新方法让神经网络自动满足多材料界面的物理连续性，无需额外损失项。
INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities
- 用变分形式隐式处理诺伊曼边界与界面条件，避免显式约束。
- 在尖锐界面和复杂几何下，精度更高、收敛更快更平滑。
- 适合解决含混合边界条件的多材料问题，代码开源可复现。
我们提出一种新型弱形式物理信息神经网络（INI-VPINN）。该方法将诺伊曼边界条件与界面条件自然融入变分框架，无需额外损失项或多个子域网络。通过使用紧支撑权函数并结合分部积分，隐式施加通量连续性约束，确保跨材料边界的物理一致性。该方法在具有尖锐界面和复杂几何的泊松方程与拉普拉斯方程上进行了测试。结果表明，相比其他基于物理信息神经网络的方案，INI-VPINN始终表现出更高精度、更平滑且更快的收敛速度。该框架为使用神经网络求解具有复杂几何与混合诺伊曼-狄利克雷边界条件的多材料问题提供了一种通用方法。实现代码已公开于GitHub仓库。
原文摘要 · Abstract (English)
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.
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