Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation

TL;DR

This paper introduces LGNet, a neural network leveraging Legendre-Galerkin basis functions to efficiently approximate solutions to both linear and nonlinear differential equations, demonstrating promising accuracy and computational benefits.

Contribution

The paper proposes a novel Legendre-Galerkin Deep Neural Network (LGNet) that predicts solution coefficients using spectral basis functions, advancing neural methods for differential equations.

Findings

LGNet accurately predicts solutions for linear and nonlinear equations.

Spectral element method effectively generates training data.

LGNet outperforms traditional neural architectures in this context.

Abstract

Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is associated with solving these systems. We explore the performance and accuracy of various neural architectures on both linear and nonlinear differential equations by creating accurate training sets with the spectral element method. Next, we implement a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to differential equations. By constructing a set of a linear combination of the Legendre basis, we predict the corresponding coefficients, $\alpha_i$ which successfully approximate the solution as a sum of smooth basis functions $u \simeq \sum_{i=0}^{N} \alpha_i \varphi_i$. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.