PINN-FEM: A Hybrid Approach for Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks

TL;DR

PINN-FEM is a hybrid method combining physics-informed neural networks with finite element methods to accurately and reliably enforce Dirichlet boundary conditions in solving PDEs, especially in complex domains.

Contribution

This paper introduces PINN-FEM, a novel hybrid approach that uses FEM to enforce boundary conditions strongly within PINNs, improving accuracy and convergence.

Findings

PINN-FEM outperforms standard PINNs in accuracy and robustness.

The method effectively enforces Dirichlet boundary conditions in complex geometries.

Experiments demonstrate superior convergence and reliability across various problems.

Abstract

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.