DiffusionPDE: Generative PDE-Solving Under Partial Observation
Jiahe Huang, Guandao Yang, Zichen Wang, Jeong Joon Park
2024-06-26
Summary
This paper introduces DiffusionPDE, a new method for solving partial differential equations (PDEs) using generative diffusion models. It focuses on situations where we don't have all the necessary information to solve these equations traditionally.
What's the problem?
Many existing methods for solving PDEs struggle when they don't have complete data or when some information is missing. This is a common issue in real-world scenarios, making it difficult to accurately solve these equations. When the data is incomplete, traditional approaches can perform poorly, leading to incorrect or unreliable results.
What's the solution?
The authors propose DiffusionPDE, which can fill in the missing information while solving the PDE at the same time. This method uses generative models to understand the relationship between the solution and the coefficients (the values that define the equation). By modeling both aspects together, DiffusionPDE can provide accurate solutions even when only a small portion of the data is available. The framework has been shown to work well for various types of PDEs, significantly outperforming existing methods.
Why it matters?
This research is important because it offers a new way to solve complex mathematical problems that often arise in fields like physics and engineering. By improving how we handle incomplete data in PDEs, DiffusionPDE can lead to better predictions and analyses in real-world applications, making it a valuable tool for scientists and engineers.
Abstract
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.