Fractional is Better: Learnable Derivative Orders in Neural Operator Learning
Abstract
Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs $(x, u(x))$. We ask whether inputs aligned with PDE differential structure can improve learning. Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden. We prove this intuition: providing derivative features improves approximation bounds from O(W−(s−m)/d) to O(W−s/d), where W is parameter count, s is input regularity, m is the PDE order, and d is spatial dimension. Our central finding, however, is a surprise: the optimal derivative order β* is strictly less than the PDE order m. This gap arises from a bias-variance tradeoff in spectral space that we characterize in closed form. Learning β from data achieves automatic spectral regularization. We introduce ∂-NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone. Across benchmark problems and architectures, ∂-NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.
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BibTeX
@misc{mehouachi2026fractional,
title = {{Fractional is Better: Learnable Derivative Orders in Neural Operator Learning}},
author = {Fares B. Mehouachi and Saif Eddin Jabari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20363},
eprint = {2610.20363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20363},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fractional is Better: Learnable Derivative Orders in Neural Operator Learning
AU - Fares B. Mehouachi
AU - Saif Eddin Jabari
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20363
UR - https://arxiv.org/abs/2610.20363
AB - Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs $(x, u(x))$. We ask whether inputs aligned with PDE differential structure can improve learning. Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden. We prove this intuition: providing derivative features improves approximation bounds from O(W^−(s−m)/d) to O(W^−s/d), where W is parameter count, s is input regularity, m is the PDE order, and d is spatial dimension. Our central finding, however, is a surprise: the optimal derivative order β^* is strictly less than the PDE order m. This gap arises from a bias-variance tradeoff in spectral space that we characterize in closed form. Learning β from data achieves automatic spectral regularization. We introduce ∂-NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone. Across benchmark problems and architectures, ∂-NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.
ER -