  In this study, we present a novel computational framework that integrates the
finite volume method with graph neural networks to address the challenges in
Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility
of graph neural networks to adapt to various types of two-dimensional
unstructured grids, enhancing the model's applicability across different
physical equations and boundary conditions. The core innovation lies in the
development of an unsupervised training algorithm that utilizes GPU parallel
computing to implement a fully differentiable finite volume method
discretization process. This method includes differentiable integral and
gradient reconstruction algorithms, enabling the model to directly solve
partial-differential equations(PDEs) during training without the need for
pre-computed data. Our results demonstrate the model's superior mesh
generalization and its capability to handle multiple boundary conditions
simultaneously, significantly boosting its generalization capabilities. The
proposed method not only shows potential for extensive applications in CFD but
also establishes a new paradigm for integrating traditional numerical methods
with deep learning technologies, offering a robust platform for solving complex
physical problems.
