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J. Nathan Kutz Department of Applied Mathematics University of - - PowerPoint PPT Presentation

Dec 13, 2023 634 likes •1.07k views

DATA-DRIVEN MODEL DISCOVERY AND COORDINATE EMBEDDINGS FOR PHYSICAL SYSTEMS ICERM 2019 J. Nathan Kutz Department of Applied Mathematics University of Washington Email: kutz@uw.edu Web: faculty.washington.edu/kutz Model Discovery Steven

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DATA-DRIVEN MODEL DISCOVERY AND COORDINATE EMBEDDINGS FOR PHYSICAL SYSTEMS ICERM 2019

  • J. Nathan Kutz

Department of Applied Mathematics University of Washington Email: kutz@uw.edu Web: faculty.washington.edu/kutz

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Model Discovery

Steven Brunton

Mechanical Engineering University of Washington

Joshua Proctor

Institute for Disease Modeling

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Mathematical Framework

x

Dynamics Measurement

State-space Parameters Stochastic effects Measurement model Measurement noise Dynamics

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What Could the Right Side Be?

Limited by your imagination 2nd degree polynomials

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Sparse Identification of Nonlinear Dynamics (SINDy)

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Identifying ROMs

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Discovery Innovations

Schaeffer et al -- corrupt data, PDEs, integral formulation, convergence Dongbin Xiu & co-workers (2018) – Sampling strategies Guang Lin & co-workers (2018) -- Uncertainty Metrics Hod Lipson and co-workers (2006) — Symbolic/genetic regression Karniadakis, Raissai, Perdikaris …. — Neural Nets

Zheng, Askham, Brunton, Kutz & Aravkin (2018) – SR3 sparse relaxed regularized regression (for SINDy, LASSO, CS, TV, Matrix Completion …)

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Manifolds and Embeddings Observables & Coordinates

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Bernard Koopman 1931

Mezic (2004) Coifman, Kevrekidis, co-workers

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Koopman Invariant Subspaces

Brunton, Proctor & Kutz, PLOS ONE (2018)

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Burgers’ Equation

Cole-Hopf

Kutz, Proctor & Brunton, Complexity (2018)

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Dynamic Mode Decomposition

Travis Askham

Askham & Kutz, SIADS (2018)

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Approximate Dynamical Systems

Linear dynamics (equation-free) Eigenfunction expansion Least-square fit

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DMD with Control

Input Input Snapshots DMD generalization

Proctor, Brunton & Kutz, SIADS (2016)

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Koopman vs DMD: All about Observables!

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Neural Nets

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The Zoo NN Zoo

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NNs for Koopman Embedding

Bethany Lusch

Lusch et al. Nat. Comm (2018)

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NNs for Koopman Embedding

Bethany Lusch

Lusch et al. Nat. Comm (2018)

Handling the Continuous Spectra

Lusch, Kutz & Brunton, arxiv (2017)

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The Pendulum

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Flow Around a Cylinder

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Autoencoder + SINDy

Kathleen Champion

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Fast Learning

Charles Delahunt

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Moth Olfactory System

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Learning New Odors

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Sparsity for Learning

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Rapid Learning in NNs

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Comparisons

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Decoder Networks

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Structure of Mapping

Linear Maps: SVD (left singular vector) defines layer Erichson, Mathelin, Brunton, Mahoney & Kutz (2019)

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Measurements State space Mapping Approximate the full state space from limited measurements Optimization

Mathematical Framework

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Linear Maps

Singular value decomposition Data Linear measurements H Optimize (least-squares)

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Optimal Placement

Point measurements Optimal Sensors via QR pivots Manohar, Kutz & Brunton (2018) IEEE Control Systems Magazine Clark, Askham, Brunton & Kutz (2019) IEEE Sensors

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Nonlinear Mapping

General Form: Compositional Layers Universal Approximators: Hornik 1990

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Shallow Layer Mapping

Two Layers Composition Erichson, Mathelin, Brunton, Mahoney & Kutz (2019) SIAM

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Activation Functions

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Linear vs Nonlinear Maps

Improved Interpretability of Modes

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Improved Performance

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Robustness to Noise

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Improved Performance

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Super Resolution Analysis

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Conclusion

Model Discovery: Sparse regression provides parsimonious dynamical models Coordinates: Learning Koopman embeddings can provide optimal basis for dynamics Neural Networks: Structure and function matter

  • Connect discovery and coordinates
  • Structure can lead to fast (one-shot) learning with limited data
  • Discovery of improved coordinate embeddings through decoding

COMING SOON: A multi scale physics challenge set