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

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 $u\simeq {\sum }_{i=0}^N{\alpha }_i{\phi }_i$. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.

arXiv comment19 pages, 7 figures, 1 table

References in corpus (10)

• Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
• An overview of gradient descent optimization algorithms
• Deep Residual Learning for Image Recognition
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Convolutional Neural Networks for Sentence Classification
• Searching for Activation Functions
• Multi-Scale Convolutional Neural Networks for Time Series Classification
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Nonlinear Systems Identification Using Deep Dynamic Neural Networks
• Learned discretizations for passive scalar advection in a 2-D turbulent flow