analiza aiqbench
This research establishes a rigorous functional-analytic framework for Fourier Neural Operators, proving that their approximation error in Sobolev norms follows an explicit complexity-scaling law. Empirical validation on the Burgers equation confirms these theoretical bounds and reveals optimization instabilities inherent to large-scale models when trained with derivative-aware losses. The work bridges abstract operator theory with practical deep learning scaling behavior, offering a clearer path for designing stable neural operators that generalize beyond function values alone.
Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
arXiv:2605.08170v2 Announce Type: replace Abstract: Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_\theta\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_\theta\|_{H^{1}} \approx C N^{-\alpha}$ with empirical exponent $\alpha \approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.
Abstract:Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_\theta\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_\theta\|_{H^{1}} \approx C N^{-\alpha}$ with empirical exponent $\alpha \approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.
Machine Learning (cs.LG); Functional Analysis (math.FA)
arXiv:2605.08170 [cs.LG]
(or arXiv:2605.08170v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.08170 Focus to learn more arXiv-issued DOI via DataCite
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Text integral sub licența deschisă a publicației · 319 cuvinte
Subiecte conexe
Mixtures of Neural Operators Reduce Active Complexity in Operator Learning
arXiv · 10 iun. 2026
Measure Neural Mappings
arXiv · 14 oct. 2026
Disentanglement with Holographic Reduced Representations
arXiv · 9 iun. 2026
A Mike's-Eye View of ARC's Research
AI Alignment Forum · 9 iun. 2026