Comparative analysis of finite element method and physics-informed neural network for solving 3D electrostatics problems

Abstract

Accurate three-dimensional (3D) electrostatic (ES) simulation is crucial but computationally challenging using traditional finite element methods (FEM) due to meshing bottlenecks. This study aims to systematically evaluate physics-informed neural networks (PINNs) against FEM for solving 3D ES problems. Two 3D benchmark problems were solved: a homogeneous cubic domain governed by Laplace’s equation, and a multi-material domain with discontinuous properties. Both methods were evaluated to assess accuracy, computational efficiency and adaptability to geometric and material complexities. FEM demonstrated robustness in well-meshed domains, achieving a −4.8 dB mean error, though mesh generation created computational bottlenecks. PINNs achieved comparable accuracy without meshing, yielding −5.7 dB error in the homogeneous case and −7.0 dB in the multi-material case. PINNs excelled at enforcing interface continuity in multi-material problems, though they required careful loss function tuning. FEM remains superior for static, well-defined problems, whereas PINNs offer significant advantages for mesh-free environments and inverse scenarios. The study proposes hybrid methodologies combining FEM’s reliability with PINNs’ flexibility to advance 3D electromagnetic simulations.