Score-Based Stabilization for Time-Dependent Problems

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.

Machine Learning (cs.LG); Numerical Analysis (math.NA)

arXiv:2607.25119 [cs.LG]

(or arXiv:2607.25119v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.25119 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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