Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving

The study introduces the hard-soft physics-informed neural network (HSPINN) framework, which enhances the conventional PINN approach by enforcing Dirichlet and periodic boundary conditions exactly while treating other constraints as soft. Key features include adaptive loss weighting through an inverse-share softmax strategy, which improves convergence speed and stability, and applications to elliptic, parabolic, and hyperbolic PDEs show significant improvements in accuracy and robustness. This advancement provides a more efficient and reliable method for practitioners solving boundary-constrained PDEs in various scientific and engineering contexts.