Neural operators approximate strongly continuous convex monotone semigroups

Abstract:We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Analysis of PDEs (math.AP); Probability (math.PR); Machine Learning (stat.ML)

arXiv:2609.02727 [math.NA]

(or arXiv:2609.02727v1 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2609.02727 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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