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

Abstract

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 $12\times$ 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.