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

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.