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：我们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。该框架特别关注经典求解器所需场景信息不完备的实际情况。当数据观测值或底层系数不完整时（现实测量中的常见假设），现有大多数正向或逆向PDE求解方法表现欠佳。本研究提出的DiffusionPDE框架通过对解空间与系数空间的联合分布建模，能够同步补全缺失信息并求解PDE。研究表明，习得的生成先验可构建通用框架，在部分观测条件下精确求解多种PDE，其正向与逆向求解性能均显著超越现有最优方法。