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This paper introduces a score-based stabilization framework for the numerical simulation of partial differential equations, utilizing a learned score model to define a stabilization operator that corrects provisional numerical updates. By enforcing structural and physical consistency, this operator ensures that iterates converge toward the manifold of admissible states, effectively enhancing the stability of time-stepping schemes. Numerical experiments across various equations, including Advection and Burgers', reveal significant improvements in robustness and the suppression of nonphysical instabilities while maintaining qualitative dynamics.
A learned stabilization operator can dramatically enhance the stability of numerical simulations for complex partial differential equations, preventing nonphysical instabilities.
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.