Physics-informed neural network with variable scaling method for radiative-conductive heat transfer problems

Radiative–conductive coupled heat transfer in participating media is a fundamental problem in high-temperature engineering. Physics-informed neural networks offer a mesh-free framework for forward prediction and inverse parameter identification, yet their application to radiative–conductive coupling remains limited, largely due to training instability induced by stiffness. Here, we analyze the stiffness mechanism and show that, under typical high-temperature conditions, the dimensionless governing equations can contain coefficients differing by 10²–10³ in magnitude, which leads to severe residual imbalance during optimization. To address this issue, we propose a scaling strategy that rescales the dimensionless spatial coordinate to reduce coefficient disparity and improve training robustness. Seventeen benchmark cases representative of classical engineering conditions are considered for both forward and inverse problems. The conventional PINN and the proposed scaled PINN are evaluated against Finite-Element Method reference solutions. For the forward problem, the scaled PINN reduces the final training loss, the maximum temperature error, and the maximum radiative heat-flux error by at least one order of magnitude, and by up to four orders in the most challenging cases. For the inverse problem, the scaled PINN enables accurate retrieval of the extinction coefficient and single-scattering albedo from sparse temperature data, achieving identification errors as low as 3.68%. These results suggest that the primary role of scaling is to keep the effective coefficients of residual terms at O(10), thereby alleviating stiffness and improving convergence. Overall, the proposed scaled PINN provides an efficient and physics-consistent approach for forward and inverse radiative–conductive problems across stiff regimes.