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.
Abstract (translated)
本论文提出了一种新的方法,利用人工智能软件库中的工具来解决通过标准数值方法离散化的偏微分方程(PDEs)。我们特别描述了如何通过在神经网络内部预先确定卷积层权重来代表数值离散化,因为权重是由离散化方案定义的,因此不需要网络训练,而得到的解决方案与通常用Fortran或C++编写的标准代码得到的解决方案相同(考虑到求解器容忍度)。我们还解释了如何使用AI库中的函数来实现贾奇方法和多网格求解器,对于后者,我们使用了U-Net架构,能够代表齿状多网格方法。使用这种方法的好处是可以利用人工智能库的力量和内置技术。例如,它们的执行已经针对不同的计算机架构进行了优化,无论是CPU、GPU还是新一代人工智能处理器。在本文中,我们应用了 proposed 方法来解决反应堆物理学中的积分问题,其中 neutron 传输用扩散理论来描述。对于燃料组件基准,我们证明了我们新方法得到的解决方案与使用Fortran标准代码编码的相同离散化解决方案相同(考虑到求解器容忍度)。然后,我们使用新方法解决了反应堆核心基准问题。
URL
https://arxiv.org/abs/2301.09939
