Physics Informed Neural Networks for the mixed dual form of the neutron diffusion equation with heterogeneous coefficients

Physics-Informed Neural Networks (PINNs), a popular deep learning framework for numerical approximations of Partial Differential Equations (PDEs), are investigated in this work to approximate the solution of the neutron diffusion equation, which is used in simulations of nuclear reactor cores. Moreover, this equation may have low regularity solution due to heterogeneous coefficients, which presents a challenge for the PINNs approach based on the primal form of the neutron diffusion equation. In this work, we study the PINNs approach for the mixed dual form of the neutron diffusion equation and aim to demonstrate that it can significantly improve the accuracy of the approximate solution, especially in cases with heterogeneous coefficients, compared to the primal approach. Besides, neural networks are typically based on the inverse power method for the k-eigenvalue problem, and it is well-known that this algorithm converges very slowly if the dominance ratio is high, as is commonly the case in several reactor physics applications. Therefore, we also discuss some acceleration methods for the PINNs approach applied to the k-eigenvalue problem. Several numerical test cases for the source and k-eigenvalue problems illustrate our purposes.