Title: Integrating Solving Forward and Inverse Problems in PDEs with Flow-based Models
Primary area: generative models
Keywords: operator learning, inverse problems, flow-based models, partial differential equations

Abstract: Solving partial differential equations (PDEs) given input parameters (forward problem) and inferring unknown parameters from partially observed solutions (inverse problem) are two critical problems in scientific computing. Most existing approaches treat forward problems and inverse problems as separate ones. In this article, we present a novel method based on rectified flow that integrates the solution of both problems in a single model. Specifically, in the training stage, we approximate the velocity field in rectified flow with fixed pairs $z(0) = a$ and $z(T) = u$, where $a$ is the input parameter and $u$ is the corresponding solution. In the inference stage, the forward problem can be solved by feeding $z(0)$ with $a$ and running the forward pass of the flow model, and the inverse problem can be solved by feeding $z(T)$ with $u$ and running the reverse pass of the flow model. Numerical results on various equations demonstrate that the proposed method achieves competitive accuracy in solving forward problems and shows better performance in solving inverse problems within a single model.