Approximation of functions and their derivatives: A neural network implementation with applications

A neural network implementation for the numerical approximation of functions of several variables and their first and second order partial derivatives can result in improved numerical methods for solving partial differential equations by eliminating the need to discretise the volume of the analysis domain.

Abstract

This paper reports a neural network (NN) implementation for the numerical approximation of functions of several variables and their first and second order partial derivatives. This approach can result in improved numerical methods for solving partial differential equations by eliminating the need to discretise the volume of the analysis domain. Instead only an unstructured distribution of collocation points throughout the volume is needed. An NN approximation of relevant variables for the whole domain based on these data points is then achieved. Excellent test results are obtained. It is shown how the method of approximation can then be used as part of a boundary element method (BEM) for the analysis of viscoelastic flows. Planar Couette and Poiseuille flows are used as illustrative examples.