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.

翻译：我们提出了一种引导式随机采样方法，该方法通过从偏微分方程残差和观测约束导出的物理引导机制增强扩散模型的采样过程，确保生成的样本保持物理可容性。我们将此采样过程嵌入到新的序贯蒙特卡洛框架中，构建了一个可扩展的生成式偏微分方程求解器。在多个基准偏微分方程系统以及多物理场与相互作用偏微分方程系统上的实验表明，本方法生成的解场比现有最先进的生成式方法具有更低的数值误差。