![Future Extension
Chebyshev Polynomials for System Identification
𝑇0 𝑥 = 1, 𝑇1 𝑥 = 𝑥
𝑇𝑛+1 𝑥 = 2 ∗ 𝑥 ∗ 𝑇𝑛 𝑥 − 𝑇𝑛−1 𝑥
𝑥𝜖[−1,1]
𝑦 𝑥 = 𝛼0𝑇0 𝑥 + 𝛼1𝑇1 𝑥 + 𝛼2𝑇2 𝑥 + ⋯ + 𝛼𝑛𝑇𝑛 𝑥
Why Chebyshev Polynomials?
• Orthogonality
• Numerical Stability
• Efficient Approximation
Application to System Identification:
• Use high-order Chebyshev polynomials to create candidate basis functions.
• Each function becomes a weighted sum of these polynomials:](https://image.slidesharecdn.com/pptsindyautoencoder-250609190349-023e80eb/85/SindyAutoEncoder-Interpretable-Latent-Dynamics-via-Sparse-Identification-48-320.jpg)
SindyAutoEncoder: Interpretable Latent Dynamics via Sparse Identification
- 1. Data-Driven Discovery of Dynamical Systems and Governing Equations SINDy Autoencoders: A Hybrid Approach Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019). Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences, 116(45), 22445-22451.
- 3. Personal Motivation – 3 levels of ML usage Decide Interpret Reveal
- 4. Personal Motivation – 3 levels of Murder Mystery Resolution Who? How? Why?
- 5. Agenda • Ambition • Key Assumptions • Example: Lotka-Volterra (Predator-Prey) Equations • Generalized SINDy • Transforming Co-ordinates • SINDy Autoencoder • Applications & Challenges • Future Extension
- 6. Ambition Given the current state of the system, we want to learn the governing equations that can estimate the the system's rate of change.
- 7. Ambition of Sparse Identification of Nonlinear Dynamical Systems (SINDy) Given the current state of the system, we want to learn the sparse governing equations that accurately estimate the system's rate of change. 𝑑𝑥(𝑡) 𝑑𝑡 = 𝑓(𝑥 𝑡 )
- 8. Ambition of SINDy Given the current state of the system, we want to learn the sparse governing equations that accurately estimate the system's rate of change. 𝑑𝑥(𝑡) 𝑑𝑡 = 𝑓(𝑥 𝑡 ) Turn high-dimensional scientific data into parsimonious dynamical models
- 9. Key Assumptions • Causality: The Current State Encapsulates Future Behavior oKnowing the current state 𝑥(𝑡) allows us to predict how the system will evolve in the next instant. • Markov Property: Future Depends Only on Present
- 10. Key Assumptions SINDy models the dynamics of a system using only the current state 𝑥(𝑡) 𝑑𝑥(𝑡) 𝑑𝑡 = 𝑓(𝑥 𝑡 ) Simplicity Limitation Mathematical Tractability Non-Markovian Systems Applicability to a Wide Range of Systems Incomplete State Representation Computational Efficiency: Extensions Required for Complex Systems
- 11. Example: Lotka-Volterra (Predator-Prey) Equations 𝑑𝑥 𝑑𝑡 = 𝛼𝑥 − 𝛽𝑥𝑦 𝑑𝑦 𝑑𝑡 = 𝛿𝑥𝑦 − 𝛾𝑦 Where, 𝑥: Prey population 𝑦: Predator population 𝛼, 𝛽, 𝛿, 𝛾: Positive constants representing interaction rates
- 12. Generating data • Initial Conditions
- 13. Generating data • Simulated data X(t)
- 14. Generating data • Simulated data X(t) ሶ 𝑋
- 15. Learning the governing Equations X(t) ሶ 𝑋 Learn a governing set of equations so that 𝑓 𝑋 𝑡 = ሶ 𝑋
- 16. Learning the governing equations X(t) ሶ 𝑋 Library Functions Θ(𝑋)
- 17. Learning the governing equations X(t) ሶ 𝑋 Library Functions Θ(𝑋) 𝑑𝑥 𝑑𝑡 = 𝛼𝑥 − 𝛽𝑥𝑦 𝑑𝑦 𝑑𝑡 = 𝛿𝑥𝑦 − 𝛾𝑦 0 0 0 −𝛾 𝛼 0 0 0 𝛽 𝛿 Ξ
- 18. Generalized SINDy ሶ 𝑋 = Θ(𝑋)Ξ Brunton, S. (2017, March). Discovering governing equations from data by sparse identification of nonlinear dynamics. In APS March Meeting Abstracts (Vol. 2017, pp. X49-004).
