On quadrature rules for solving Partial Differential Equations using Neural Networks

• Published in: Computer methods in applied mechanics and engineering 2022-04, Vol.393, p.114710, Article 114710
• Main Authors: Rivera, Jon A., Taylor, Jamie M., Omella, Ángel J., Pardo, David
• Format: Article
• Language: English
• Subjects: Approximation; Deep learning; Least-Squares method; Mathematical analysis; Monte Carlo simulation; Neural Networks; Numerical integration; Partial differential equations; Polynomials; Quadrature rules; Quadratures; Regularization; Ritz method
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2022.114710

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 several 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 numerical integration scheme. 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 spatial dimension; however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks. •We analyze quadrature errors arising when solving PDEs using NNs.•We propose four different alternatives to overcome quadrature problems.•We test numerically three of the proposed alternatives and discuss their viability.•For dimension higher than three, we propose the use of Monte Carlo integration.•For dimension three or lower, we propose the use of an adaptive integration method.