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

arXiv: https://arxiv.org/abs/2010.12975 · HTML: https://mlsanity.com/paper/2010.12975/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs](https://mlsanity.com/paper/2211.08900/) (2022)
- [Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equations](https://mlsanity.com/paper/2207.10241/) (2022)
- [Bayesian deep learning framework for uncertainty quantification in high dimensions](https://mlsanity.com/paper/2210.11737/) (2022)

## Abstract (arXiv)

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, αi which successfully approximate the solution as a sum of smooth basis functions u simeq sumi=0N αi varphii. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.