用物理方程残差引导扩散模型生成更精确的偏微分方程解。
Particle-Guided Diffusion Models for Partial Differential Equations
- 结合物理残差与观测约束，指导扩散模型采样。
- 在多类偏微分方程上误差低于现有生成方法。
- 适合需要高精度物理模拟的科学计算场景。
我们提出一种基于物理的引导随机采样方法，将偏微分方程（PDE）残差和观测约束融入扩散模型采样过程，确保生成样本保持物理可接受性。该方法嵌入新型序列蒙特卡洛（SMC）框架，构建出可扩展的生成式PDE求解器。在多个基准PDE系统以及多物理场与相互作用的PDE系统上，本方法生成的解场数值误差低于现有最先进生成方法。
原文摘要 · Abstract (English)
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples remain physically admissible. We embed this sampling procedure within a new Sequential Monte Carlo (SMC) framework, yielding a scalable generative PDE solver. Across multiple benchmark PDE systems as well as multiphysics and interacting PDE systems, our method produces solution fields with lower numerical error than existing state-of-the-art generative methods.
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