Neural network-based discretization of nonlinear differential equations

This article presents a discretization scheme for a nonlinear differential equation using a regression analysis technique. As with many other numerical solvers such as finite difference methods and finite element methods, the presented scheme discretizes a simulation field into a finite number of points. Although other solvers “directly and mathematically” discretize a differential equation governing the field, the presented scheme “indirectly and statistically” constructs the discretized equation using regression analyses. The regression model learns a pre-prepared training data including dependent-variable values at neighboring points in the simulation field. Each trained regression model expresses the relation between the variables and returns a variable value for a point (output) referring to the variables for its surrounding points (input). In other words, the regression model performs a role as the discretized equation of the variable. The presented approach can be applied to many kinds of nonlinear problems. This study employs artificial neural networks and polynomial functions as the regression models. The main aim here is to assess whether the neural network-based spatial discretization approach can solve a nonlinear problem. In this work, I apply the presented scheme to nonlinear steady-state heat conduction problems. The computational accuracy of the presented technique was compared with a standard finite difference method and a homogenization method that is one of representative multiscale modeling approaches.