An Evaluation of Adaptive Latin Hypercube Sampling (ALHS) for PINNs in Solving 1D Convection–Diffusion Problems

Solving the convection-diffusion equation with high Peclet numbers presents a significant challenge for Physics-Informed Neural Networks (PINNs) due to the formation of sharp boundary layers. Traditional static sampling strategies often struggle with numerical instability and physical inconsistency. This study proposes an Adaptive Latin Hypercube Sampling (ALHS) strategy designed to enhance the robustness of PINNs training. By dynamically redistributing collocation points based on the local residual, the ALHS method effectively addresses what we identify as the Loss-MSE Paradox- a phenomenon where low global loss values fail to guarantee physical accuracy. Our results, based on 20 independent simulations for an advection-dominated regime D=0.001, demonstrate that while static methods exhibit high variance, ALHS achieves superior statistical stability with a Final MSE of 1.915´10-4 and a significantly lower standard deviation of 1.211´10-4. Furthermore, qualitative analysis confirms that ALHS accurately captures the boundary layer at t=1.0 without the non-physical oscillations prevalent in Monte Carlo or standard LHS approaches.