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

Vugar Ismailov

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.

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

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 defined on a compact set (typically a compact subset of a Banach space), and the operator maps to an output function defined on a compact Euclidean domain . In this paper, we develop a topological extension in which the operator input lies in an arbitrary Hausdorff locally convex space . We construct topological feedforward neural networks on using continuous linear functionals from the dual space and introduce topological DeepONets whose branch component acts on through such linear measurements, while the trunk component acts on the Euclidean output domain. Our main theorem shows that continuous operators , where and 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.
Paper Structure (5 sections, 6 theorems, 115 equations, 1 figure)

This paper contains 5 sections, 6 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Topological networks on locally convex spaces
  3. DeepONets on locally convex spaces and operator approximation
  4. Examples illustrating Theorem \ref{['thm:operator-separable']}
  5. Conclusion

Key Result

Theorem 2.1

Let $X$ be a locally convex topological vector space and assume that the activation function $\sigma$ is a Tauber--Wiener function. Then for every compact set $K\subset X$, every function $g\in C(K;\mathbb{R}^m)$, and every $\varepsilon>0$, there exists a topological neural network $H\in\mathcal{S}_ In other words, the class $\mathcal{S}_\sigma(X;\mathbb{R}^m)$ is dense in $C(X;\mathbb{R}^m)$ with

Figures (1)

  • Figure 1: Topological DeepONet architecture. The branch network encodes the input element $u\in X$, where $X$ is a locally convex space, through finitely many linear measurements $f_1(u),\dots,f_r(u)$ with $f_j\in X^*$. The trunk network takes $y\in\mathbb{R}^d$ as input and yields $[t_1(y),\dots,t_p(y)]^T\in\mathbb{R}^p$. The outputs of the branch and trunk networks are then multiplied to produce the final output. If $X$ is the space of continuous functions and the functionals $f_j$ are point evaluation functionals, $f_j(u)=u(x_j)$, then the classical DeepONet architecture is recovered.

Theorems & Definitions (22)

  • Definition 2.1: Topological neural network on a locally convex space
  • Definition 2.2: Deep topological neural network on $X$
  • Remark 2.1
  • Definition 2.3: Tauber--Wiener function
  • Theorem 2.1
  • proof
  • Theorem 2.2: Chen and Chen Chen2
  • proof
  • Definition 3.1: Topological DeepONet
  • Remark 3.1
  • ...and 12 more