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

arXiv: https://arxiv.org/abs/2606.01244 · HTML: https://mlsanity.com/paper/2606.01244/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 52 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 29 of 29)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [The Parametric Complexity of Operator Learning](https://mlsanity.com/paper/2306.15924/) (2023)
- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [Nonparametric regression using over-parameterized shallow ReLU neural networks](https://mlsanity.com/paper/2306.08321/) (2023)
- [Deep Operator Network Approximation Rates for Lipschitz Operators](https://mlsanity.com/paper/2307.09835/) (2023)
- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study](https://mlsanity.com/paper/2410.00357/) (2024)
- [The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures](https://mlsanity.com/paper/2410.23440/) (2024)
- [Weighted variation spaces and approximation by shallow ReLU networks](https://mlsanity.com/paper/2307.15772/) (2023)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [Stochastic Gradient Descent for Operator Learning in Hilbert Spaces: Convergence Rates and Minimax Lower Bounds](https://mlsanity.com/paper/2402.04691/) (2024)
- [Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels](https://mlsanity.com/paper/2504.18184/) (2025)
- [A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning](https://mlsanity.com/paper/2509.11070/) (2025)

## Built on by (most cited first, 1 of 1)

- [Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces](https://mlsanity.com/paper/2606.17419/) (2026)

## Abstract (arXiv)

Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation. We characterize its unit ball as the closed convex hull of a vector-valued single-neuron dictionary in Bochner spaces. For the ReLU activation, the bounded linear operators in this space are precisely the Schatten-1 operators, with equivalent norms. For operators in this space, we establish encoder--decoder approximation bounds in the Bochner Lq-norm, where the error decomposes into input and output encoding errors and a finite-width term of order N-1/2. Under sub-Gaussian assumptions on the input and noise, we further derive high-probability generalization bounds for empirical least squares over path-norm-constrained encoder--decoder networks; the finite-sample contribution to the squared prediction error is of order K-1/2 up to logarithmic factors. The finite-width and finite-sample constants are independent of the encoding dimensions and bases, with the latter also independent of the network width. When the encoding errors decay algebraically, these bounds yield algebraic approximation and learning rates, in contrast to the complexity barriers for Lipschitz and Fr'echet differentiable operator classes.