DiffusionPDE：在部分觀測下生成偏微分方程求解

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

我們提出了一個使用生成擴散模型解決偏微分方程（PDEs）的通用框架。特別是，我們專注於在沒有足夠了解場景的完整知識以應用傳統求解器的情況下。大多數現有的正向或反向PDE方法在觀測數據或基礎係數不完整時表現不佳，這是對於現實世界測量的常見假設。在這項工作中，我們提出了DiffusionPDE，它可以同時填補缺失信息並通過建模解和係數空間的聯合分佈來解決PDE。我們展示了學習的生成先驗導致了一個多才多藝的框架，可以準確解決各種在部分觀測下的PDE，明顯優於當前正向和反向方法的最新技術。

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.