OptiPINN: A two-stage NSGA-II multi-objective optimization framework for solving 2D transient non-linear heat transfer problems with coupled convection and radiation

Transient non-linear heat transfer with coupled convection, radiation, and localized heat generation is challenging due to strong non-linearities, sharp thermal gradients, and asymmetric boundary conditions. Conventional Physics-Informed Neural Networks (PINNs) typically require manual tuning of the network architecture. The adjustment of weight coefficients for various loss terms is also time-consuming and can lead to unstable convergence and limited prediction accuracy, particularly for complex two-dimensional asymmetric non-linear heat transfer problems. To address these limitations, an optimized PINNs framework, termed OptiPINN, is proposed based on a two-stage NSGA-II multi-objective optimization strategy: (i) Stage 1: NSGA-II optimizes the PINNs architecture by determining the number of hidden layers, neurons per layer, and learning rate; and (ii) Stage 2: the optimized architecture is retained, and NSGA-II is employed to adaptively determine Pareto-optimal weights for the governing equation, initial condition, and mixed boundary-condition loss terms. The resulting OptiPINN is subsequently trained using a hybrid Adam/L-BFGS strategy. Three transient two-dimensional benchmark cases involving coupled convection–radiation effects and a localized Gaussian heat source are considered, with the method of lines used as the reference solution for accuracy assessment. Compared with the baseline PINNs, OptiPINN reduces the final total training loss by 80.93%–92.25% and the mean relative L 2 error by 21.62%–64.33% across the three cases. These results demonstrate that the proposed two-stage optimization strategy improves both convergence and prediction accuracy while maintaining compact network architectures, providing an effective approach for solving complex transient non-linear heat transfer problems.