Stabilizing PDE–ML coupled systems

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 may help with more complex systems. In particular, we focus on a viscous Burgers’-ML system and, after identifying the cause of the instabilities, propose strategies to stabilize it. To improve the accuracy of the stabilized system, we next explore methods based on the Mori–Zwanzig formalism. We show that the memory-based corrections from this approach help considerably in yielding accurate results. Finally, we draw analogies with more complex systems and how these strategies may generalize to those settings.