MUMS Opening Workshop -Data-Driven Discovery of Governing Physical Laws and their Parametric Dependencies in Engineering, Physics and Biology - Nathan Kutz , August 21, 2018

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The Statistical and Applied Mathematical Sciences Institute

A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive but provide insight into the nature of the underlying dynamical system that generated the data. This problem is made more difficult by the fact that many systems of interest exhibit diverse behaviors across multiple time scales. We introduce several data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales. Specifically, we can discover distinct governing equations at slow and fast scales. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates, can be used to obtain a linear model and Koopman invariant measurement system that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. We introduce two strategies for using HAVOK on systems with multiple time scales. Together, our approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.

Education

Data-driven discovery of governing
equations of physical systems
J. Nathan Kutz
Department of Applied Mathematics
University of Washington
Email: kutz@uw.edu
August 1, 2018 – AMAZON
- Model discovery
- Manifolds & Embeddings
- Measurement & Sensors

Data Science Today
=
=Under
Over
-
- pinv
- Lasso
- Ridge
- Elastic net
- Robust fit

Model Discovery
Brunton,Proctor & Kutz, PNAS (2016)
Steven Brunton
Mechanical Engineering
University of Washington
Joshua Proctor
Institute for Disease Modeling

Governing Dynamical Systems
Generic
nonlinear
,
time-‐dependent,
parametric
system
Measurements
(assimilation)

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

Sparse Identification of Nonlinear Dynamics (SINDy)

Modifications: Implicity-SINDy
Niall Mangan

Model Selection and Information Theory
Mangan, Kutz, Brunton & Proctor, PRSA (2017)
Model Selection
- 1950s KL divergence
- Early 70s AIC (Akaike)
- 78 BIC (G. Schwarz)
- BIC/AIC limited # of models

Discovering PDEs
Rudy,Brunton, Proctor & Kutz, Sci Adv (2017)Sam Rudy

KEY CHALLENGES
- Limited measurements & data
- Noise
- Multi-scale physics
- Latent variables
- Pametric dependencies
- Stochastic systems
Champion et al, arxiv (2018)
Rudy et al, arxiv (2018)

Limits of Model Discovery
Champion,Brunton, & Kutz, arxiv (2018)

Parametric Discovery
Group
LASSO
vs
Sequential
Group
Threshold
Regression
(SGTR)

Our
innovation:

SGTR
(works
amazingly
well!)

Manifolds and Embeddings
Observables & Coordinates

Bernard Koopman 1931
Koopman (1931)
Meziz (2005)

Dynamic Mode Decomposition
Schmid (2010)
Rowley, Mezic et al (2009)
Tu et al (2014)
Travis Askham& Kutz (2017)

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

Some Applications
Video Processing + RPCA
ECOG recordings
Erichson, Brunton & Kutz(2017)
Brunton,Johnson, Ojemann & Kutz (2017)

DMD with Control
Input
Input

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

Koopman Invariant Subspaces
Brunton , Brunton,Proctor, Kutz, PLOS One (2016)

Burgers’ Equation
Cole-‐Hopf
Kutz, Proctor, Brunton,arxiv (2016)

Neural Networks for Koopman Embedding
Lusch, Kutz& Brunton,arxiv (2017)Bethany Lusch

Conclusions
Model
Selection
&
Sparse
Regression
Matter
- Principled approach to determining dynamics
- (i) classification, (ii) reconstruction, (iii) future state prediction
- Sensors should be maximally informative

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