Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation
This work addresses computational challenges in solving differential equations for physical systems, presenting an incremental improvement by combining neural networks with spectral methods.
The paper tackles solving linear and nonlinear differential equations by proposing a Legendre-Galerkin Deep Neural Network (LGNet) algorithm that predicts coefficients for Legendre basis functions to approximate solutions, demonstrating performance on models with various boundary conditions.
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, $α_i$ which successfully approximate the solution as a sum of smooth basis functions $u \simeq \sum_{i=0}^{N} α_i \varphi_i$. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.