Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation

In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes the setting to operator-valued kernels. This unified framework enables a rigorous theoretical analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In particular, it allows us to establish optimal learning rates and provides a good understanding of how many neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space. These results sharpen and complete earlier findings for specific kernel algorithms.

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.

Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces. In this study, we focus on the approximation of continuous operators between infinite-dimensional normed or metric spaces in the topology of uniform convergence on compact sets. Unlike standard results in the operator learning literature, we additionally investigate the case where the approximating sequence of encoder-decoder architectures can be chosen independently of the compact sets. Taking a topological perspective, we point out that compact-set-independent approximation is a strictly stronger property in most relevant operator learning frameworks. To establish our results, we introduce new approximation properties of input and output spaces tailored to encoder-decoder architectures. These properties enable us to prove a universal operator approximation theorem ensuring uniform convergence on every compact subset of the input space. Our results unify and extend existing universal operator approximation theorems for various encoder-decoder architectures, including classical DeepONets, BasisONets, MIONets, architectures based on frames and other related approaches. A notable feature of our framework is that it also applies to metric spaces beyond the normed setting. In particular, it allows the consideration of $p$-Wasserstein spaces of probability measures as input or output spaces, and Skorohod spaces of càdlàg functions as input spaces. This generality also opens up potential applications in optimal transport.