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

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 Brunton Mechanical Engineering University of Washington Joshua Proctor Institute for Disease Modeling

Mathematical Framework State-space Dynamics Parameters x Stochastic effects Dynamics Measurement Measurement noise Measurement model

What Could the Right Side Be? Limited by your imagination 2 nd degree polynomials

Sparse Identification of Nonlinear Dynamics (SINDy)

Identifying ROMs

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 …)

Manifolds and Embeddings Observables & Coordinates

Bernard Koopman 1931 Mezic (2004) Coifman, Kevrekidis, co-workers

Koopman Invariant Subspaces Brunton, Proctor & Kutz, PLOS ONE (2018)

Burgers’ Equation Cole-Hopf Kutz, Proctor & Brunton, Complexity (2018)

Dynamic Mode Decomposition Travis Askham Askham & Kutz, SIADS (2018)

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

DMD with Control Input Input Snapshots DMD generalization Proctor, Brunton & Kutz, SIADS (2016)

Koopman vs DMD: All about Observables!

Neural Nets

The Zoo NN Zoo

NNs for Koopman Embedding Bethany Lusch Lusch et al. Nat. Comm (2018)

Handling the Continuous Spectra NNs for Koopman Embedding Bethany Lusch Lusch, Kutz & Brunton, arxiv (2017) Lusch et al. Nat. Comm (2018)

The Pendulum

Flow Around a Cylinder

Autoencoder + SINDy Kathleen Champion

Fast Learning Charles Delahunt

Moth Olfactory System

Learning New Odors

Sparsity for Learning

Rapid Learning in NNs

Comparisons

Decoder Networks

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

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

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

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

Nonlinear Mapping General Form: Compositional Layers Universal Approximators: Hornik 1990

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

Activation Functions

Linear vs Nonlinear Maps Improved Interpretability of Modes

Improved Performance

Robustness to Noise

Improved Performance

Super Resolution Analysis

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