Solving the discretised neutron diffusion equations using neural networks
Solving the discretised neutron diffusion equations using neural networks
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DOI：
10.1002/nme.7321
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发表时间：
2023-01
影响因子：
2.9
通讯作者：
Toby R. F. Phillips;C. Heaney;Boyang Chen;A. Buchan;C. Pain
中科院分区：
文献类型：
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作者：
Toby R. F. Phillips;C. Heaney;Boyang Chen;A. Buchan;C. Pain
本文提出了一种新方法，该方法使用人工智能 (AI) 软件库中的工具作为求解已使用标准数值方法离散化的偏微分方程 (PDE) 的替代方法。特别是，我们描述了如何通过预先确定神经网络中卷积层的权重来表示由有限体积和有限元方法产生的数值离散。由于权重是由离散化方案定义的，因此不需要对网络进行训练，并且获得的解决方案与使用通常用 Fortran 或 C++ 编写的标准代码获得的解决方案相同（考虑求解器容差）。我们还解释了如何使用 AI 库中提供的函数来实现雅可比方法和多重网格求解器。对于后者，我们使用能够表示锯齿多重网格方法的 U-Net 架构。以这种方式使用人工智能库的一个好处是，人们可以利用其内置技术使相同的代码无需任何修改即可在不同的计算机架构（例如中央处理单元、图形处理单元或新一代人工智能处理器）上运行。在本文中，我们将所提出的方法应用于反应堆物理学中的特征值问题，其中中子输运由扩散理论描述。对于燃料组件基准，我们证明从我们的新方法获得的解决方案与使用 Fortran 以标准方式编码的相同离散化获得的解决方案相同（考虑求解器容差）。然后，我们继续使用新方法解决反应堆堆芯基准测试。对于这两个基准测试，我们给出了在 CPU 和 GPU 上运行的神经网络实现以及在 CPU 上运行的串行 Fortran 代码的时序。
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.