ReLU Neural Network Approximation to Smooth Functional Operator: Dimensional Decay and Error Analysis

• URL: http://arxiv.org/abs/2609.15355v1
• Date: Mon, 14 Sep 2026 10:39:15 GMT
• Title: ReLU Neural Network Approximation to Smooth Functional Operator: Dimensional Decay and Error Analysis
• Abstract summary: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural networks.<n>This is the first work to characterize neural network approximation error for infinite-dimensional functional inputs explicitly through the joint dimensional decay of coordinate magnitudes and directional sensitivities.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural networks. Writing the functional input as $X(t)=\sum_{d\geq1}ξ_dν_d(t)$, we quantify the importance of coordinate $d$ through $w_ds_d$, where $s_d$ bounds the magnitude of the corresponding basis score and $w_d$ controls the directional Fréchet sensitivity of the target functional. Our constructive analysis combines coordinate truncation, anisotropic partitioning, local Taylor approximation, and ReLU network realization, while allowing unrestricted interactions among the retained coordinates. We establish a general nonasymptotic upper bound for the uniform approximation error and a complementary pseudo-dimension-based lower bound for the worst-case approximation error. Under generalized exponential coordinate decay $w_ds_d\asymp\exp(-cd^ρ)$, with $ρ>0$, the upper and lower bounds match at the leading order and thus yield the nearly optimal approximation rate, which is stretched-exponential in the logarithm of the network budget. This is the first work to characterize neural network approximation error for infinite-dimensional functional inputs explicitly through the joint dimensional decay of coordinate magnitudes and directional sensitivities.

Related papers

• ReLU neural network approximation to piecewise constant functions [3.5928501649873326]
  We show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function. If the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided.
  arXiv  Detail & Related papers  (2024-10-21T20:58:34Z)
• Approximation Error and Complexity Bounds for ReLU Networks on Low-Regular Function Spaces [0.0]
  We consider the approximation of a large class of bounded functions, with minimal regularity assumptions, by ReLU neural networks. We show that the approximation error can be bounded from above by a quantity proportional to the uniform norm of the target function.
  arXiv  Detail & Related papers  (2024-05-10T14:31:58Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• Approximation Results for Gradient Descent trained Neural Networks [0.0]
  The networks are fully connected constant depth increasing width. The continuous kernel error norm implies an approximation under the natural smoothness assumption required for smooth functions.
  arXiv  Detail & Related papers  (2023-09-09T18:47:55Z)
• Optimal Approximation Complexity of High-Dimensional Functions with Neural Networks [3.222802562733787]
  We investigate properties of neural networks that use both ReLU and $x2$ as activation functions. We show how to leverage low local dimensionality in some contexts to overcome the curse of dimensionality, obtaining approximation rates that are optimal for unknown lower-dimensional subspaces.
  arXiv  Detail & Related papers  (2023-01-30T17:29:19Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Deep neural network approximation of analytic functions [91.3755431537592]
  entropy bound for the spaces of neural networks with piecewise linear activation functions. We derive an oracle inequality for the expected error of the considered penalized deep neural network estimators.
  arXiv  Detail & Related papers  (2021-04-05T18:02:04Z)
• Finding Global Minima via Kernel Approximations [90.42048080064849]
  We consider the global minimization of smooth functions based solely on function evaluations. In this paper, we consider an approach that jointly models the function to approximate and finds a global minimum.
  arXiv  Detail & Related papers  (2020-12-22T12:59:30Z)
• Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions [84.49087114959872]
  We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonsmooth functions. In particular, we study Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions.
  arXiv  Detail & Related papers  (2020-02-10T23:23:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.