QuadNorm: Resolution-Robust Normalization for Neural Operators

TL;DR

QuadNorm introduces a resolution-robust normalization method for neural operators, reducing transfer errors across different discretizations and improving performance on PDE problems.

Contribution

The paper proposes a quadrature-based normalization technique, QuadNorm, that enhances discretization robustness and reduces resolution dependence in neural operators.

Findings

QuadNorm achieves $O(h^2)$ consistency across discretizations.

Experiments confirm predicted transfer-error scaling with resolution gap and network depth.

QuadNorm improves cross-resolution performance on PDE benchmarks.

Abstract

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.