Exact SO(3)-Equivariant Isotropic Kernels for Rotation-Robust Neural Dynamics

Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within the model are Fourier layers, which apply linear transformations directly to the Fourier modes, with parameters depending on the wave numbers. However, most physical systems are isotropic, with the results being independent of the coordinate system chosen, but the linear transformations do not necessarily respect these symmetries. We propose a modification to the linear transformations that ensures spatial symmetries are respected, called the Isotropic Fourier Neural Operator, which both improves model performance and reduces the number of parameters by up to a factor of 16 in 2D and 96 in 3D.

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

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.