Learning Inside Finite-Volume Computations with Effective Interface States

Finite-volume methods provide a powerful framework for physical simulation by expressing conservation laws through local exchanges on a mesh. Building directly on their reconstruction–flux–update structure, we introduce the Finite-Volume Graph Neural Operator (FV-GNO), which learns the interface approximation within a prescribed numerical update rather than the full state transition. A graph network decodes effective two-sided interface states, while prescribed numerical fluxes, source treatment, and geometric assembly advance the solution. Trained end-to-end on state observations alone, these interfaces are optimized for prediction over the observation interval, without requiring interface or flux labels. The paired formulation inherits local balance for arbitrary network parameters, while a constructive analysis illustrates how retaining known nonlinear computation can provide a representation advantage beyond conservation alone. Experiments across diverse nonlinear equations on regular and irregular meshes demonstrate improved prediction over the evaluated neural baselines. On regular grids, FV-GNO also achieves higher accuracy and lower rollout cost than the tested second-order finite-volume schemes. Coarse-grid pretraining further enables accurate fine-grid prediction with few target trajectories.