Solving the discretised neutron diffusion equations using neural networks
Author(s)
Phillips, Toby
Heaney, Claire
Chen, Boyang
Buchan, Andrew
Pain, Christopher
Type
Journal Article
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 built-in technologies to enable the same code to run on different
computer architectures (such as Central Processing Units, Graphics Processing Units or new-generation
AI processors) without any modification. 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. For both benchmarks we give timings for the neural network
implementation run on a CPU and a GPU, and a serial Fortran code run on a CPU.
in this way is that one can exploit their built-in technologies to enable the same code to run on different
computer architectures (such as Central Processing Units, Graphics Processing Units or new-generation
AI processors) without any modification. 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. For both benchmarks we give timings for the neural network
implementation run on a CPU and a GPU, and a serial Fortran code run on a CPU.
Date Issued
2023-11-15
Date Acceptance
2023-06-25
Citation
International Journal for Numerical Methods in Engineering, 2023, 124 (21), pp.4659-4686
ISSN
0029-5981
Publisher
Wiley
Start Page
4659
End Page
4686
Journal / Book Title
International Journal for Numerical Methods in Engineering
Volume
124
Issue
21
Copyright Statement
© 2023 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
License URL
Subjects
convolutional neural network
finite difference method
finite volume method
multigrid solver
neutron diffusion equation
numerical solution of partial differential equations
reactor physics
U-net
Publication Status
Published
Date Publish Online
2023-07-12