Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)

TL;DR

This paper introduces a kernel-based method for learning interpretable, robust models of high-dimensional nonlinear dynamical systems, improving upon traditional DMD by effectively disambiguating linear and nonlinear effects.

Contribution

The authors develop a kernel learning framework that enhances dynamic mode decomposition for high-dimensional nonlinear systems, allowing for accurate linear and nonlinear disambiguation and incorporating partial physical knowledge.

Findings

Kernel method effectively handles high-dimensional data.

Accurately recovers linear contributions amidst nonlinear effects.

Demonstrated on various nonlinear PDEs from physical sciences.

Abstract

Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics, and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modeling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. In contrast, sparse identification of nonlinear dynamics (SINDy) learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems. In this work, we present a kernel method that learns interpretable data-driven models for high-dimensional, nonlinear systems. Our method performs kernel regression on a sparse dictionary of samples that appreciably contribute to the underlying dynamics. We show that this kernel method efficiently handles high-dimensional data and is flexible enough to incorporate partial knowledge of system physics. It is possible to accurately recover the linear model contribution with this approach, disambiguating the effects of the implicitly defined nonlinear terms, resulting in a DMD-like model that is robust to strongly nonlinear dynamics. We demonstrate our approach on data from a wide range of nonlinear ordinary and partial differential equations that arise in the physical sciences. This framework can be used for many practical engineering tasks such as model order reduction, diagnostics, prediction, control, and discovery of governing laws.