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This research introduces INI-VPINN, a novel weak-form Physics-Informed Neural Network framework that implicitly handles Neumann boundary conditions and material interfaces without requiring additional loss terms. By leveraging compact support weighting functions and integration by parts, the method achieves superior accuracy and convergence rates on complex multimaterial domains compared to existing PINN formulations. The publicly available implementation offers researchers a robust tool for solving Poisson and Laplace problems with geometric singularities using neural networks.
INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities
arXiv:2606.18032v1 Announce Type: cross 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.
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.
Preprint version. Under peer review. Code available at: this https URL
Numerical Analysis (math.NA); Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
arXiv:2606.18032 [math.NA]
(or arXiv:2606.18032v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2606.18032 arXiv-issued DOI via DataCite (pending registration)
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