INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities

2026-06-16

AI Summary

This work addresses the challenge of solving partial differential equations in multi-material domains featuring geometric singularities and mixed Neumann–Dirichlet boundary conditions. The authors propose a novel variational physics-informed neural network, termed INI-VPINN, which implicitly embeds Neumann boundary conditions and enforces flux balance and continuity constraints at material interfaces through a variational formulation incorporating integration by parts and compactly supported weight functions. This approach eliminates the need for additional loss terms or multiple subdomain networks. Numerical experiments on Poisson and Laplace problems demonstrate that INI-VPINN significantly outperforms existing PINN methods in accuracy, solution smoothness, and convergence rate, while rigorously preserving physical consistency across material boundaries.

Abstract

We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN). INI-VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks-based formulations, the INI-VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

Problem

• multi-material domains
• geometric singularities
• Neumann boundary conditions
• interface handling
• physics-informed neural networks

Innovation

• Variational Physics-Informed Neural Network
• Implicit Neumann condition
• Interface handling
• Multi-material domains
• Geometric singularities