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 L2 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.