<p>This work presents a physics-infused reduced-order modeling (PIROM) framework toward the design optimization of non-linear dynamical systems. It is demonstrated via the modeling of transient thermal behavior in multi-layered hypersonic thermal protection systems. The PIROM architecture integrates a reduced-physics backbone, based on the lumped capacitance model, with data-driven correction dynamics formulated via a coarse-graining approach rooted in the Mori–Zwanzig formalism. While the lumped capacitance model captures the dominant heat transfer mechanisms, the correction terms compensate for residual dynamics arising from higher-order non-linear interactions and heterogeneities across material layers. The proposed PIROM is benchmarked against a non-intrusive ROM (i.e., operator inference) and a surrogate model (i.e., neural ordinary differential equations). The PIROM consistently achieves errors below 1% for a wide range of extrapolative settings of design parameters involving time- and space-dependent boundary conditions and temperature-varying material property perturbations. In contrast, the baseline ROM exhibits moderate degradation, and the baseline surrogate model suffers substantial loss of accuracy due to its lack of embedded physics. The PIROM approach delivers online evaluations of two orders of magnitude faster than the full-order model. These results demonstrate that PIROM effectively reconciles the trade-offs between accuracy, generalizability, and efficiency, providing a promising framework for optimizing non-linear dynamical systems, such as hypersonic thermal protection systems under diverse operating conditions.</p>