On the under-reaching phenomenon in message passing neural PDE solvers: Revisiting the CFL condition

• This paper proposes sharp lower bounds for the number of message passing iterations required in graph neural networks (GNNs) when solving partial differential equations (PDE). • For systems governed by a hyperbolic PDE, it is shown that it is sufficient for the message to be ahead of the wave that characterises these systems, in the style of the CFL condition for finite differences. • In elliptic and parabolic systems, on the other hand, the phenomenon of infinite-speed information transmission occurs, which forces the number of passes to be at least the number necessary for the message to reach the furthest corner of the mesh. We propose a sharp lower bound for the number of message passing iterations that guarantees the stability of the simulations for hyperbolic, elliptic and parabollic problems. (a) Our proposed lower bound is depicted with the dashed line. Below this value, see (b), predictions rapidly degenerate, while above the predicted lower bound, see (c), the stability of the simulation is guaranteed. This paper proposes sharp lower bounds for the number of message passing iterations required in graph neural networks (GNNs) when solving partial differential equations (PDE). This significantly reduces the need for exhaustive hyperparameter tuning. Bounds are derived for the three fundamental classes of PDEs (hyperbolic, parabolic and elliptic) by relating the physical characteristics of the problem in question to the message-passing requirement of GNNs. In particular, we investigate the relationship between the physical constants of the equations governing the problem, the spatial and temporal discretisation and the message passing mechanisms in GNNs. When the number of message passing iterations is below these proposed limits, information does not propagate efficiently through the network, resulting in poor solutions, even for deep GNN architectures. In contrast, when the suggested lower bound is satisfied, the GNN parameterisation allows the model to accurately capture the underlying phenomenology, resulting in solvers of adequate accuracy. Examples are provided for four different examples of equations that show the sharpness of the proposed lower bounds.