DiffusionPDE: Generative PDE-Solving Under Partial Observation
Authors: name: Jiahe Huang, name: Guandao Yang, name: Zichen Wang, name: Jeong Joon Park

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.