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.
