DSP-PIGAN: A Precision-Consistency Machine Learning Algorithm for Solving Partial Differential Equations

Abstract

Partial differential equations (PDEs) are the most ubiquitous tool for modeling problems in nature. In recent years, machine learning techniques are adopted to solve PDEs. However, the prediction errors of existing machine learning methods vary widely on different subdomains of PDEs. How to achieve precision-consistency is a crucial and complex issue for machine learning methods for solving PDEs. To tackle this issue, we propose DSP, an adaptive framework for solving PDEs. DSP is composed of domain decomposition, searching for singular subdomains, and prediction. Furthermore, a novel generative model, physics-informed generative adversarial network (PIGAN), is designed to solve PDEs. In addition, we introduce points with high-precision labels into the training process of the model to improve model accuracy. We test the effectiveness of our approach on three real physical equations: Poisson equation, Helmhotz equation and Eikonal equation. Through experiments, we prove that the combination of DSP and PIGAN outperforms various state-of-the-art baselines.