Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving
This study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting to address the slow convergence and inaccurate boundary enforcement of conventional PINNs. The framework enforces Dirichlet and periodic boundary conditions exactly through analytical lifting or masking, while treating PDE residuals and initial conditions as soft constraints balanced by an inverse-share softmax strategy.