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

2020/10/24 by Bryce Chudomelka, Chudomelka, Bryce, Youngjoon Hong +6 · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #65Kxx #65Lxx #65Mxx #Applied mathematics #Artificial intelligence #Artificial neural network #Computational Physics and Python Applications #Computer science #Differential (mechanical device) #Differential equation #FOS: Computer and information sciences #FOS: Mathematics #Finite element method #Galerkin method #Legendre polynomials #Legendre wavelet #Legendre's equation #Machine Learning (cs.LG) #Mathematical analysis #Mathematics #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA) #Physics #cs.LG #cs.NA #math.NA #msc:65Kxx #msc:65Lxx #msc:65Mxx

paper · pdf · doi:10.48550/arxiv.2010.12975

published in arXiv (Cornell University) (Cornell University) · 19 pages, 7 figures, 1 table

arxiv created 2020/10/24 · openalex publication_date 2020/10/24 · arxiv updated 2020/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, α_i which successfully approximate the solution as a sum of smooth basis functions u ≃ ∑_i=0^N α_i φ_i. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.

Citations

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