Score-Based Stabilization for Time-Dependent Problems

2026-07-27

AI Summary

This work proposes a stabilization framework based on score-based generative models to address non-physical instabilities and structural distortions commonly encountered in the numerical solution of time-dependent partial differential equations (PDEs). For the first time, score-based generative models are introduced into PDE numerical stabilization, where a conditional stabilizing operator with manifold-contracting properties is constructed by learning the physically admissible solution manifold. This operator corrects intermediate solutions during time integration. Numerical experiments demonstrate that the method significantly enhances robustness for convection, Korteweg–de Vries (KdV), nonlinear Schrödinger, and Burgers equations, effectively suppressing spurious oscillations while preserving essential dynamical features.

Abstract

We propose a score-based stabilization framework for numerical simulation of partial differential equations, in which a learned score model defines a stabilization operator applied to provisional numerical updates. This operator augments standard time-stepping schemes by enforcing structure and physical consistency through a correction that drives iterates toward the manifold of admissible states. We show that the stabilization operator acts as a contraction toward this manifold, yielding a correction mechanism with basin-conditional stability. Numerical experiments on Advection, Korteweg-de Vries (KdV), Nonlinear Schrodinger (NLS), and Burgers' equations demonstrate improved robustness, suppression of nonphysical instabilities, and preservation of qualitative dynamics.

Problem

• time-dependent PDEs
• numerical instability
• physical consistency
• stabilization
• nonphysical oscillations

Innovation

• score-based stabilization
• learned score model
• manifold projection
• time-dependent PDEs
• numerical stability