Deep surrogate model for learning Green's function associated with linear reaction-diffusion operator

TL;DR

This paper introduces a deep learning-based surrogate model using U-Net architecture to efficiently learn Green's functions for reaction-diffusion PDEs, enabling rapid solutions with different sources and boundary conditions.

Contribution

The paper proposes a novel deep surrogate model with a custom loss function and boundary encoding to accurately learn Green's functions without labeled data.

Findings

Effective learning of Green's functions demonstrated through numerical examples

Fast PDE solver developed from the learned Green's function

Model captures boundary conditions accurately

Abstract

In this paper, we present a deep surrogate model for learning the Green's function associated with the reaction-diffusion operator in rectangular domain. The U-Net architecture is utilized to effectively capture the mapping from source to solution of the target partial differential equations (PDEs). To enable efficient training of the model without relying on labeled data, we propose a novel loss function that draws inspiration from traditional numerical methods used for solving PDEs. Furthermore, a hard encoding mechanism is employed to ensure that the predicted Green's function is perfectly matched with the boundary conditions. Based on the learned Green's function from the trained deep surrogate model, a fast solver is developed to solve the corresponding PDEs with different sources and boundary conditions. Various numerical examples are also provided to demonstrate the effectiveness of the proposed model.