Graph neural networks (GNNs) have established themselves as powerful tools for leveraging the geometric relationships encapsulated in data that rely on spatial discretizations[1]. This approach has significantly enhanced the universal predictor function inherent in neural networks. However, the mechanisms associated with message passing within GNNs can influence their performance and hinder their interpretability. We study different partial differential equations with the goal of understanding how the physical propagation of information is linked to the computational message passing within the graph processor. Therefore, the relationship between message passing and the predictive capacity of a GNN for an hyperbolic differential equation is studied. The proposed network demonstrates a high capacity to approximate the numerical solution without requiring prior knowledge of the problem’s nature or boundary conditions. However, its simple architecture allows us to delve into the learning and attention mechanisms, as well as define metrics that, similarly to the Courant–Friedrichs–Lewy condition (CFL) [2], indicate whether the network is capable of converging to an approximation that adequately reflects the defined behavior. The results of this work highlight the crucial importance of hyperparameterizing the number of steps in the processor, emphasizing that message passing is the primary mechanism that determines the network’s performance. This adjustment must be considered based on the specific problem and the type of differential equation being addressed. REFERENCES [1] Q. Hernández, A. Badías, F. Chinesta and E. Cueto, Thermodynamics-Informed Graph Neural Networks, IEEE Transactions on Artificial Intelligence, 5, 3, 967-976, (2024). [2] Gnedin, N.Y., Semenov, V.A. and Kravtsov, A.V. Enforcing the Courant-Friedrichs-Lewy Condition in Explicitly Conservative Local Time Stepping Schemes. Journal of Computational Physics 359, 93–105, (2018).