Hidden-State Transfer for Multi-Fidelity Mesh PDE Surrogates via Coarse Propagation and Fine Decoding

High-resolution mesh PDE surrogates face a mismatch between geometric resolution and the resolution required for physical propagation. Fine meshes are often necessary to represent boundaries and localized structures, while global communication over all fine nodes can introduce substantial computational cost. This raises a key question in multi-fidelity surrogate modeling: what information should be transferred across fidelity levels to reconstruct fine-resolution fields? We introduce MUFI, a coarse-propagation/fine-decoding framework that studies hidden-state transfer across mesh resolutions. MUFI runs an existing mesh surrogate backbone on a coarsened mesh, transfers its intermediate hidden representations through a cached geometry-aware sparse operator, and reconstructs the solution on the original mesh using a FiLM-conditioned neural field decoder. Unlike coarse field transfer approaches that only propagate predicted outputs, MUFI preserves pre-readout backbone states generated during physical propagation. A matched coarse-field transfer control demonstrates that hidden-state transfer provides additional reconstruction benefits beyond coarse propagation and decoder capacity alone. Across CFD and CSD benchmarks from the PLAID datasets, multiple mesh surrogate backbones, and a varying-geometry surface benchmark, MUFI improves the accuracy-cost trade-off and provides controllable operating points through fidelity selection. These results indicate that the choice of transferred representation is an important design factor in multi-fidelity mesh surrogate learning.