Stabilizing PDE--ML coupled systems

TL;DR

The paper tackles instabilities in PDE--ML surrogates for large PDEs by first stabilizing the diffusion term in divergence form $w_t = \partial_z(-\frac{1}{2} w^2 + \nu \mathcal{N}[w])$ and applying a low-pass filter to suppress high-frequency errors, addressing spectral bias. It then augments the stabilized system with memory terms derived from the Mori--Zwanzig formalism to recover accuracy, using a projection-based reduced-order model with memory kernels $K_{jk}(t)$ computed via Hermite quadrature. The authors derive a practical ROM framework and demonstrate that memory-free stabilization improves stability but memory corrections—including linear and cubic terms—substantially improve the $L^2$ accuracy relative to the exact projected solution for the viscous Burgers' equation. The work provides a scalable route to stable and more accurate PDE--ML coupling, with potential applicability to more complex PDE 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 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.

Figures (6)

• Figure 1: The $L^{2}$ error evolution with time for different values of $M$, the cut-off for the low-pass filter.
• Figure 2: Comparison of the exact solution to \ref{['NLexample']} and those obtained by expanding the memory kernels up to linear polynomials. We employ $N_q = 20$ Hermite quadrature nodes translated to $[1,4]$.
• Figure 3: Comparison of the exact solution to \ref{['NLexample']} and those obtained by expanding the memory kernels up to quadratic polynomials. We employ $N_q = 30$ Hermite quadrature nodes spaced over $[1,4]$.
• Figure 4: Comparison of the exact solution to \ref{['NLexample']} and those obtained by expanding the memory kernels up to cubic polynomials. We employ $N_q = 40$ Hermite quadrature nodes translated to $[1,4]$.
• Figure 5: Absolute values of the memory kernels $K_{l1}(t)$ for the nonlinear system \ref{['NLexample']} for various values of $d$, the highest polynomial degree allowed in the expansion \ref{['MemKerExp']}.