Stabilizing PDE--ML coupled systems
This addresses a long-standing obstacle in using machine-learnt surrogates for PDE systems, which is incremental as it builds on existing efforts with a prototype approach.
The paper tackled the problem of instabilities in PDE-ML coupled systems by analyzing a viscous Burgers'-ML prototype, identifying instability causes, and prescribing stabilization strategies, then explored Mori-Zwanzig methods to improve accuracy.
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.