Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization

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.

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a $\log\log/\log$-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{1}} \approx C N^{-α}$ with empirical exponent $α\approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.