用扩散模型自适应学习随机偏微分方程解，提升不确定性下的精度。
A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions
- 基于得分扩散模型，在递归贝叶斯框架中融合物理规律与观测数据。
- 在稀疏噪声观测下仍保持高精度，数值实验验证了鲁棒性。
- 提出无训练的集成得分滤波器，适合高维实时推理场景。
我们提出一种新框架，利用得分扩散模型在递归贝叶斯推断设置下自适应学习随机偏微分方程（SPDE）的时间演化解。SPDE 在不确定性环境下的复杂物理系统建模中起核心作用，但其数值解常因物理知识不全和环境变化导致模型误差与精度下降。为此，我们通过仿真数据将控制物理规律编码到扩散模型的得分函数中，并在反向时间随机微分方程中引入似然修正以融合观测信息，实现随新数据迭代优化的自适应学习。为提升高维场景下的计算效率，提出无需训练的集成得分滤波器，用于实时推断。在基准 SPDE 上的数值实验表明，该方法在稀疏且含噪声观测下仍具高准确性和鲁棒性。
原文摘要 · Abstract (English)
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.
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