Neural operators are often reported to exhibit zero-shot super-resolution, a phenomenon in which a model trained on coarse grids produces accurate predictions on finer testing grids without additional retraining. Despite strong empirical evidence, the theoretical foundations of this phenomenon remain unclear. In this work, we provide a systematic theoretical study of zero-shot super-resolution in operator learning. We first show that zero-shot super-resolution can be information-theoretically impossible even in benign settings such as when the input functions are available over the entire continuum and the ground truth is a simple rank-one linear operator. We then identify H{" o}lder smoothness of the output functions as a sufficient condition for zero-shot super-resolution and derive corresponding generalization bounds. Finally, we also validate the identified failure modes through experimental results.
Single-Image Super-Resolution (SISR) aims to reconstruct a High-Resolution (HR) image from a Low-Resolution (LR) observation, a fundamentally ill-posed problem where high-frequency details are severely degraded at large upscaling factors. Recent advances have been driven by transformer-based architectures and diffusion models improve global context modeling and perceptual quality at the cost of increased computational complexity. In contrast, this work focuses on enhancing the expressivity of local operators under minimal resources. We propose SRGAN--CKAN, a hybrid super-resolution framework that integrates Convolutional Kolmogorov--Arnold Networks (CKAN) into an adversarial learning setting reformulating convolution as a nonlinear patch-based transformation. The proposed operator replaces linear local mappings with spline-based functional representations, allowing expressive modeling of complex local structures and high-frequency textures using minimal hardware resources. Experimental results demonstrate that the proposed approach improves perceptual quality while preserving reconstruction fidelity, achieving a favorable balance between distortion-based and perceptual metrics. These results are obtained under constrained computational settings, highlighting the efficiency of the proposed formulation. Overall, this work introduces a complementary direction to existing approaches by improving the representational power of local transformations, providing an efficient and scalable alternative to globally intensive architectures.
Roberto Isai Navaro-Aviña, Eduardo Said Merin-Martinez, Andres Mendez-Vazquez +1
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 learn mappings between function spaces, but are typically developed with dense input-output training fields and fully observed inputs at inference. Many scientific problems require instead predicting solution fields from sparse, irregular, or partial observations under uncertainty. We introduce Neural Operator Processes (NOPs), a framework that unifies neural-process conditioning with neural-operator decoding to predict full output fields from limited context. NOPs condition on sparse joint input-output observations and support deterministic and probabilistic prediction within a shared encoder-decoder architecture. We study two conditioning strategies, convolutional pooled summaries and query-aligned attention, and analyze how their interaction with latent stochastic variables depends on PDE geometry. Across function regression and three PDE benchmarks, we find that sparse conditional operator learning is viable and can match dense-grid behavior in several regimes, that preserving local context-query geometry is essential in non-periodic settings but less so in spectrally smooth periodic regimes, and that uncertainty-aware operator learning succeeds when latent conditioning complements rather than overwrites the local geometric pathway. These results provide a basis for probabilistic operator learning under partial observations and help bridge operator learning and probabilistic meta-learning in function space.