标题： 基于得分的扩散模型方法用于随机偏微分方程解的自适应学习 显示英文标题

标题： A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions

摘要： 我们提出了一种新颖的框架，通过在递归贝叶斯推理设置中使用基于评分的扩散模型，自适应地学习随机偏微分方程（SPDEs）的时间演化解。 SPDEs 在不确定性下的复杂物理系统建模中起着核心作用，但其数值解通常由于不完整的物理知识和环境变化而受到模型误差和精度降低的影响。 为了解决这些挑战，我们利用仿真数据将控制物理编码到扩散模型的评分函数中，并通过反向时间随机微分方程中的似然校正来整合观测信息。 这使得在新数据可用时能够通过迭代精炼解决方案实现自适应学习。 为了在高维设置中提高计算效率，我们引入了集合评分滤波器，这是一种无需训练的评分函数近似方法，专为实时推理设计。 在基准 SPDE 上的数值实验表明，在稀疏和噪声观测下，所提出的方法具有准确性和鲁棒性。 显示英文摘要

摘要： We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.