arXiv：cs.LG· Fares B. Mehouachi, Saif Eddin Jabari·· 4 小时前
∂-NO：可学习分数阶导数让神经算子更优
Fractional is Better: Learnable Derivative Orders in Neural Operator Learning
AI 导读
研究者提出 ∂-NO（del-NO）架构无关增强方法，为任意神经算子骨干提供可学习的分数阶导数特征。理论证明导数特征可将逼近界从 O(W^{-(s-m)/d}) 改善至 O(W^{-s/d})，并发现最优导数阶 β* 严格小于 PDE 阶数 m。在多个基准和架构上，∂-NO 持续提升预测精度，该工作已被 ICML 2026 接收。
正文
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 $\beta^*$ 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 $\beta$ from data achieves automatic spectral regularization. We introduce $\partial$-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, $\partial$-NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.
| Comments: | Accepted at ICML 2026. 37 pages, 11 figures, 13 tables. Code: this https URL |
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.20363 [cs.LG] |
| (or arXiv:2610.20363v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.20363 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Fares B. Mehouachi [view email]
[v1]
Wed, 14 Oct 2026 15:37:48 UTC (621 KB)
来源：arXiv：cs.LG · arxiv.org