QuadNorm: Resolution-Robust Normalization for Neural Operators

2026-05-08

AI Summary

This work identifies normalization layers in neural operators as a primary source of resolution dependence, owing to their reliance on uniform-grid averaging, which introduces significant errors under cross-resolution settings or non-uniform discretizations. To address this, the authors propose QuadNorm and its blended variant BlendQuadNorm—novel normalization schemes that replace uniform averaging with high-order (O(h²)) consistent quadrature rules based on numerical integration. These methods achieve robustness across arbitrary discretization schemes and are applicable to non-periodic PDEs and non-spectral architectures. Evaluated on Darcy flow problems and real-world datasets, the proposed approaches substantially improve cross-resolution transfer performance while simultaneously enhancing accuracy at the original resolution, thereby approaching resolution invariance.

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.

Problem

• normalization
• neural operators
• resolution robustness
• discretization dependence
• transfer error

Innovation

• QuadNorm
• neural operators
• resolution robustness
• numerical quadrature
• normalization