用学习的评分模型稳定时间依赖方程的数值求解，提升计算稳定性。
Score-Based Stabilization for Time-Dependent Problems
- 通过学习得分模型构建修正算子，引导数值解回归物理可行状态流形。
- 在对流、KdV、非线性薛定谔和伯格斯方程上均实现非物理不稳定性抑制。
- 适合需要长期稳定模拟的物理建模与工程仿真研究者使用。
我们提出一种基于得分模型的时间依赖偏微分方程数值模拟稳定化框架，其中学习得到的得分模型定义了一个施加于临时数值更新的稳定化算子。该算子通过将迭代解拉向允许状态流形，增强标准时间步进方案的结构一致性和物理合理性。我们证明该稳定化算子在流形上具有收缩性，从而实现具有盆地条件稳定性的修正机制。在对流方程、Korteweg-de Vries (KdV) 方程、非线性薛定谔 (NLS) 方程和伯格斯方程上的数值实验表明，该方法显著提升了鲁棒性，有效抑制了非物理不稳定性，并保持了定性动力学特性。
原文摘要 · Abstract (English)
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.
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