On quadrature rules for solving Partial Differential Equations using Neural Networks

Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose different alternatives...

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Bibliographic Details
Published in:arXiv.org 2021-10
Main Authors: Rivera, Jon A, Taylor, Jamie M, Omella, Ángel J, Pardo, David
Format: Article
Language:English
Subjects:
Approximation
Mathematical analysis
Monte Carlo simulation
Neural networks
Partial differential equations
Polynomials
Quadratures
Regularization
Online Access:Get full text
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Summary:Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose different alternatives to overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of the Neural Network output, and the inclusion of regularization terms in the loss. We also discuss the advantages and limitations of each proposed alternative. We advocate the use of Monte Carlo methods for high dimensions (above 3 or 4), and adaptive integration or polynomial approximations for low dimensions (3 or below). The use of regularization terms is a mathematically elegant alternative that is valid for any spacial dimension, however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks.
ISSN:2331-8422
DOI:10.48550/arxiv.2111.00217