Physics-Regularized Heat-Source-Conditioned FNO-LA Surrogate for Coupled Neutronic–Thermal Field Prediction in a Steady-State Single-Rod Benchmark

High-fidelity neutronic–thermal simulations provide detailed field information, but remain expensive for repeated evaluation. This study develops a physics-regularized heat-source-conditioned Fourier neural operator with linear attention (HSC-FNO-LA) for predicting the volumetric heat-source field Q and temperature field T in a steady-state pressurized-water-reactor single-rod benchmark. Unlike parallel multi-output surrogates, the proposed model first reconstructs Q and uses Q-derived features to condition the thermal decoder. Global heat-source energy regularization and a soft thermal-consistency term are included during three-stage training. With complete physical cases given equal statistical weight, the frozen reference model achieved a validation temperature mean absolute error of 16.26 K (95% bootstrap confidence interval: 15.35–17.22 K), a temperature relative error of 1.997%, a Q-field relative error of 2.366%, and a Q-energy error of 0.291%. Across five independently trained initializations, temperature mean absolute error was 12.46 ± 3.31 K, indicating non-negligible initialization sensitivity. Post-training interventions showed that shuffled and training-set-mean Q inputs increased temperature error by 25.64 and 16.40 K on average, respectively, across all five initializations, whereas replacing predicted Q with reference Q did not reduce mean temperature error. An additional 48 previously unused OpenFOAM cases yielded a temperature mean absolute error of 13.41 K. These results support explicit Q-conditioned thermal decoding for the tested benchmark, while broader geometric, transient, and experimental validation remains necessary.