Frequency Reduction and Order Elevation (FROE): A physics-informed deep learning method for ultra-fine group resonance self-shielding in heterogeneous systems

Resonance self-shielding calculation in complex heterogeneous geometries is essential for reactor physics and core design. While the ultra-fine group method solves the neutron slowing-down equation with high accuracy, its adoption has been limited by low computational efficiency and geometric constraints. Recent progress in deep learning has demonstrated strong potential in neutron transport calculations, offering new solutions to longstanding challenges in resonance self-shielding. This paper proposes a novel deep learning method termed "Frequency-Reduction and Order-Elevation(FROE)." First, frequency reduction is achieved by defining a composite function as the product of the slowing-down spectrum and the total cross-section. This transformation effectively smooths high-frequency oscillations induced by resonance peaks, facilitating neural network approximation. Second, order elevation involves reformulating the integro-differential slowing-down equation into an exact differential form by converting integral terms into their corresponding antiderivatives, thereby improving computational efficiency. Deep neural networks are then employed to represent the composite function and the antiderivatives. The slowing-down equation is solved by training the networks through an alternating iterative scheme to minimize this loss function. Numerical results for several heterogeneous problems, including cases with multi-nuclide resonance interference, validate the proposed method. Our approach accurately yields continuous slowing-down spectra in both space and energy. This work opens a promising new avenue for tackling the enduring challenge of resonance self-shielding in reactor physics.