxpVcyYKS+9kvlMS7NFskIUVdTmc6ioTQSUDyExfL6Jxq9nfOeFuZAtIIrdGP9tDZuq6pudoiZCTjUB4wYWfcK9stsoQs3BbQgdmoAZW/DdqZU2Ad239jxkH9ALWxI9thb5hIZ4wnvTHH+tyftzs5YkzlB2AfcxOJEdlQhT9YM7gAhxOiMkxDy0YYqoirhOyK4FnolVdJFr4kyW5akk+8LBVJxIUiG76EvcDmlqnT7ynTySp45YGfLvUPdORfC3mYQFMpQhkz3yie1iJdtL8Od/KJrt8zM/Xc+qqZH1yF1kUBPYBUlsRiRFaORe0fpLxdkNljYE1F/FZUJee3yyXLzBDYVc5X4t7kHGHcByRdmV4lljIUYByuACtxU5bM/KENlqiVNVT52rXhIgi00+gSqTTJQ9jDgFwCIfpa8D2yPDlRUWJl6rJL4PWS8TxLu5JU+DamTTzNz9wFFcmlNosrMCtkba6ZtYtmkIolmPp/a+54n7el2x76cNeSVZdTEoIIO4VrTUQTEHDwoJrBZxkfrH0cALbrgt9tu75DvC4K5dngKCT8OgCQskgNMtfrwaOO9yXgfDeLAFHIfxK2NnIVxFvYOQFg+r72e5mQrlxHfCq9uhK3Gwqm0UAboMl6nCdV5hRn6eoEJxOZ6zHQD2Zrs6Bq8kfySvtWV13XknZDJBbIGSpR95wlf0ivwGQJEmqNTfdMJ31TqQBD6NF6NJFwKrmUACv/3qVFFYhhOqW+mBkN1CnuTpFX2PCjmcOPGaPHGWsAPiVCgZAMEDHUDgDczvM6eleYRQtUHPrZjt6NDVwsBKi3tIdQ45HPxKnLsRiCp2SUy2pQvvilYmxLiM8AFoGja7pMAZaBJmuATrGLQ9GOr2yTmPzudbspaxXq9KGWIs8oqAzWB8NxseM9zoS73QlcbSVk8pDrsb8yayvqmSPFMmme7WH4fvrToSZM6+HAGhiZfE2dMvR0rodmmeLnnb+cS222k8Yb+OeywPFfC1M+jdkxqI/DwEldloQh2pchffWhQWRxY8YE24nmFd2Ap9wFJ0R+WcKU0QbFf7J+fdlbQWi1vz7DLp0QVbDyQ3dMhf3mqcFXt1274xb6MeuHFikl2FCeRWvk8+e++uuUp5Yb7D1st11t7DhmpMd9917NnjddQ5VCWiXS2DH7UVTL7GI+xtN8HWUvHQPsLz0Qj+twnylK3NlTJuNc9xrLCj7cJzPXTl84zB8xan1L6a7S0dclXBt40/wyPwTJx20q+1PXNUalhXFeqgIVRnqb5VVH8OUZccMiEHeVxW6DRnf8uAOb+/o6igeb/3HW7QBozG3hZmaXTlw/uaukrdqBRvgGck8FYLTWPrP5bYwNKPznIUCWGr1GmLVrqVw8ZYtBILGW4voi3uvwI6GVm5yFlAivdh1PIbR0KWK7jopTXJKztpq1Gzw/yCbWcOR76oxBMEMaphQg6M+BymBn6ZD1sAlB+O0V8nzeZ7YZitQybAvxERels2YLjfXOK4FJGfE7sDSc/cIZA5N1d2dszAlkqyj5oY/BDVfIOa/WsCHGFIIm8yg9wq4x31gyZAABaOopw1RDMeDJWBox7S7lHuupzHcEHPW7b7hLjCwHxIaXwZlKw8OpcJdQwrOdJeRTmGwa5fngdKk9x+ach/zMd0xUEeRTXT2GoA/a+bMMYztglLvf/cvgJw+PrxKRZL/usmcCb5CDMpQOSOw/iPhJntnmwliVe3sgOsRvrSmEE6x/FI1zK75UJoi7MqYplXVVyimNEf20E4njmSQzYstbALi5/5niN6URvq9Jj+ApWLuKIHmMGf9hCCZnGlzo8duyDyvtyKXKxk9ozt4SEez1mKXQNqtKCbFTIrdpmxuvUSZbN67m6Fa+8lPi1ZN6w94kMWPYXurAOwp00QVKAb8ss0/YuGfnCWAWpcfMbvp7yzXZPFRTXPa7F++M5Z+KcS3mbTPJ2o/bF3uE3TOgei2Wk7odIuDYmxhR4HhWrvlPW9ANME2uKfB8Xz7UYh+oUXZIt9ERQrViak4JMAzBZ9okS7N/FJLWJViuLZW2XY5flAhblcxDd0QfSSo8h3cihC/DYXEQC8VoFc9anwdBFnQ/xuFn/pnomjV4UAoBu9YsxVnFow8eY/WPkVGlNwoly09B3pFeswxnj/J++KihhcbG6AIF+3Z8khHSX9AUSkb2s15Qx3HjtfP+ztPPy6t7O3/4f91sTpa68cvff1fm/n6z9881VvZ3fPgtMdEY7D6eR/FoK/YZ5j9r/Canf/uP9Nb2d//48YWiIz9FkkKe7up/Eypb91NRWfLz7b2LN/Htn9cPZgZ293Z++H/Y3v/53/dPJv1v517d/Wttb21r5Z+37txVp/7e3aeO0/1/5r7b/X/uf2f2//8vFXH3/NoL/8Bef8y5rx7+Nv/x8tC4lf</latexit> 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kr( ˆx; µ)k2 [µ;ˆrg]‣ Proposed features : + low noise variance + extremely cheap: only compute 10 elements of the residual
