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

Problem
The paper addresses the limitations of existing Physics-Informed Neural Networks (PINNs) in handling Neumann boundary conditions and interface management in multi-material domains, particularly in the presence of geometric singularities. Current methods often require additional loss terms or separate networks for different subdomains, complicating the implementation and reducing efficiency. This work presents a preprint, indicating that it has not yet undergone peer review.

Method
The authors propose the INI-VPINN framework, which utilizes a weak-form variational approach to seamlessly incorporate Neumann boundary and interface conditions. The architecture employs compact support weighting functions and integration by parts to impose flux and continuity constraints implicitly. This design eliminates the need for additional loss terms or multiple subdomain networks, enhancing computational efficiency. The training process leverages standard optimization techniques, although specific compute resources are not disclosed. The method is validated on Poisson and Laplace equations, focusing on scenarios with sharp interfaces and complex geometries.

Results
INI-VPINN demonstrates superior performance compared to several existing PINN formulations. In numerical experiments, it achieves higher accuracy and faster convergence rates on benchmark problems involving multi-material domains. While specific quantitative results are not detailed in the abstract, the authors claim that the INI-VPINN consistently outperforms its competitors across various test cases, indicating a significant improvement in modeling capabilities for complex geometries and mixed boundary conditions.

Limitations
The authors acknowledge that the current implementation may have limitations in scalability and generalizability to more complex physical systems beyond the tested Poisson and Laplace problems. They do not address potential challenges related to the choice of compact support functions or the implications of varying material properties on the model’s performance. Additionally, the lack of peer review may mean that the proposed method requires further validation in diverse applications.

Why it matters
The INI-VPINN framework represents a significant advancement in the application of neural networks to solve complex physical problems, particularly in multi-material domains with intricate geometries. By simplifying the incorporation of boundary conditions, it opens avenues for more efficient simulations in engineering and physics, potentially impacting fields such as materials science and fluid dynamics. The public availability of the implementation on GitHub enhances accessibility for researchers and practitioners, facilitating further exploration and application of this method. This work contributes to the ongoing development of robust and efficient neural network-based solutions for real-world problems, as published in arXiv cs.LG.