Solving the Discretised Neutron Diffusion Equations using Neural Networks

TL;DR

This paper introduces a novel method that leverages AI library tools to solve discretised neutron diffusion equations without training, matching standard numerical solutions while exploiting AI hardware optimizations.

Contribution

The paper presents a way to represent discretised PDEs within neural networks using fixed weights, eliminating training and enabling direct solutions with AI library efficiencies.

Findings

Solutions match standard numerical methods within tolerances.

The approach efficiently solves reactor physics eigenvalue problems.

Utilizes AI hardware optimizations for PDE solving.

Abstract

This paper presents a new approach which uses the tools within Artificial Intelligence (AI) software libraries as an alternative way of solving partial differential equations (PDEs) that have been discretised using standard numerical methods. In particular, we describe how to represent numerical discretisations arising from the finite volume and finite element methods by pre-determining the weights of convolutional layers within a neural network. As the weights are defined by the discretisation scheme, no training of the network is required and the solutions obtained are identical (accounting for solver tolerances) to those obtained with standard codes often written in Fortran or C++. We also explain how to implement the Jacobi method and a multigrid solver using the functions available in AI libraries. For the latter, we use a U-Net architecture which is able to represent a sawtooth multigrid method. A benefit of using AI libraries in this way is that one can exploit their power and their built-in technologies. For example, their executions are already optimised for different computer architectures, whether it be CPUs, GPUs or new-generation AI processors. In this article, we apply the proposed approach to eigenvalue problems in reactor physics where neutron transport is described by diffusion theory. For a fuel assembly benchmark, we demonstrate that the solution obtained from our new approach is the same (accounting for solver tolerances) as that obtained from the same discretisation coded in a standard way using Fortran. We then proceed to solve a reactor core benchmark using the new approach.