Solving Multi-Group Neutron Diffusion Eigenvalue Problem with Decoupling Residual Loss Function

TL;DR

This paper proposes two residual loss function called Decoupling Residual loss function and Direct Iterative loss function that can deal with multi-group eigenvalue problem, and also single-group eigenvalue problem.

Abstract

In the midst of the neural network's success in solving partial differential equations, tackling eigenvalue problems using neural networks remains a challenging task. However, the Physics Constrained-General Inverse Power Method Neural Network (PC-GIPMNN) approach was proposed and successfully applied to solve the single-group critical problems in reactor physics. This paper aims to solve critical problems in multi-group scenarios and in more complex geometries. Hence, inspired by the merits of traditional source iterative method, which can overcome the ill-condition of the right side of the equations effectively and solve the multi-group problem effectively, we propose two residual loss function called Decoupling Residual loss function and Direct Iterative loss function. Our loss function can deal with multi-group eigenvalue problem, and also single-group eigenvalue problem. Using the new residual loss functions, our study solves one-dimensional, two-dimensional, and three-dimensional multi-group problems in nuclear reactor physics without prior data. In numerical experiments, our approach demonstrates superior generalization capabilities compared to previous work.

Figures (19)

• Figure 1: Schematic diagram of the problem physical meaning.
• Figure 2: Neural network to solve PDE flow chart.
• Figure 3: Left: the algorithm flowchart for traditional methods of solving multi-group neutron diffusion problems. Right: the flowchart for solving the same problem using neural networks. The orange dashed boxes represent the decoupling process in both algorithms.
• Figure 4: Computational domain of 1-D Swedish Ringhals-4 pressurized water reactor, where $a = 279.5\text{cm}, b = 161.25\text{cm}$.
• Figure 5: Results of 1-D Swedish Ringhals-4 pressurized water reactor problem: Left: reference solution. Mid: neural network solution. Right: absolute error. First row: information of $\phi_1(x)$. Second row: information of $\phi_2(x)$.