QuadNorm: Resolution-Robust Normalization for Neural Operators

Fourier Neural Operators are often assumed to generalize across spatial resolutions, enabling training on a coarse grid and deployment on a finer grid. We test this assumption by contrasting two inference-time choices when moving from training resolution $s$ to test resolution $S>s$: running FNO directly at $S$, or running at $s$ and upsampling the prediction to $S$ via Fourier zero-padding. On Darcy flow, we observe that direct fine-grid inference is not reliably beneficial and can be worse than the low-grid-plus-upsampling baseline. We further analyze layerwise spectra and find that, under Fourier truncation, intermediate representations increasingly concentrate energy in low frequencies, with high-frequency output produced mainly by late nonlinear/decoder stages. This offers a mechanistic explanation for why FNO can perform well while retaining few modes, yet remain sensitive under resolution shifts. Our findings highlight a simple but strong baseline for cross-resolution evaluation and point to nonlinear aliasing as a key obstacle to zero-shot resolution equivariance.

Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

Pre-norm is the standard normalization placement in modern Transformers because it facilitates joint optimization of full-depth models. We ask whether this preference persists when depth is introduced through a curriculum. In curriculum depth growth, each appended block receives the boundary representation produced by a trained prefix, making normalization placement relevant to forward conditioning. We therefore test whether placement and training curriculum interact. In a controlled distillation study with a Qwen3-8B teacher and a nine-layer student, pre-norm and post-norm are indistinguishable under joint training, differing by $0.0004$ validation CE, while post-norm improves over pre-norm by $0.0328$ under curriculum growth, an order of magnitude larger. A post-joint control matched by student active-layer tokens remains worse than post-grow, which rules out compute as the sole explanation. The ranking crosses over during the curriculum: post-norm takes the lead once blocks are appended. Single-block and freeze controls localize the ranking change to block appending rather than shallow-block quality or retraining. Boundary diagnostics associate post-norm with stable residual scales and pre-norm with structural-token scale drift; on a fixed batch, the final pre-grow block is also nearly identity-mapped. Together with the phase-wise crossover, these observations are consistent with boundary-scale conditioning after new blocks are appended. The results motivate treating normalization placement and training curriculum as coupled design choices in this distillation setting.