- 19. Generalized SINDy ሶ 𝑋 = Θ(𝑋)Ξ 𝑋 = 𝑥𝑇(𝑡1) 𝑥𝑇(𝑡2) ⋮ 𝑥𝑇(𝑡𝑚) = 𝑥1(𝑡1) 𝑥2(𝑡1) … 𝑥𝑛(𝑡1) 𝑥1(𝑡2) 𝑥2(𝑡2) … 𝑥𝑛(𝑡2) ⋮ ⋮ ⋱ ⋮ 𝑥1(𝑡𝑚) 𝑥2(𝑡𝑚) … 𝑥𝑛(𝑡𝑚) Where, Θ 𝑋 = | | | | | | | | 1 𝑋 𝑋𝑃2 𝑋𝑃3 … sin(𝑋) cos(𝑋) … | | | | | | | | Ξ = 𝜉1 𝜉2 … 𝜉𝑛
- 20. Generalized SINDy ሶ 𝑋 = Θ(𝑋)Ξ 𝑋 = 𝑥𝑇(𝑡1) 𝑥𝑇(𝑡2) ⋮ 𝑥𝑇(𝑡𝑚) = 𝑥1(𝑡1) 𝑥2(𝑡1) … 𝑥𝑛(𝑡1) 𝑥1(𝑡2) 𝑥2(𝑡2) … 𝑥𝑛(𝑡2) ⋮ ⋮ ⋱ ⋮ 𝑥1(𝑡𝑚) 𝑥2(𝑡𝑚) … 𝑥𝑛(𝑡𝑚) Where, Θ 𝑋 = | | | | | | | | 1 𝑋 𝑋𝑃2 𝑋𝑃3 … sin(𝑋) cos(𝑋) … | | | | | | | | Ξ = 𝜉1 𝜉2 … 𝜉𝑛 Once Ξ has been determined ሶ 𝑥𝑘 = 𝑓𝑘 𝑥 = Θ(𝑥𝑇)𝜉𝑘
- 21. Co ordinates … Choosing the right coordinates to simplify dynamics has always been important … Brunton, S. (2017, March). Discovering governing equations from data by sparse identification of nonlinear dynamics. In APS March Meeting Abstracts (Vol. 2017, pp. X49-004).
- 22. Simplifying co-ordinate systems Example Complex Coordinate System Simplified Coordinate System Celestial Mechanics Geocentric (Earth-centered) Heliocentric (Sun-centered) Fourier Transform (Heat Equation) Time Domain Frequency Domain (Fourier space) Principal Component Analysis (PCA) Original high-dimensional space Principal Component Space Polar Coordinates Cartesian Coordinates (x, y) Polar Coordinates (r, θ) x y
- 23. Transforming Co-ordinates SVD (Shallow Linear) Deep AutoEncoder (Deep Non- Linear)
- 24. Losses
- 25. Loss Term I Training the Sindy model Ensure that we get a good representation of ሶ 𝒛 • Capturing Core Dynamics • Efficient Prediction • Simplified Understanding
- 26. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ
- 27. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ 𝑧 = 𝜑 𝑥 ሶ 𝑧 = ?
