Physics-Constrained Neural Reduced-Order Models for Thermal Systems Under Operating Condition Shift: A Reproducible Benchmark Study

This study presents a reproducible numerical benchmark investigating physics-constrained neural reduced-order models under operating-condition shift. The benchmark considers a dimensionless, one-dimensional steady heat-conduction problem with spatially varying thermal conductivity and a sinusoidal heat source. A proper orthogonal decomposition (POD) basis with eight modes is combined with a multilayer perceptron (MLP) to predict reduced-order solution coefficients. A supervised baseline is compared with physics-regularized variants using four residual-penalty weights. Model performance is evaluated on 300 in-range test cases and 300 shifted cases in which the conductivity-variation parameter falls outside the training interval. Three neural-network training seeds are evaluated using a fixed data split. The baseline model achieves a mean median relative L2 error of 1.39% on in-range cases and 23.62% under the specified parameter shift. With a physics-residual penalty weight of 1.0, the mean shifted-case 95th-percentile error decreases from 69.94% to 62.64%. Residual magnitude is positively associated with solution error, suggesting potential value as an error-ranking indicator, although no calibrated reliability threshold is established. The findings are limited to the specified synthetic benchmark and do not establish general superiority, experimental validation, or industrial readiness. The study provides a basis for further investigation of physics regularization, reduced-order modelling, and model reliability under distribution shift.