Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks

TL;DR

This paper introduces two neural network-based methods for solving elliptic PDEs using small linear convolutional networks, achieving accuracy comparable to traditional methods with fewer parameters and improved interpretability.

Contribution

The paper presents novel supervised and unsupervised neural network approaches for discontinuous Galerkin solutions, reducing parameter count and enhancing interpretability over existing neural PDE solvers.

Findings

Achieves comparable accuracy to traditional DG solutions.

Uses significantly fewer parameters than similar neural methods.

Provides both supervised and unsupervised learning frameworks.

Abstract

In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such as physics-informed neural networks, depend on automatic differentiation and the sampling of collocation points, which can result in a lack of interpretability and lower accuracy compared to traditional numerical methods. To address this issue, we propose two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks. Our first approach is supervised and depends on labeled data, while our second approach is unsupervised and does not rely on any training data. In both cases, our methods use substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.