QuadNorm: Resolution-Robust Normalization for Neural Operators

• URL: http://arxiv.org/abs/2605.07375v1
• Date: Fri, 08 May 2026 07:30:43 GMT
• Title: QuadNorm: Resolution-Robust Normalization for Neural Operators
• Abstract summary: We introduce a quadrature normalization family that replaces existing uniform averaging in normalization layers with numerical quadrature: QuadNorm and BlendQuadNorm.<n>On endpoint-inclusive uniform grids, the proposed quadrature moments are $O(h2)$-consistent across discretizations.<n>A transfer-error bound then predicts how normalization-induced mismatch scales with both the resolution gap and network depth.
• Score: 41.429437803093485
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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