Neural operators approximate strongly continuous convex monotone semigroups
TL;DR
This work introduces the general class of so-called Chernoff-neural operators and shows in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well, and introduces the more specialized class of envelope-neural operators for envelope semigroups which allows for quantitative approximation rates.
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 H\"older 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.
