MeshFiLM: Fast Physics Surrogates on Unstructured Meshes via Node-Wise Modulation

Neural surrogates for physics on unstructured meshes usually exchange information between nodes in every layer, by message passing along mesh edges or through a small set of slice tokens. We study how far a surrogate can go when this exchange happens once. MeshFiLM embeds every node independently, pools the embeddings once into a context vector for each node, and decodes the fields with a pointwise MLP whose hidden features are modulated in every layer by feature-wise linear modulation, computed from the node's own embedding and its context. We compare two pooling operators: statistics pooling gives every node the same summary, and routed pooling lets each node read its own mixture of learned region summaries. On five PLAID benchmarks and the surface and volume fields of AhmedML, we compare models in one training pipeline using the same node features and hardware. MeshFiLM trains $4.9$-$8.4\times$ faster per epoch than the fastest baseline on PLAID and $1.7$-$2.6\times$ faster on AhmedML, with fewer parameters. At equal epochs, its routed variant has lower mean error than the strongest baseline in four of seven settings and comparable error in a fifth. In four PLAID settings with equal-time runs, MeshFiLM-R has $21$-$64$% lower error than the strongest baseline on three; on the fourth, both MeshFiLM variants remain about $4$% higher.