This work investigates the integration of geometric inductive bias within Physics-Informed Neural Networks (PINNs) by utilizing Graph Neural Network (GNN) architectures. The primary objective is to combine the structural advantages of graphs—which naturally represent irregular domains and non-structured meshes—with the capability of PINNs to be trained without labelled data through the minimization of partial differential equation (PDE) residuals. The methodology focuses on evaluating the feasibility of this "Graph-PINN" approach for solving steady-state problems, such as the Poisson and Helmholtz equations, on complex 2D geometries. Unlike traditional meshless PINNs that often struggle with spatial point distribution and long-range dependencies, the proposed framework exploits the mesh topology as a graph. This structural prior allows the model to enforce local physical constraints more effectively through message-passing mechanisms. Specifically, this study compares two distinct paradigms: the GPINN [1], representing a "PINN with geometric bias" through graph embeddings and strong-form residuals, and the PI-GGalerkinN [2], framed as a "Graph with physics bias" constrained by the Galerkin variational principle. A comparative performance analysis will be conducted to evaluate how these distinct inductive biases influence convergence and accuracy. Numerical benchmarks derived from standard finite element formulations will provide the necessary ground truth for validation across irregular domains. This investigation seeks to determine whether augmenting continuous solvers with geometric embeddings or constraining discrete graph solvers with variational physics offers the most robust path for mesh-based deep learning in computational mechanics. REFERENCES [1] Miao, Y., Li, H., & Mandic, D. (2024). GPINN: Physics-Informed Neural Network with Graph Embedding. 2024 International Joint Conference on Neural Networks (IJCNN), 1–8. https://doi.org/10.1109/IJCNN60899.2024.10651053. [2] Gao, H., Zahr, M. J., & Wang, J. X. (2022). Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 390. https://doi.org/10.1016/j.cma.2021.114502.