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

Nahil Sobh, Rini Jasmine Gladstone, Hadi Meidani

TL;DR

The authors address the difficulty of enforcing Dirichlet boundary conditions in Physics-Informed Neural Networks by proposing PINN-FEM, a hybrid method that uses a boundary-focused finite element mesh to enforce essential BCs exactly while solving the interior with a PINN. The method relies on a domain decomposition and a loss function based on the principle of minimum potential energy, ensuring stability and seamless coupling at the interface between FE and PINN regions. Through six elasticity-based experiments with increasing geometric and boundary-condition complexity, PINN-FEM consistently outperforms standard PINNs with soft enforcement and other exact-BC PINN variants, particularly in scenarios with discontinuous or point boundaries and cracks. The approach demonstrates strong potential for industrial applications where accurate BC enforcement is critical and suggests a generalizable framework for combining FE rigor with PINN flexibility.

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.

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

TL;DR

The authors address the difficulty of enforcing Dirichlet boundary conditions in Physics-Informed Neural Networks by proposing PINN-FEM, a hybrid method that uses a boundary-focused finite element mesh to enforce essential BCs exactly while solving the interior with a PINN. The method relies on a domain decomposition and a loss function based on the principle of minimum potential energy, ensuring stability and seamless coupling at the interface between FE and PINN regions. Through six elasticity-based experiments with increasing geometric and boundary-condition complexity, PINN-FEM consistently outperforms standard PINNs with soft enforcement and other exact-BC PINN variants, particularly in scenarios with discontinuous or point boundaries and cracks. The approach demonstrates strong potential for industrial applications where accurate BC enforcement is critical and suggests a generalizable framework for combining FE rigor with PINN flexibility.

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.
Paper Structure (21 sections, 45 equations, 3 figures, 8 tables)

This paper contains 21 sections, 45 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Physics-informed neural networks
  4. Method
  5. Formulation of PINN-FEM Method
  6. One-dimensional boundary value problem
  7. Dirichlet BC at one end
  8. Dirichlet BC at both the ends
  9. Two- and three-dimensional boundary value problems
  10. Numerical Results
  11. Mesh Generation and Collocation Points
  12. Baseline models
  13. Experiments
  14. Square plate with constant stress
  15. Square plate with a circular hole
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Domain Decomposition with complementary Neumann and Dirichlet boundary conditions at the two ends for a one-dimensional boundary value problem and the corresponding exact BC imposition using the proposed PINN-FEM method.
  • Figure 2: Domain Decomposition with Dirichlet boundary conditions at both the ends for a one-dimensional boundary value problem and the corresponding exact BC imposition using the proposed PINN-FEM method.
  • Figure 3: Domain decomposition for the proposed PINN-FEM method for a two-dimensional boundary value problem.