Fast data-driven model reduction for nonlinear dynamical systems

TL;DR

This paper introduces simplified, fast algorithms for data-driven model reduction of nonlinear dynamical systems onto spectral submanifolds, achieving accurate, sparse models with significant computational speedup.

Contribution

The authors develop two explicit algorithms that simplify and accelerate the process of fitting spectral submanifolds and computing normal forms from data, improving upon previous implicit methods.

Findings

Algorithms produce accurate, sparse models for nonlinear dynamics.

Significant speedup over previous methods, several orders of magnitude.

Validated on numerical and experimental datasets.

Abstract

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). We use observed data to locate a low-dimensional, attracting slow SSM and compute a maximally sparse approximation to the reduced dynamics on it. The recently released SSMLearn algorithm uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to the normal form. Here, we present two simplified algorithms, which reformulate manifold fitting and normal form computation as explicit problems under certain assumptions. We show on both numerical and experimental datasets that these algorithms yield accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics. The new algorithms are significantly simplified and provide a speedup of several orders of magnitude.