A weak-form discrete physical-residual PINN method for solving multigroup neutron diffusion equations

To address the substantial computational burden of high-order spatial derivatives, the difficulty of residual optimization near material interfaces, and the optimization stiffness associated with direct space-time training when conventional strong-form physics-informed neural networks (PINNs) are applied to heterogeneous multigroup neutron diffusion problems, a PINN method based on weak-form discrete physical residuals is developed. Finite-element shape functions are used to construct the flux interpolation field and assemble the discrete physical operators, while the neural network predicts nodal fluxes and neural source iteration drives the coupled convergence of the fluxes and effective multiplication factor. P1 triangular elements are adopted for the two-dimensional problem and Q1 hexahedral elements for the three-dimensional problem, eliminating the need to evaluate high-order spatial derivatives of the flux by automatic differentiation during training. The method is progressively verified using the TWIGL and IAEA-3D PWR benchmarks. For the steady-state TWIGL problem, the PINN yields an effective multiplication factor of 0.9132141, differing from the finite element method (FEM) result by 0.04 pcm, while the root-mean-square errors (RMSEs) of both fast- and thermal-group fluxes are below 2.4 × 10 −4 . When this steady-state flux is used to initialize the FEM/Crank–Nicolson (C-N) transient calculation, the RMSEs of relative power for the step and ramp perturbations are 5.27 × 10 −4 and 4.57 × 10 −4 , respectively, relative to the pure FEM/C-N results; the maximum absolute relative deviations from the published POLCA-T reference values are both below 0.1%. For the IAEA-3D PWR problem, the PINN produces an effective multiplication factor of 1.0289669, differing from the same-mesh FEM result by 3.67 pcm. The RMSEs of the fast- and thermal-group fluxes are 3.008 × 10 −3 and 1.487 × 10 −3 , respectively. For the assembly relative power, the mean and maximum absolute relative errors with respect to published reference values are 0.1779% and 0.4647%, respectively. These results demonstrate that the proposed method avoids the dependence of strong-form PINNs on high-order spatial automatic differentiation and achieves accurate solutions of heterogeneous two-dimensional P1 and three-dimensional Q1 multigroup core-diffusion problems within a unified framework, providing a common methodological basis for differentiable solution of complex reactor-physics problems.