# Neural operators approximate strongly continuous convex monotone semigroups

arXiv: https://arxiv.org/abs/2609.02727 · HTML: https://mlsanity.com/paper/2609.02727/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 66 references that are in the map.
- First posted on arXiv 2 Sep 2026.

## Builds on (most cited first, 30 of 30)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [The Parametric Complexity of Operator Learning](https://mlsanity.com/paper/2306.15924/) (2023)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Universal approximation property of Banach space-valued random feature models including random neural networks](https://mlsanity.com/paper/2312.08410/) (2023)
- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning](https://mlsanity.com/paper/2304.12231/) (2023)
- [Computational aspects of robust optimized certainty equivalents and option pricing](https://mlsanity.com/paper/1706.10186/) (2017)
- [Universal approximation results for neural networks with non-polynomial activation function over non-compact domains](https://mlsanity.com/paper/2410.14759/) (2024)
- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)
- [Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations](https://mlsanity.com/paper/2103.05744/) (2021)
- [Convex monotone semigroups and their generators with respect to Gamma-convergence](https://mlsanity.com/paper/2202.08653/) (2022)
- [Weighted universal approximation of differentiable maps on infinite-dimensional manifolds](https://mlsanity.com/paper/2606.09820/) (2026)
- [Upper envelopes of families of Feller semigroups and viscosity solutions to a class of nonlinear Cauchy problems](https://mlsanity.com/paper/1906.04430/) (2019)
- [Limits of random walks with distributionally robust transition probabilities](https://mlsanity.com/paper/2007.08815/) (2020)
- [Discrete-time approximation for stochastic optimal control problems under the G-expectation framework](https://mlsanity.com/paper/2104.02268/) (2021)
- [Convex Increasing Functionals on Cb(X) Spaces](https://mlsanity.com/paper/2209.08140/) (2022)
- [Convergence of infinitesimal generators and stability of convex monotone semigroups](https://mlsanity.com/paper/2305.18981/) (2023)
- [Hopf-Lax approximation for value functions of L'evy optimal control problems](https://mlsanity.com/paper/2501.16846/) (2025)

## Abstract (arXiv)

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.