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.

翻译：神经算子中的归一化层通常通过均匀平均离散网格值来计算统计量，这使得归一化本身具有离散化依赖性，从而成为跨不同分辨率或网格传递误差的来源。为实现离散化鲁棒性，我们提出了一族求积归一化方法，用数值求积替代归一化层中现有的均匀平均：QuadNorm和BlendQuadNorm。在包含端点的均匀网格上，所提出的求积矩在离散化间具有$O(h^2)$一致性，即跨分辨率失配随网格间距呈二次衰减。传递误差界进而预测了由归一化引起的失配如何随分辨率差距和网络深度变化。实验显示了与传递误差界预测相同的差距和深度缩放趋势。在Darcy问题上，QuadNorm在从$64^2$到$256^2$的每个测试目标分辨率下均实现了最佳跨分辨率性能；在真实数据基准测试中，结合QuadNorm的Transolver几乎实现了分辨率不变的传递。最大的增益出现在非周期PDE和非谱架构上，其原生分辨率改进也同时显现。我们还验证了BlendQuadNorm，其行为接近LayerNorm，可作为周期FNO设置中的保守默认选择。这些结果确定了归一化是神经算子中此前被忽视的分辨率依赖性来源。