- 28. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ 𝑧 = 𝜑 𝑥 ሶ 𝑧 = ? Utilize the Jacobian, where Δ𝑓 = 𝐽Δ𝑥
- 29. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ ሶ 𝑧 = ? Δ𝑓 = 𝐽Δ𝑥 𝑧 = 𝜑 𝑥
- 30. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ ሶ 𝑧 = ? ሶ 𝑧= 𝐽𝜑 ሶ 𝑥 Δ𝑓 = 𝐽Δ𝑥 𝑧 = 𝜑 𝑥
- 31. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ ሶ 𝑧 = ? ሶ 𝑧= 𝐽𝜑 ሶ 𝑥 ሶ 𝑧= ∇𝑥(𝑧) ሶ 𝑥 Δ𝑓 = 𝐽Δ𝑥 𝑧 = 𝜑 𝑥
- 32. Loss Term I • Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝑧𝑇 Ξ = 𝜃 𝜑 𝑥 𝑇 Ξ ሶ 𝑧 = ? ሶ 𝑧= 𝐽𝜑 ሶ 𝑥 ሶ 𝑧= ∇𝑥(𝑧) ሶ 𝑥 ሶ 𝑧= ∇𝑥(𝜑 𝑥 ) ሶ 𝑥 Δ𝑓 = 𝐽Δ𝑥 𝑧 = 𝜑 𝑥
- 33. Loss Term I Minimize ( ሶ 𝑧 − 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧) 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 ሶ 𝑧 = 𝜃 𝜑 𝑥 𝑇 Ξ ሶ 𝑧 = ? ሶ 𝑧= 𝐽𝜑 ሶ 𝑥 ሶ 𝑧= ∇𝑥(𝑧) ሶ 𝑥 ሶ 𝑧= ∇𝑥(𝜑 𝑥 ) ሶ 𝑥 Δ𝑓 = 𝐽Δ𝑥 𝑧 = 𝜑 𝑥 ሶ 𝑧= ∇𝑥(𝜑 𝑥 ) ሶ 𝑥 ℒ𝑑𝑧/𝑑𝑡 = ∇𝑥 𝜑 𝑥 ሶ 𝑥 − 𝜃 𝜑 𝑥 𝑇 Ξ 2 2
- 34. Loss Term II • Predicting Time Evolution • Capturing Data and Dynamics • Improving Model Accuracy with Time Derivatives Minimize ( ሶ 𝑥 − ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠)
- 35. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧)
- 36. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧) ሶ 𝑥 = 𝐽𝜓 ( ሶ 𝑧)
- 37. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧) ሶ 𝑥 = 𝐽𝜓 ( ሶ 𝑧) = ∇𝑧 ො 𝑥 ( ሶ 𝑧)
- 38. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧) ሶ 𝑥 = 𝐽𝜓 ( ሶ 𝑧) = ∇𝑧 ො 𝑥 ( ሶ 𝑧) = ∇𝑧 𝜓(𝑧) ( ሶ 𝑧)
- 39. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧) ሶ 𝑥 = 𝐽𝜓 ( ሶ 𝑧) = ∇𝑧 ො 𝑥 ( ሶ 𝑧) = ∇𝑧 𝜓(𝑧) ( ሶ 𝑧) = ∇𝑧 𝜓(𝜑 𝑥 ) ( ሶ 𝑧)
- 40. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧) ሶ 𝑥 = 𝐽𝜓 ( ሶ 𝑧) = ∇𝑧 ො 𝑥 ( ሶ 𝑧) = ∇𝑧 𝜓(𝑧) ( ሶ 𝑧) = ∇𝑧 𝜓(𝜑 𝑥 ) ( ሶ 𝑧) = ∇𝑧 𝜓(𝜑 𝑥 ) (𝜃 𝑧𝑇 Ξ)
- 41. Loss Term II Calculating ሶ 𝑥 𝑓𝑟𝑜𝑚 𝑆𝐼𝑁𝐷𝑦 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 ො 𝑥 = 𝜓(𝑧) ሶ 𝑥 = 𝐽𝜓 ( ሶ 𝑧) = ∇𝑧 ො 𝑥 ( ሶ 𝑧) = ∇𝑧 𝜓(𝑧) ( ሶ 𝑧) = ∇𝑧 𝜓(𝜑 𝑥 ) ( ሶ 𝑧) = ∇𝑧 𝜓(𝜑 𝑥 ) (𝜃 𝑧𝑇 Ξ) ℒ𝑑𝑥/𝑑𝑡 = ሶ 𝑥 − ∇𝑧 𝜓(𝜑 𝑥 ) (𝜃 𝑧𝑇 Ξ) 2 2
- 42. Loss Term III • Recreating the input ℒ𝑟𝑒𝑐𝑜𝑛 = 𝑥 − 𝜓(𝜑 𝑥 2 2
- 43. Total Loss Overall, Loss = ℒ𝑟𝑒𝑐𝑜𝑛 + 𝜆1ℒ𝑑𝑥/𝑑𝑡 + 𝜆2ℒ𝑑𝑧/𝑑𝑡 + 𝜆3ℒ𝑟𝑒𝑔 ℒ𝑟𝑒𝑐𝑜𝑛 = 𝑥 − 𝜓(𝜑 𝑥 2 2 ℒ𝑑𝑥/𝑑𝑡 = ሶ 𝑥 − ∇𝑧 𝜓(𝜑 𝑥 ) (𝜃 𝑧𝑇 Ξ) 2 2 ℒ𝑑𝑧/𝑑𝑡 = ∇𝑥 𝜑 𝑥 ሶ 𝑥 − 𝜃 𝜑 𝑥 𝑇 Ξ 2 2 ℒ𝑟𝑒𝑔 = Ξ 1 1
- 44. Applications: Nonlinear Pendulum Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019). Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences, 116(45), 22445-22451.
