Steven Brunton, UWashington
Abstract:
This work will discuss several key challenges and opportunities in the use of machine learning for nonlinear system identification. In particular, I will describe how machine learning may be used to develop accurate and efficient nonlinear dynamical systems models for complex natural and engineered systems. I will emphasize the need for interpretable and generalizable data-driven models, such as the sparse identification of nonlinear dynamics (SINDy) algorithm, which identifies a minimal dynamical system model that balances model complexity with accuracy, avoiding overfitting. I will also introduce several key benchmark problems in dynamical systems and fluid dynamics that provide a diversity of metrics to assess modern system identification techniques. Because fluid dynamics is central to transportation, health, and defense systems, we will emphasize the importance of machine learning solutions that are interpretable, explainable, generalizable, and that respect known physics.
Bio:
Dr. Steven L. Brunton is a Professor of Mechanical Engineering at the University of Washington. He is also Adjunct Professor of Applied Mathematics, Aeronautics and astronautics, and Computer science, and he is also a Data Science Fellow at the eScience Institute. He is Director of the AI Center for Dynamics and Control (ACDC) at UW and is Associate Director for the NSF AI Institute in Dynamic Systems. Steve received the B.S. in mathematics from Caltech in 2006 and the Ph.D. in mechanical and aerospace engineering from Princeton in 2012. His research combines machine learning with dynamical systems to model and control systems in fluid dynamics, biolocomotion, optics, energy systems, and manufacturing. He received the Army and Air Force Young Investigator Program (YIP) awards and the Presidential Early Career Award for Scientists and Engineers (PECASE). Steve is also passionate about teaching math to engineers as co-author of four textbooks and through his popular YouTube channel, under the moniker “eigensteve”.
Summary:
Focus: ML for Scientific Discovery and Optimization (Models from Data via Optimization
Fluid Dynamics:
Very common and valuable
Also very challenging to simulate
Nonlinear
Multiscale
High-dimensional
Non-convex
Tasks:
Compression
Reduced Modeling
Smart Sensing
Estimation
Control
Need to speed up by 1e6-1e9 times to enable real-time control
E.g. dynamically control wings in response to sensor readings of air flow
Same major challenges for other types of models/scientific fields
ML:
Good news:
Deep learning models are challenging in the same ways as fluid dynamics and we are making steady progress there
Despite complexity fluid dynamics show consistent, low-dimensional patterns that drive the overall behavior
Such patterns enable
Sparse sensing
Reduced-order modeling
Fast/efficient control
A reduced order model is a simpler equation that captures the dominant dynamics that is solvable
Digital Twin: data-driven surrogate model
Hierarchy of multi-fidelity models
Hybrid physics-ML/AI
End-to-end adjust-free optimization
To ensure the models work when we need them to, we need ML models that are Interpretable and Generalizable
Intuition: “physical” models are as simple as possible but no simpler
Simple equations
Low-dimensional
Sparse
Example: Lorenz 1963 model 3-variable system that approximates chaotic thermal convection
Sparse Identification of Nonlinear Dynamics (SINDy)
Given a time series representation of the evolution of a non-linear system
Simplest model: learn linear system that approximates it
Too simple: doesn’t capture real modes
Better: linear model of a fixed library of non-linear terms (e.g. polynomials, trig, wavelets)
Seek to use the fewest non-linear columns (L1 norm optimization)
Can expand to Partial Differential Equations (PDEs)
Physics involves many deep symmetries, key challenge is to discover them or enforce them
Equivariance: same effect on different systems
Invariance: different things act the same
Find the largest symmetry group that my data is equivariant to
Stages of ML
Decide on problem
Curate Data
Design an architecture: family of functions to be trained
Craft loss function
Employ optimization to train the function in the family
(Can embed physics at every stage)
Dominant Balance
A full equation has many complex terms
But if you break it down, you can find that in different points of space/time different terms dominate
Allows you to simplify the model by dominant physics
E.g. Buckingham Pie Network (BuckiNet)