AMR-VNO: A Native-Leaf Surrogate for Block-Structured Adaptive Meshes

Adaptive mesh refinement (AMR) solvers store solutions on nested levels of cells, yet most grid-based neural operators first rasterize these fields onto a uniform grid. We introduce AMR-VNO (AMR V-cycle Neural Operator), whose computational graph is the solver's block-AMR hierarchy, extended where needed by virtual coarse levels built from the inputs. It exchanges direction-aware messages within each level, moves information between levels through learned restriction and prolongation shared across levels, and predicts directly on active leaf cells. On the evaluated held-out 2D Darcy and advection-diffusion sets, AMR-VNO has 14 to 20% lower mean error against the solver's discrete targets than a tuned multigrid neural operator, with about 34 times fewer learned parameters, and lower error than graph and sparse-convolution models given the same hierarchy. It also improves over same-hierarchy message passing on in-distribution 3D conduction. On public Quokka turbulence, AMR-VNO and hierarchy message passing both outperform dense baselines but are not statistically distinguishable. In Darcy validation ablations, direction-aware aggregation and a second traversal each reduce error. Accuracy degrades on deeper hierarchies and changed refinement policies, so sharing weights across levels does not by itself give transfer across hierarchy structure.