AI 中文总结
该研究针对偏微分方程数值模拟提出基于分数的稳定化框架，通过学习的分数模型定义稳定化算子，增强标准时间步长方案，经数值实验验证该方法能提高鲁棒性、抑制非物理不稳定性并保持定性动力学。
AI 中文摘要
我们提出了一种用于偏微分方程数值模拟的基于分数的稳定化框架，其中一个学习到的分数模型定义了一个应用于临时数值更新的稳定化算子。该算子通过一种校正来增强标准时间步长方案，这种校正通过驱动迭代朝着可允许状态的流形来强制结构和物理一致性。我们表明稳定化算子朝着该流形起到收缩作用，产生一种具有盆地条件稳定性的校正机制。对流、Korteweg-de Vries（KdV）、非线性薛定谔（NLS）和伯格斯方程的数值实验证明了其具有更高的鲁棒性、抑制非物理不稳定性以及保持定性动力学。
英文摘要
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.
