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.
Related
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BibTeX
@misc{blessing2026neural,
title = {{Neural operators approximate strongly continuous convex monotone semigroups}},
author = {Jonas Blessing and Philipp Schmocker and Alessandro Sgarabottolo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02727},
eprint = {2609.02727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02727},
}
RIS
TY - UNPB
TI - Neural operators approximate strongly continuous convex monotone semigroups
AU - Jonas Blessing
AU - Philipp Schmocker
AU - Alessandro Sgarabottolo
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02727
UR - https://arxiv.org/abs/2609.02727
AB - 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.
ER -