FC-PINNs: Physics-informed neural networks for solving neutron diffusion eigenvalue problem with interface considerations

Abstract

In this study, a refined methodology for Physics-informed Neural Networks (PINNs), referred to as fixed-point constraint PINNs (FC-PINNs), is introduced, which leverages fixed-point transformation and computational domain concatenation method to address eigenvalue problems and contact boundary conditions in differential eigenvalue equations, with application to solving the neutron diffusion equation, a classic eigenvalue problem in neutron transport theory. Conventional PINNs may face challenges in computing differential eigenvalue equations and handling contact boundaries, which sometimes leads to difficulties in converging to non-trivial solutions during the training process. For the eigenvalue, known as the effective multiplication factor $k_{\text{eff}}$, two methods are proposed applying hard or soft-constraint fixed-point transformation, ensuring accurate predictions of eigenvalue and eigenfunction distribution. For the contact boundary, a computational domain concatenation technique is employed, combining fixed-point constraints with extrapolated boundary constraints, which effectively stitches together multiple outputs to handle calculations at contact boundaries. FC-PINNs enable high-precision computation of differential eigenvalue equations through solving the inverse problem of PDEs, significantly enhancing the efficiency of eigenvalue computation. FC-PINNs are tested across one-dimensional, two-dimensional, and specific engineering problems, comparing the results with analytical and numerical solutions obtained from the outer-inner iteration method, with all test cases showing errors below 0.3 %.