Limits of Resolution Equivariance in Fourier Neural Operators

Sparse representations are often expected to make models smaller and also reduce inference cost. For Fourier Neural Operators (FNOs), these objectives are not equivalent or do not always align: removing parts of the learned operator can leave the underlying transforms and dense computations unchanged, while changing the grid on which the model is evaluated can introduce overhead of its own. We therefore distinguish sparsity in the representation, in the stored parameters, in the theoretical operation count, and in measured runtime, and present an empirical study of several routes toward sparse FNOs that tests each transition between them separately. Coarsening the execution grid reduces the theoretical cost without reducing measured latency, and adding a correction term recovers accuracy at the cost of making the model slower. Even an 83% parameter reduction remains slower than the dense baseline under ordinary execution. These results motivate a stricter definition of useful sparsity: the deployed operator must preserve solution accuracy and map its reduced support to a genuinely cheaper execution path.

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.

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 $O(h^2)$-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 $64^2$ to $256^2$; 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.