arXiv Machine Learning
Jun 2
Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces
arXiv:2606. 01244v1 Announce Type: cross Abstract: We study operator learning using encoder--decoder neural networks.
New universal operator approximation theorem for encoder-decoder architectures
arXiv:2503. 24092v2 Announce Type: replace-cross Abstract: 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.
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arXiv Machine Learning
Sep 3
Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization
arXiv:2606. 01244v2 Announce Type: replace-cross Abstract: 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.
arXiv Machine Learning
Sep 15
Resolution-Independent Analysis of Encoder--Decoder Operator Learning via Limiting Kernels
The paper studies operator learning on function spaces using encoder–decoder architectures. It shows that as input and output resolutions grow, the induced kernels converge to a limiting kernel, enabling regularity assumptions independent of resolution. The authors derive upper and lower bounds for regularized stochastic gradient descent, extend the analysis to neural networks via the limiting neural tangent kernel, and provide error bounds and complexity guarantees for various kernel and encoding constructions.
arXiv AI
Sep 2
Universal Approximation of Nonlinear Operators and Their Derivatives
The paper establishes the first Universal Approximation Theorems for k‑times differentiable nonlinear operators and their derivatives in general Banach spaces, extending classical results to infinite‑dimensional settings. It introduces Derivative‑Informed Operator Learning (DIOL) and formulates Bastiani–Sobolev training for this framework, covering architectures such as DeepONets, Deep‑H‑ONets, and PCA‑Nets. The work also outlines applications to high‑order accuracy in operator learning, constrained optimization in Banach spaces, and numerical methods for infinite‑dimensional PDEs.
arXiv AI
Jul 13
Projection Methods for Operator Learning and Universal Approximation
arXiv:2406. 12264v5 Announce Type: replace-cross Abstract: We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping.
arXiv Machine Learning
Sep 4
A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations
arXiv:2609. 03129v1 Announce Type: cross Abstract: Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks.
arXiv Machine Learning
Jun 16
Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy
arXiv:2606. 16975v1 Announce Type: cross Abstract: In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks.
arXiv Statistics ML
Aug 25
Q-Learning with Stable Infinite-Dimensional Linear Function Approximation
arXiv:2608.22636v1 Announce Type: cross Abstract: Q-learning with linear function approximation can be unstable because an arbitrary approximation architecture need not preserve the Bellman contracti...
arXiv Machine Learning
Aug 10
Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces
arXiv:2608. 06428v1 Announce Type: new Abstract: Deep Operator Networks (DeepONets; arXiv:1910.
arXiv Machine Learning
Jun 25
Structured Approximations of Measures
arXiv:2310. 09149v3 Announce Type: replace-cross Abstract: We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints.
arXiv Machine Learning
Aug 18
Operator-Theoretic Generalization Bounds for Multitask Deep Learning
arXiv:2608. 15982v1 Announce Type: new Abstract: We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces.
arXiv Machine Learning
Jun 16
Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks
arXiv:2606. 14954v1 Announce Type: cross Abstract: We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers.