Physics-Aligned Canonical Equivariant Fourier Neural Operator under Symmetry-Induced Shifts

2026-05-18

AI Summary

Neural operators struggle to simultaneously model coordinate alignment and physical evolution in out-of-distribution scenarios, leading to degraded generalization. To address this, this work proposes the PACE-FNO framework, which explicitly decouples symmetry alignment from physical modeling by mapping input fields to a reference coordinate system, applying a standard Fourier Neural Operator (FNO) for prediction, and transforming the result back to the target coordinates. The approach enforces equivariance solely through input/output coordinate transformations without altering the FNO backbone, and further incorporates a Lie algebra–based coordinate estimator, symmetry-aware perturbation training, and a low-dimensional frame refinement strategy during inference. Evaluated on various periodic-domain PDE tasks, PACE-FNO maintains in-distribution accuracy while reducing out-of-distribution relative error by up to 12× compared to FNO with data augmentation.

Abstract

Neural operators approximate PDE solution maps, but they need not respect the symmetries of the governing equation. In out-of-distribution (OOD) regimes, a standard neural operator must often learn coordinate alignment and physical evolution within a single map, which can hurt generalization. We use known continuous symmetries of evolution equations on periodic domains to separate these two roles. We propose the Physics-Aligned Canonical Equivariant Fourier Neural Operator (PACE-FNO), which estimates the input frame with a Lie-algebra coordinate estimator, maps the field to a reference frame, applies a standard Fourier Neural Operator (FNO), and restores the prediction to the target frame. We train alignment and operator prediction jointly using bounded symmetry perturbations, with an optional low-dimensional refinement step that updates the estimated frame at inference. Equivariance is enforced by the input and output transformations, while the FNO architecture remains unchanged. Across 1-D and 2-D Burgers, shallow-water, and Navier-Stokes equations on periodic domains, PACE-FNO matches the in-distribution (ID) accuracy of standard neural operators and reduces out-of-distribution (OOD) relative error by up to 12x over FNO with symmetry augmentation (FNO+Aug) under translations and Galilean shifts, with smaller gains for coupled rotation-translation shifts. Ablations show that aligning the input and restoring the output frame account for most OOD gains; inference-time refinement provides a smaller correction.

Problem

• neural operators
• symmetry
• out-of-distribution generalization
• equivariance
• PDE solution maps

Innovation

• equivariant neural operator
• symmetry-induced shifts
• Fourier Neural Operator
• physics-aligned learning
• Lie-algebra coordinate estimation