Learning-based Reduced Order Model Stabilization for Partial Differential Equations: Application to the Coupled Burgers Equation

TL;DR

This paper introduces a learning-based method to stabilize reduced order models of PDEs by automatically tuning closure model parameters using an extremum seeking algorithm, demonstrated on the coupled Burgers' equation.

Contribution

It develops a model-free extremum seeking approach for auto-tuning closure models in ROMs, enhancing stabilization for PDEs.

Findings

Successful auto-tuning of linear closure models for ROM stabilization.

Extension to combined linear and nonlinear closure models improves performance.

Validated on the coupled Burgers' equation as a test case.

Abstract

We present results on stabilization for reduced order models (ROM) of partial differential equations using learning. Stabilization is achieved via closure models for ROMs, where we use a model-free extremum seeking (ES) dither-based algorithm to learn the best closure models' parameters, for optimal ROM stabilization. We first propose to auto-tune linear closure models using ES, and then extend the results to a closure model combining linear and nonlinear terms, for better stabilization performance. The coupled Burgers' equation is employed as a test-bed for the proposed tuning method.