  • 72. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Summary 56 Accurate, low-cost, structure-preserving, reliable, cer;fied nonlinear model reduc;on ‣ accuracy: LSPG projecGon [C., Bou-Mosleh, Farhat, 2011; C., Barone, AnGl, 2017] ‣ low cost: sample mesh [C., Farhat, CorGal, Amsallem, 2013] ‣ low cost: reduce temporal complexity [C., Ray, van Bloemen Waanders, 2015; C., Brencher, Haasdonk, Barth, 2017; Choi and C., 2017] ‣ structure preserva(on [C., Tuminaro, Boggs, 2015; Peng and C., 2017; C. and Choi, 2017] ‣ reliability: adapGvity [C., 2015] ‣ cer(fica(on: machine learning error models [Drohmann and C., 2015; Trehan, C., Durlofsky, 2017; Freno and C., 2017]
  • 73. /38 Kevin CarlbergAdvances in nonlinear model reduc4on Questions? 57 Machine-learning error models: ‣ Freno, C. “Machine-learning error models for approximate solu4ons to parameterized systems of nonlinear equa4ons,” arXiv e-Print, 1808.02097 (2018). ‣ Trehan, C, and Durlofsky. “Error modeling for surrogates of dynamical systems using machine learning,” InternaGonal Journal for Numerical Methods in Engineering, Vol. 112, No. 12, p. 1801–1827 (2017). ‣ Drohmann and C. “The ROMES method for staGsGcal modeling of reduced-order- model error,” SIAM/ASA Journal on Uncertainty QuanGficaGon, Vol. 3, No. 1, p.116– 145 (2015). LSPG reduced-order model: ‣ C, Barone, and An4l. “Galerkin v. least-squares Petrov–Galerkin projec4on in nonlinear model reduc4on,” Journal of Computa4onal Physics, Vol. 330, p. 693– 734 (2017). ‣ C, Farhat, CorGal, and Amsallem. “The GNAT method for nonlinear model reducGon: EffecGve implementaGon and applicaGon to computaGonal fluid dynamics and turbulent flows,” Journal of ComputaGonal Physics, Vol. 242, p. 623– 647 (2013). ‣ C, Bou-Mosleh, and Farhat. “Efficient non-linear model reducGon via a least- squares Petrov–Galerkin projecGon and compressive tensor approximaGons,” InternaGonal Journal for Numerical Methods in Engineering, Vol. 86, No. 2, p. 155– 181 (2011).