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

• URL: http://arxiv.org/abs/2603.11972v1
• Date: Thu, 12 Mar 2026 14:23:35 GMT
• Title: Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem
• Abstract summary: We provide a branch-trunk neural architecture for approxing nonlinear operators acting between function spaces.<n>Our main theorem shows that continuous operators $G:Vto C(K;mathbbRm)$, where $Vsubset X$ and $KsubsetmathbbRd$ are compact, can be uniformly approximated by such topological DeepONets.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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