Composite Function Antiderivative Transformation Iterative Deep Learning Solution Method for Hyper-fine Group Resonance Self-shielding Calculation in Homogeneous System

Resonance self-shielding calculations are crucial in reactor physics analysis and core design computation, as they serve as a preliminary step for conducting neutron transport or diffusion calculations based on macroscopic cross sections. In resonance self-shielding calculation methods, the equivalence theory and subgroup method are commonly used in engineering for their high efficiency, but they have the problem of insufficient accuracy. While the traditional hyper-fine group method is recognized for its high accuracy, it faces challenges such as low computational efficiency and limited geometric adaptability. At present, with the rapid advancement of artificial intelligence technology, deep learning methods have emerged as effective surrogate models in reactor physics calculations. At the same time, they have demonstrated exceptional performance in directly solving neutron diffusion/transport equations without requiring prior data acquisition, while offering innovative technical methodologies for resonance self-shielding calculations. This paper proposed a composite function antiderivative transformation iterative deep learning solution method (C-AIM) for hyper-fine group resonance self-shielding calculations in homogeneous systems. The C-AIM involved forming composite function by multiplying the slowing-down spectrum by the continuous energy total cross section. It also transformed all integral terms in the equation into antiderivatives of the integrand functions, converting the integral-form slowing-down equation into an exact differential equation. Then, deep neural networks were used to represent the composite function and different antiderivatives, with weighted loss functions constructed for each neural network function. These loss functions included those derived from the neutron slowing-down equation, the antiderivative transformation equation, the antiderivative solution constraints, and the boundary conditions. At last, the C-AIM conducted alternating iterative training to minimize the loss function. When the loss function’s value was below the preset minimal threshold, the numerical solution to the slowing-down equation was obtained. This solution could be used to condense the effective resonance self-shielding cross sections. The paper carried out numerical verification on a number of examples, including single-resonance nuclide problems and multi-nuclide resonance interference problems, leveraging a physics-informed neural network (PINN) deep learning architecture with fully connected neural networks. The continuous-energy slowing-down spectrum was obtained, and the multi-group cross section condensed using this spectrum was accurate compared to the result of the traditional hyper-fine group method. This lays a solid technical foundation for future resonance self-shielding calculations in complex heterogeneous systems. C-AIM represents not only a novel resonance self-shielding computation technique in theory but also demonstrates potential engineering application value in the generation of resonance integrals, subgroup parameters, and interference factors within the framework of traditional equivalence theory and subgroup methods.