Novel Method to Improve the Convergence of Physics-Informed Neural Networks for Complex Thermal Simulations

In the context of developing PINN methods for real-time digital twins in manufacturing processes, we propose a new approach that combines two complementary weighting strategies to significantly improve their convergence. The first method, called SD-PINN, balances the loss terms associated with the governing equations, boundary conditions, and initial conditions, ensuring that their contributions are dimensionally consistent and therefore comparable in magnitude. The second method, called SDFEET-PINN, rescales the terms of the governing equations during the early stages of training. This facilitates learning by temporarily modifying the equations to make terms comparable in amplitude, and then progressively restoring the original formulation, thereby preserving the influence of lower-magnitude terms that are often neglected in standard PINN approaches. We apply these methods to transient thermal problems, which are critical for predicting defects in Powder Bed Fusion (PBF). A range of 2D configurations with complex boundary conditions is used to test robustness, and a practical case study is carried out on heat transfer in a complex 3D geometry previously investigated both numerically and experimentally in PBF. Results show that the combined SD-PINN and SDFEET-PINN approach achieves higher predictive accuracy and stability compared to classical PINNs. Furthermore, we introduce an Adaptive Learning Rate strategy that reduces the step size after initial stabilization, further enhancing predictive performance and enabling efficient convergence across all test cases.