Finite Difference Neural Networks: Fast Prediction of Partial Differential Equations

TL;DR

FDNet is a neural network framework inspired by finite difference methods that efficiently learns and predicts the behavior of partial differential equations from data, demonstrating superior performance over traditional numerical methods.

Contribution

Introduces FDNet, a novel neural network architecture that learns PDEs from data using finite difference principles, enabling fast and accurate predictions with minimal parameters.

Findings

Successfully predicts heat equation dynamics with noise and forcing.

Outperforms Forward Euler method in predictive accuracy.

Uses Hessian-Free Trust Region for improved training stability.

Abstract

Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.