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

TL;DR

This paper extends DeepONets to operate on inputs in arbitrary Hausdorff locally convex spaces, providing a universal approximation theorem for continuous operators in this broader topological setting.

Contribution

It introduces topological DeepONets that generalize classical operator approximation results to locally convex spaces, expanding the theoretical foundation of DeepONets.

Findings

Topological DeepONets can approximate continuous operators on locally convex spaces.

The work generalizes the Chen-Chen operator approximation theorem.

Provides a branch-trunk approximation theorem beyond Banach spaces.

Abstract

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 $u\in C(K_1)$ defined on a compact set $K_1$ (typically a compact subset of a Banach space), and the operator maps $u$ to an output function $G(u)\in C(K_2)$ defined on a compact Euclidean domain $K_2\subset\mathbb{R}^d$. 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 $G:V\to C(K;\mathbb{R}^m)$, where $V\subset X$ and $K\subset\mathbb{R}^d$ 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.