We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical solvers. Most existing forward or inverse PDE approaches perform poorly when the observations on the data or the underlying coefficients are incomplete, which is a common assumption for real-world measurements. In this work, we propose DiffusionPDE that can simultaneously fill in the missing information and solve a PDE by modeling the joint distribution of the solution and coefficient spaces. We show that the learned generative priors lead to a versatile framework for accurately solving a wide range of PDEs under partial observation, significantly outperforming the state-of-the-art methods for both forward and inverse directions.

翻译：本文提出了一种利用生成扩散模型求解偏微分方程（PDE）的通用框架。我们特别关注那些缺乏经典求解器所需完整场景信息的场景。当数据观测值或底层系数不完整时（现实世界测量中的常见假设），大多数现有的正向或逆向PDE方法表现欠佳。本工作中，我们提出DiffusionPDE方法，通过对解空间与系数空间的联合分布进行建模，能够同时填补缺失信息并求解PDE。研究表明，学习得到的生成先验可形成一个通用框架，能在部分观测条件下精确求解多种PDE，其性能在正向与逆向求解任务上均显著优于现有最优方法。