Cell-average based neural network method for high dimensional parabolic differential equations

TL;DR

This paper presents a cell-average based neural network (CANN) method for efficiently solving high-dimensional parabolic PDEs, leveraging integral formulations and supervised training to achieve convergence without strict CFL constraints.

Contribution

The paper introduces a novel neural network approach that utilizes cell averages and integral formulations, enabling effective high-dimensional PDE solutions with relaxed CFL conditions.

Findings

Neural network trained to optimality for high-dimensional problems

CANN method's errors relate to spatial mesh size, not time step

Method does not strictly require CFL condition

Abstract

In this paper, we introduce cell-average based neural network (CANN) method to solve high-dimensional parabolic partial differential equations. The method is based on the integral or weak formulation of partial differential equations. A feedforward network is considered to train the solution average of cells in neighboring time. Initial values and approximate solution at $t=\Delta t$ obtained by high order numerical method are taken as the inputs and outputs of network, respectively. We use supervised training combined with a simple backpropagation algorithm to train the network parameters. We find that the neural network has been trained to optimality for high-dimensional problems, the CFL condition is not strictly limited for CANN method and the trained network is used to solve the same problem with different initial values. For the high-dimensional parabolic equations, the convergence is observed and the errors are shown related to spatial mesh size but independent of time step size.