- 45. Applications: Lorenz System Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019). Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences, 116(45), 22445-22451.
- 46. Applications: Reaction Diffusion System Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019). Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences, 116(45), 22445-22451.
- 47. Challenges • Clean, noise free measurement data • Limited interpretability of deep learning models • Coordinate transformation limitations • Need for integration with domain knowledge
- 48. Future Extension Chebyshev Polynomials for System Identification 𝑇0 𝑥 = 1, 𝑇1 𝑥 = 𝑥 𝑇𝑛+1 𝑥 = 2 ∗ 𝑥 ∗ 𝑇𝑛 𝑥 − 𝑇𝑛−1 𝑥 𝑥𝜖[−1,1] 𝑦 𝑥 = 𝛼0𝑇0 𝑥 + 𝛼1𝑇1 𝑥 + 𝛼2𝑇2 𝑥 + ⋯ + 𝛼𝑛𝑇𝑛 𝑥 Why Chebyshev Polynomials? • Orthogonality • Numerical Stability • Efficient Approximation Application to System Identification: • Use high-order Chebyshev polynomials to create candidate basis functions. • Each function becomes a weighted sum of these polynomials:
- 49. Extending Chebyshev Polynomials to Multiple Features • Generalizing to Multiple Features: • For features 𝑥1, 𝑥2,… 𝑥𝑚 ∶ 𝑦 𝑥1, 𝑥2 = 𝑖=0 𝑛 𝑗 𝑛 𝛼𝑖,𝑗𝑇𝑖 𝑥1 𝑇𝑗 𝑥2 • Including Interactions Between Features • Tensor Products • For Two Features 𝑥1 and 𝑥2 𝑦 𝑥1, 𝑥2, … 𝑥𝑚 = 𝑖,𝑗,…,𝑘 𝛼𝑖,𝑗,…,𝑘𝑇𝑖 𝑥1 𝑇𝑗 𝑥2 … 𝑇𝑘 𝑥𝑚
- 50. Optimizing and Sparsifying with Cross- Feature Terms • Learn the coefficients 𝛼𝑖,𝑗,…,𝑘that best fit the system while promoting sparsity • Optimization problem: 𝑚𝑖𝑛𝛼 𝑦 − 𝑖,𝑗,…,𝑘 𝛼𝑖,𝑗,…,𝑘𝑇𝑖 𝑥1 𝑇𝑗 𝑥2 … 𝑇𝑘 𝑥𝑚 2 + 𝜆 𝛼 1 Sparsity and Interpretability • Pruning Insignificant Terms: Coefficients corresponding to unimportant polynomials shrink to zero. • Simplified Model: Results in a model that is both accurate and interpretable, using only the most significant terms.
