DiffusionPDE: Generative PDE-Solving Under Partial Observation
This addresses a common real-world problem in scientific computing where measurements are incomplete, offering a versatile solution for various PDEs.
The authors tackled solving partial differential equations (PDEs) with incomplete data by proposing DiffusionPDE, a generative diffusion model that fills in missing information and solves PDEs simultaneously, significantly outperforming state-of-the-art methods in both forward and inverse directions.
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.