扩散PDE：部分观测下的生成式偏微分方程求解

DiffusionPDE: Generative PDE-Solving Under Partial Observation

June 25, 2024

作者: Jiahe Huang, Guandao Yang, Zichen Wang, Jeong Joon Park

cs.AI

摘要

我们引入了一个通用框架，使用生成扩散模型来解决偏微分方程（PDEs）。特别是，我们关注的是在没有足够了解场景的完整知识以应用经典求解器的情况下。大多数现有的正向或反向PDE方法在观测数据或基础系数不完整时表现不佳，这是对真实世界测量的常见假设。在这项工作中，我们提出了DiffusionPDE，它可以同时填补缺失信息并通过对解空间和系数空间的联合分布建模来解决PDE。我们展示了学习到的生成先验可以导致一个多才多艺的框架，可以准确解决在部分观测下的广泛PDE范围，显著优于现有技术方法，无论是正向还是反向方向。

English

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.