- 51. Synergy with Neural Networks • Neural Network Enhancement • Adaptive Coefficients • Dynamic Modeling • Model Architecture • Input Layer: Receives features 𝑥1, 𝑥2, … , 𝑥𝑚 • Hidden Layers • Output Layer: 𝛼𝑖,𝑗,…,𝑘 • Combined Model: 𝑦 𝑥1, 𝑥2, … 𝑥𝑚 = 𝑖,𝑗,…,𝑘 𝛼𝑖,𝑗,…,𝑘 𝑥1, 𝑥2, … , 𝑥𝑚 𝑇𝑖 𝑥1 𝑇𝑗 𝑥2 … 𝑇𝑘 𝑥𝑚
- 52. Remarks Advantages • Enhanced Expressiveness • Improved Accuracy • Interpretability Practical considerations • Computational Complexity • Regularization Techniques • Training the Neural Network
- 53. Example application: Neural Networks for classification Normally Output of a neuron is σ𝑖=0 𝑚 𝑤𝑖𝑥𝑖, m is the number of features Where 𝑤𝑖 is the learnable parameter For ChebyShev, each 𝑤𝑖 is as follows 𝑤𝑖 = 𝑑=0 𝑘 𝑐𝑖,𝑑𝑇𝑑 𝑥𝑖 Where 𝑐𝑑 is the learnable parameter, k is the order of Chebyshev polynomial Thus for Chebyshev, output of a neuron is 𝑖=0 𝑚 𝑤𝑖𝑥𝑖 Which can also be written as 𝑖=0 𝑚 𝑑=0 𝑘 𝑐𝑖,𝑑𝑇𝑑 𝑥𝑖 𝑥𝑖
- 54. For k=3(0,1,2), Cheby Shev Output 𝑖=0 𝑚 𝑑=0 𝑘 𝑐𝑖,𝑑𝑇𝑑 𝑥𝑖 𝑥𝑖 = 𝑖=0 𝑚 𝑐𝑖,0𝑇0 𝑥𝑖 + 𝑐𝑖,1𝑇1 𝑥𝑖 + 𝑐𝑖,2𝑇2 𝑥𝑖 𝑥𝑖 Now, if 𝑐𝑖,1and 𝑐𝑖,2are forced to be 0, and 𝑇0 is 1, output becomes σ𝑖=0 𝑚 𝑐𝑖,0 𝑥𝑖 which is same as σ𝑖=0 𝑚 𝑤𝑖𝑥𝑖 where each 𝑐𝑖,0 term in the expression represents 𝑤𝑖 Example application: Neural Networks for classification
- 55. Results File Size Columns Number Of Classes MLP F1 Score F1 Score by ChebyShevNetwork Delta percentage krkopt.csv 28056 6 18 0.509 0.632 24.165 contraceptive.csv 1473 9 3 0.529 0.582 10.019 led7.csv 3200 7 10 0.703 0.764 8.677 cmc.csv 1473 9 3 0.536 0.58 8.209 car.csv 1728 6 4 0.898 0.967 7.684 splice.csv 3188 60 3 0.851 0.899 5.64 wine_quality_red.csv 1599 11 6 0.559 0.59 5.546 wine_quality_white.csv 4898 11 7 0.517 0.542 4.836 letter.csv 20000 16 26 0.896 0.924 3.125 satimage.csv 6435 36 6 0.86 0.886 3.023 solar_flare_2.csv 1066 12 6 0.72 0.74 2.778 page_blocks.csv 5473 10 5 0.959 0.978 1.981 nursery.csv 12958 8 4 0.984 0.997 1.321 sleep.csv 105908 13 5 0.744 0.753 1.21 allhypo.csv 3770 29 3 0.938 0.948 1.066 allhyper.csv 3771 29 4 0.981 0.988 0.714 allrep.csv 3772 29 4 0.971 0.975 0.412 ann_thyroid.csv 7200 21 3 0.982 0.986 0.407 texture.csv 5500 40 11 0.986 0.99 0.406 segmentation.csv 2310 19 7 0.972 0.974 0.206 dna.csv 3186 180 3 0.938 0.939 0.107 pendigits.csv 10992 16 10 0.994 0.994 0 yeast.csv 1479 8 9 0.588 0.587 -0.17 allbp.csv 3772 29 3 0.977 0.973 -0.409 optdigits.csv 5620 64 10 0.976 0.972 -0.41 car_evaluation.csv 1728 21 4 0.991 0.986 -0.505 led24.csv 3200 24 10 0.742 0.735 -0.943
- 56. Data Extrapolation Initial Hypothesis Generation Physical Model Simulation Prediction and Uncertainty Estimation Data Integration and Model Update Active Learning: Data Acquisition
- 57. Thank You!