Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem
This work provides a theoretical extension for operator approximation in machine learning, enabling applications in domains where input functions lie in more general topological spaces, but it is incremental as it builds on existing DeepONet frameworks.
The paper tackles the problem of approximating nonlinear operators acting between function spaces by extending Deep Operator Networks (DeepONets) to handle inputs from arbitrary Hausdorff locally convex spaces, rather than just Banach spaces, and proves a theorem showing that continuous operators can be uniformly approximated by such topological DeepONets, generalizing the classical Chen-Chen 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 $u\in C(K_1)$ 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.