PINN-FEM: A Hybrid Approach for Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks
This addresses a specific bottleneck in PINNs for solving PDEs in complex domains, making it more reliable for industrial and scientific applications, though it is incremental as it builds on existing methods.
The paper tackled the challenge of accurately enforcing Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs), which often leads to soft enforcement and compromised convergence. The result was PINN-FEM, a hybrid approach combining PINNs with finite element methods, which outperformed standard PINN models in six experiments with superior accuracy and robustness.
Physics-Informed Neural Networks (PINNs) solve partial differential equations (PDEs) by embedding governing equations and boundary/initial conditions into the loss function. However, enforcing Dirichlet boundary conditions accurately remains challenging, often leading to soft enforcement that compromises convergence and reliability in complex domains. We propose a hybrid approach, PINN-FEM, which combines PINNs with finite element methods (FEM) to impose strong Dirichlet boundary conditions via domain decomposition. This method incorporates FEM-based representations near the boundary, ensuring exact enforcement without compromising convergence. Through six experiments of increasing complexity, PINN-FEM outperforms standard PINN models, showcasing superior accuracy and robustness. While distance functions and similar techniques have been proposed for boundary condition enforcement, they lack generality for real-world applications. PINN-FEM bridges this gap by leveraging FEM near boundaries, making it well-suited for industrial and scientific problems.