Symmetry-reduced model reduction of shift-equivariant systems via operator inference

TL;DR

This paper introduces a data-driven reduced-order modeling technique for shift-equivariant PDEs that captures traveling solutions by incorporating a traveling reference frame and estimating the traveling speed, validated on the Kuramoto-Sivashinsky equation.

Contribution

The method extends operator inference to include terms for estimating traveling speed, improving modeling of shift-equivariant systems with traveling solutions.

Findings

Robustly captures traveling solutions in PDEs.

Shows improved numerical stability over standard operator inference.

Validated on Kuramoto-Sivashinsky equation.

Abstract

We consider data-driven reduced-order models of partial differential equations with shift equivariance. Shift-equivariant systems typically admit traveling solutions, and the main idea of our approach is to represent the solution in a traveling reference frame, in which it can be described by a relatively small number of basis functions. Existing methods for operator inference allow one to approximate a reduced-order model directly from data, without knowledge of the full-order dynamics. Our method adds additional terms to ensure that the reduced-order model not only approximates the spatially frozen profile of the solution, but also estimates the traveling speed as a function of that profile. We validate our approach using the Kuramoto-Sivashinsky equation, a one-dimensional partial differential equation that exhibits traveling solutions and spatiotemporal chaos. Results indicate that our method robustly captures traveling solutions, and exhibits improved numerical stability over the standard operator inference approach.