mlsanityData to 14 Oct 2026

PACE-FNO: Physics-Aligned Canonical Equivariance for Fourier Neural Operators

arXiv 2605.18606 ↗First posted 18 May 2026cs.LGcited by 0

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Abstract (arXiv)

Neural operators are often tested on states that differ physically from training data. A distinct failure occurs when the physical dynamics are unchanged but the observed coordinate frame differs from training. PACE-FNO addresses this case by estimating the frame, predicting after pulling the field to a canonical representative, and restoring the requested terminal frame. The default inference path uses one forward prediction; optional test-time adaptation (TTA) updates only the low-dimensional coordinate. On translated and Galilean-shifted Burgers and shallow-water systems, PACE-FNO lowers out-of-distribution (OOD) relative error by up to 12× relative to a data-augmented Fourier Neural Operator (FNO+Aug). A matched-estimator control supports attributing the gain to prediction in the canonical frame rather than added estimator capacity. Our analysis separates the canonical approximation error from the two alignment residuals. We also test regional WeatherBench2 time-series forecasting with the fifth-generation ECMWF reanalysis (ERA5) under natural time OOD, where PACE-FNO and FNO achieve nearly identical performance. Experiments with approximate rotation, other backbones, irregular domains, rollouts, and image data delineate the conditions under which the mechanism is effective.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 20 of its 33 references that are in the map. First posted on arXiv 18 May 2026.

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20 of 33 references.

References · 20

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2016Group Equivariant Convolutional Networkscited by 1,165
  3. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  4. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 539
  5. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  6. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
  7. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 326
  8. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
  9. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 168
  10. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 126
  11. 2022Equivariance with Learned Canonicalization Functionscited by 109
  12. 2022Clifford Neural Layers for PDE Modelingcited by 92
8 more
  1. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 77
  2. 2018Explorations in Homeomorphic Variational Auto-Encodingcited by 65
  3. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 51
  4. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 45
  5. 2024Equivariant Frames and the Impossibility of Continuous Canonicalizationcited by 43
  6. 2024A Canonicalization Perspective on Invariant and Equivariant Learningcited by 27
  7. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 24
  8. 2024Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEscited by 16

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