Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators
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Abstract (arXiv)
Recently, there has been growing interest in characterizing the function spaces underlying neural networks. While shallow and deep scalar-valued neural networks have been linked to scalar-valued reproducing kernel Banach spaces (RKBS), ℝd-valued neural networks and neural operator models remain less understood in the RKBS setting. To address this gap, we develop a notion of adjoint pairs of vector-valued RKBSs (vv-RKBS), which inherently involves an associated reproducing kernel, and prove that every vv-RKBS belongs to such a pair. Our construction extends existing kernel definitions by avoiding restrictive assumptions such as symmetric kernel domains, finite-dimensional output spaces, reflexivity, or separability, while still recovering familiar properties of vector-valued reproducing kernel Hilbert spaces (vv-RKHS). We then show that shallow ℝd-valued neural networks are elements of a specific vv-RKBS, namely an instance of an integral vv-RKBS. To also explore the functional structure of neural operators, we analyze the DeepONet and Hypernetwork architectures and demonstrate that they too belong to an integral vv-RKBS. In all cases, we establish a representer theorem, showing that optimization over these function spaces recovers the corresponding neural architectures.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 31 of its 56 references that are in the map. First posted on arXiv 30 Sep 2025.
Builds on
31 of 56 references.
References · 31
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
- 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
- 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
- 2020Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don'tcited by 95
- 2020A Measure-Theoretic Approach to Kernel Conditional Mean Embeddingscited by 86
- 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 83
- 2018A Priori Estimates of the Population Risk for Two-layer Neural Networkscited by 76
- 2023A Mathematical Guide to Operator Learningcited by 76
- 2020Representation formulas and pointwise properties for Barron functionscited by 62
19 more
- 2021What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theorycited by 60
- 2021Understanding neural networks with reproducing kernel Banach spacescited by 50
- 2023Kernel Methods are Competitive for Operator Learningcited by 50
- 2020On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamicscited by 39
- 2019On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructionscited by 29
- 2021Two-layer neural networks with values in a Banach spacecited by 26
- 2018Sparsity of solutions for variational inverse problems with finite-dimensional datacited by 25
- 2023Learning in latent spaces improves the predictive accuracy of deep neural operatorscited by 19
- 2023Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compressioncited by 18
- 2022Duality for Neural Networks through Reproducing Kernel Banach Spacescited by 15
- 2023Sparse Representer Theorems for Learning in Reproducing Kernel Banach Spacescited by 10
- 2024Neural reproducing kernel Banach spaces and representer theorems for deep networkscited by 8
- 2025Deep Networks are Reproducing Kernel Chainscited by 8
- 2023RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representationscited by 7
- 2020Reproducing Kernel Hilbert Spaces Cannot Contain all Continuous Functions on a Compact Metric Spacecited by 6
- 2024Hypothesis Spaces for Deep Learningcited by 4
- 2024A Lipschitz spaces view of infinitely wide shallow neural networkscited by 2
- 2025RONOM: Reduced-Order Neural Operator Modelingcited by 2
- 2024On extremal points for some vectorial total variation seminormscited by 1
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Citing papers per year: 2026: 2.
2 papers.