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

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.

Analysis of task-related MEG functional brain networks using dynamic mode decomposition

“…One limitation of the method is that when the duration of the brain network activity is short and window-based approaches miss capturing them. In the future, we intend to solve these problems using an incremental version of the methodology using incremental DMD and instantaneous dFCs and constrained dictionary learning approaches such as LANDO [37] to enhance resolution and extract all the dynamics in the data. The methodology described in this paper was applied to cognitive tasks, however, it would be interesting to apply the pipeline to resting-state experiments during which the brain dynamically evolves by shifting from one brain configuration to the other [7].…”

Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems

“…Given a Koopman operator K of a measure-preserving dynamical system, we use the concept of unitary extension to formally construct a related normal operator K . That is, suppose that K : L 2 (Ω, ω) → L 2 (Ω, ω) is an isometry, then there exists a unitary extension K defined on an extended Hilbert space H with L 2 (Ω, ω) ⊂ H [68, Proposition I.2.3], 6 and unitary operators are normal. Even though such an extension is not unique, it allows us to understand the spectral information of K by considering K .…”

“…However, for larger dimensions that arise in applications such as fluid dynamics, molecular dynamics, and climate forecasting, it is not possible to explicitly store a dictionary. DMD is a popular approach to studying Koopman operators associated with high-dimensional dynamics, which can yield accurate results for periodic or quasiperiodic systems but is often unable to adequately capture relevant nonlinear dynamics [6,19,110] as it implicitly seeks to fit linear dynamics. Moreover, a rigorous connection with Galerkin approximations of Koopman operators does not always hold [105].…”

Physics-informed dynamic mode decomposition (piDMD)

“…Recent successful examples of ML algorithms that have been modified to respect physical principles include neural networks [2][3][4][5][6][7][8][9][10], kernel methods [11,12], deep generative models [13], and sparse regression [14][15][16][17][18]. These examples demonstrate that incorporating partially-known physical principles into machine learning architectures can increase the accuracy, robustness, and generalizability of the resulting models, while simultaneously decreasing the required training data.…”