mlsanityData to 14 Oct 2026

Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes

arXiv 2610.20585 ↗First posted 14 Oct 2026cs.LG cs.CVcited by 0

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Abstract (arXiv)

Current approaches for steady partial differential equations surrogate models on unstructured meshes apply a single mixing mechanism to the whole solution. This paper proposes an architecture that partitions the solution among vanilla transformer experts, each focused on a different spectral scale. The partition is built from a discrete Hodge Laplacian L0=star0-1 d0topstar1 d0, that serves as a fitting layer measuring high-frequency content. A gating mechanism takes the mesh as input and produces the per-vertex weights that recombine the expert outputs. AirFRANS is a steady Reynolds-averaged Navier--Stokes (RANS) airfoil benchmark. We evaluate our approach in a reduced remeshed version of the benchmark with each unstructured mesh counting 9.6K faces and 4.9K vertices. We compare our approach with the Transolver on the same dataset and dataset slit. Our approach reduces relative L2 error by 31% and interior mean squared error by 47%.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 30 of its 42 references that are in the map. First posted on arXiv 14 Oct 2026.

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References · 30

  1. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,248
  2. 2017Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layercited by 3,509
  3. 2021Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsitycited by 3,073
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  5. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  6. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  7. 2020Learning Mesh-Based Simulation with Graph Networkscited by 868
  8. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  9. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  10. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  11. 2019Machine Learning for Fluid Mechanicscited by 385
  12. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
18 more
  1. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 326
  2. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 303
  3. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  4. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  5. 2020DiffusionNet: Discretization Agnostic Learning on Surfacescited by 164
  6. 2018Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flowscited by 150
  7. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 138
  8. 2023Scalable Transformer for PDE Surrogate Modelingcited by 128
  9. 2018Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplaciancited by 86
  10. 2022AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutionscited by 82
  11. 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 62
  12. 2025AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformerscited by 53
  13. 2023Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEscited by 18
  14. 2020DiffGCN: Graph Convolutional Networks via Differential Operators and Algebraic Multigrid Poolingcited by 17
  15. 2026Structure-Preserving Learning Improves Geometry Generalization in Neural PDEscited by 8
  16. 2026Topological Neural Operatorscited by 4
  17. 2025HodgeFormer: Transformers for Learnable Operators on Triangular Meshes through Data-Driven Hodge Matricescited by 3
  18. 2026Topology-Preserving Neural Operator Learning via Hodge Decompositioncited by 3

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