Model discovery for nonautonomous translation-invariant problems

TL;DR

This paper introduces a novel method combining Lagrangian dynamic mode decomposition with a local time-invariant Koopman approximation to discover mathematical models of nonautonomous, translation-invariant physical systems from data.

Contribution

It presents a new data-driven approach that effectively handles time dependence and translation invariance in modeling complex physical phenomena.

Findings

Accurate prediction bounds for nonautonomous systems

Effective modeling of Navier-Stokes equations

Improved system identification in challenging scenarios

Abstract

Discovery of mathematical descriptors of physical phenomena from observational and simulated data, as opposed to from the first principles, is a rapidly evolving research area. Two factors, time-dependence of the inputs and hidden translation invariance, are known to complicate this task. To ameliorate these challenges, we combine Lagrangian dynamic mode decomposition with a locally time-invariant approximation of the Koopman operator. The former component of our method yields the best linear estimator of the system's dynamics, while the latter deals with the system's nonlinearity and non-autonomous behavior. We provide theoretical estimators (bounds) of prediction accuracy and perturbation error to guide the selection of both rank truncation and temporal discretization. We demonstrate the performance of our approach on several non-autonomous problems, including two-dimensional Navier-Stokes equations.