Dimensionality Reduction and Reduced Order Modeling for Traveling Wave Physics

TL;DR

This paper presents an unsupervised machine learning approach combining sparse regression and subspace clustering to identify traveling waves and invariances in PDE-governed systems, improving reduced order models for complex spatio-temporal dynamics.

Contribution

It introduces a novel method for automatically discovering translational invariances, enhancing the construction of low-dimensional, accurate reduced order models from both numerical and experimental data.

Findings

Effective identification of traveling waves across multiple PDEs

Robustness to noise and data limitations

Improved reduced order model accuracy

Abstract

We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatio-temporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and subspace clustering to robustly identify translational invariances that can be leveraged to build improved reduced order models (ROMs). Invariances, whether translational or rotational, are well known to compromise the ability of ROMs to produce accurate and/or low-rank representations of the spatio-temporal dynamics. However, by discovering translations in a principled way, data can be shifted into a coordinate systems where quality, low-dimensional ROMs can be constructed. This approach can be used on either numerical or experimental data with or without knowledge of the governing equations. We demonstrate our method on a variety of PDEs of increasing difficulty, taken from the field of fluid dynamics, showing the efficacy and robustness of the proposed approach.