Finite Difference Nets: A Deep Recurrent Framework for Solving Evolution PDEs

TL;DR

This paper introduces a deep recurrent neural network framework inspired by finite difference methods to efficiently solve time-dependent PDEs without large datasets, demonstrating effectiveness on wave and conservation law models.

Contribution

It presents a novel architecture connecting traditional numerical schemes with deep neural networks for PDE solving, avoiding large data requirements.

Findings

Effective in solving PDEs with integral form

Shows efficiency on wave equations

Applicable to conservation laws

Abstract

There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data sets. We provide a new perspective, that is, a different type of architecture through exploring the possible connections between traditional numerical methods (such as finite difference schemes) and deep neural networks, particularly convolutional and fully-connected neural networks. Our proposed approach will show its effectiveness and efficiency in solving PDE models with an integral form, in particular, we test on one-way wave equations and system of conservation laws.