Stabilizing PDE--ML coupled systems

TL;DR

This paper investigates the instability issues in PDE-ML coupled systems, particularly viscous Burgers' equation, and proposes stabilization strategies along with methods to enhance accuracy using the Mori--Zwanzig formalism.

Contribution

It identifies the cause of instabilities in PDE-ML systems and introduces stabilization techniques, along with approaches to improve accuracy through the Mori--Zwanzig formalism.

Findings

Identified causes of instabilities in PDE-ML systems.

Developed strategies to stabilize coupled PDE-ML systems.

Explored Mori--Zwanzig formalism to enhance accuracy.

Abstract

A long-standing obstacle in the use of machine-learnt surrogates with larger PDE systems is the onset of instabilities when solved numerically. Efforts towards ameliorating these have mostly concentrated on improving the accuracy of the surrogates or imbuing them with additional structure, and have garnered limited success. In this article, we study a prototype problem and draw insights that can help with more complex systems. In particular, we focus on a viscous Burgers'-ML system and, after identifying the cause of the instabilities, prescribe strategies to stabilize the coupled system. To improve the accuracy of the stabilized system, we next explore methods based on the Mori--Zwanzig formalism.