🤖 AI Summary
This work addresses key limitations of conventional physics-informed neural networks (PINNs)—including slow convergence, sensitivity to loss weighting, and difficulty in accurately enforcing boundary conditions—by introducing a unified hard-soft PINN (HSPINN) framework. The method explicitly enforces Dirichlet and periodic boundary conditions as hard constraints through analytical or polynomial lifting, masking functions, and periodic feature mappings. Meanwhile, the PDE residuals, Neumann fluxes, and initial conditions are incorporated as soft constraints. To eliminate manual hyperparameter tuning, an inverse shared Softmax strategy is employed to adaptively balance the weights of all loss components. The proposed approach demonstrates significantly improved convergence speed, accuracy, and numerical stability across elliptic (Poisson), parabolic (Burgers), and hyperbolic (periodic convection) problems without requiring user-specified loss weights.
📝 Abstract
Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.