# Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem

arXiv: https://arxiv.org/abs/2603.11972 · HTML: https://mlsanity.com/paper/2603.11972/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 9 of 9)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [Universal Approximation Theorem for Vector- and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2401.02277/) (2024)
- [Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems](https://mlsanity.com/paper/2411.18964/) (2024)
- [Universal approximation theorem for neural networks with inputs from a topological vector space](https://mlsanity.com/paper/2409.12913/) (2024)
- [Deep Operator Neural Network Model Predictive Control](https://mlsanity.com/paper/2505.18008/) (2025)
- [On shallow feedforward neural networks with inputs from a topological space](https://mlsanity.com/paper/2504.02321/) (2025)

## Built on by (most cited first, 2 of 2)

- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces](https://mlsanity.com/paper/2608.06428/) (2026)

## Abstract (arXiv)

Deep Operator Networks (DeepONets) provide a branch-trunk neural architecture for approximating nonlinear operators acting between function spaces. In the classical operator approximation framework, the input is a function u∈ C(K1) defined on a compact set K1 (typically a compact subset of a Banach space), and the operator maps u to an output function G(u)∈ C(K2) defined on a compact Euclidean domain K2subsetℝd. In this paper, we develop a topological extension in which the operator input lies in an arbitrary Hausdorff locally convex space X. We construct topological feedforward neural networks on X using continuous linear functionals from the dual space X^* and introduce topological DeepONets whose branch component acts on X through such linear measurements, while the trunk component acts on the Euclidean output domain. Our main theorem shows that continuous operators G:V→ C(K;ℝm), where Vsubset X and Ksubsetℝd are compact, can be uniformly approximated by such topological DeepONets. This extends the classical Chen-Chen operator approximation theorem from spaces of continuous functions to locally convex spaces and yields a branch-trunk approximation theorem beyond the Banach-space setting.