mlsanityData to 7 Oct 2026

QuadNorm: Resolution-Robust Normalization for Neural Operators

arXiv 2605.07375 ↗First posted 8 May 2026cs.LG cs.CE cs.NAcited by 0

Not on the map. Why?

Abstract (arXiv)

Normalization layers in neural operators usually compute statistics by uniformly averaging discrete grid values, making the normalization itself discretization-dependent and thereby a source of transfer error across different resolutions or meshes. To enable discretization robustness, we introduce a quadrature normalization family that replaces existing uniform averaging in normalization layers with numerical quadrature: QuadNorm and BlendQuadNorm. On endpoint-inclusive uniform grids, the proposed quadrature moments are O(h2)-consistent across discretizations, meaning that their cross-resolution mismatch decays quadratically with grid spacing. A transfer-error bound then predicts how normalization-induced mismatch scales with both the resolution gap and network depth. The experiments show the same gap- and depth-scaling trends predicted by the transfer-error bound. On Darcy, QuadNorm delivers the best cross-resolution performance at every tested target resolution from 642 to 2562; on real-data benchmarks, Transolver with QuadNorm achieves nearly resolution-invariant transfer. The largest gains appear on nonperiodic PDEs and nonspectral architectures, where native-resolution improvements also emerge. We also validate BlendQuadNorm, which stays close to LayerNorm behavior and serves as a conservative default for periodic FNO settings. These results identify normalization as a previously overlooked source of resolution dependence in neural operators.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 39 of its 53 references that are in the map. First posted on arXiv 8 May 2026.

Builds on

39 of 53 references.

References · 39

  1. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  2. 2016Layer Normalizationcited by 5,579
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  5. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  6. 2018Group Normalizationcited by 1,514
  7. 2019Root Mean Square Layer Normalizationcited by 1,425
  8. 2016Instance Normalization: The Missing Ingredient for Fast Stylizationcited by 1,187
  9. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  10. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  11. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  12. 2016Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networkscited by 541
27 more
  1. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  2. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  3. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  4. 2022Message Passing Neural PDE Solverscited by 368
  5. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  6. 2021Choose a Transformer: Fourier or Galerkincited by 333
  7. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  8. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 299
  9. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
  10. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  11. 2021Factorized Fourier Neural Operatorscited by 237
  12. 2023Neural General Circulation Models for Weather and Climatecited by 232
  13. 2022U-NO: U-shaped Neural Operatorscited by 219
  14. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  15. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  16. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  17. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 137
  18. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 128
  19. 2024Latent Neural Operator for Solving Forward and Inverse PDE Problemscited by 76
  20. 2023Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learningcited by 57
  21. 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 56
  22. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 47
  23. 2024AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fieldscited by 38
  24. 2023Domain Agnostic Fourier Neural Operatorscited by 36
  25. 2024A Library for Learning Neural Operatorscited by 31
  26. 2024Discretization Error of Fourier Neural Operatorscited by 20
  27. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 4

Built on by

Not cited yet.