Unlocking AI's Legal Future: Sobolev Norms & Neural Operators

AI & Technology Law Update — Unlocking AI's Legal Future: Sobolev Norms & Neural Operators

arXiv:2605.08170v1 Announce Type: new
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}} \l

Chapters:
0:00 Introduction
1:17 The Core Development
2:35 The Key Facts
3:53 The Legal Frame
5:11 The Business Impact
6:28 The Expert View
7:46 What Happens Next
9:04 Conclusion & Next Steps

Source: https://arxiv.org/abs/2605.08170

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