Universal Approximation of Continuous Functionals on Compact Subsets via Linear Measurements and Scalar Nonlinearities

2026-02-03

AI Summary

This study addresses the universal approximation of continuous functionals defined on compact subsets of Hilbert space products. It establishes that architectures comprising finite continuous linear measurements, scalar nonlinear activations, and a fusion step can uniformly approximate any such functional. The result is further extended to Banach space–valued mappings. To the best of our knowledge, this work provides the first rigorous theoretical guarantee for this widely adopted network structure in operator learning and imaging, specifically on compact sets. By proving a universal approximation theorem for continuous functionals over compact domains, the study validates the theoretical soundness of deep operator network designs commonly used in practice.

Technology Category

Computer Vision: Learning & Optimization for CVMachine Learning: Deep Learning TheorySearch and Optimization: Non-convex Optimization

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Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear measurements of the inputs and then combine these measurements through continuous scalar nonlinearities. We also extend the approximation principle to maps with values in a Banach space, yielding finite-rank approximations. These results provide a compact-set justification for the common ``measure, apply scalar nonlinearities, then combine''design pattern used in operator learning and imaging.

Problem

• universal approximation
• continuous functionals
• compact subsets
• Hilbert spaces
• operator learning

Innovation

• universal approximation
• continuous functionals
• linear measurements
• scalar nonlinearities
• operator learning