Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems
Abstract
The purpose of this paper is to investigate neural network capability systematically. The main results are: 1) every Tauber-Wiener function is qualified as an activation function in the hidden layer of a three-layered neural network; 2) for a continuous function in S'(R(1 )) to be a Tauber-Wiener function, the necessary and sufficient condition is that it is not a polynomial; 3) the capability of approximating nonlinear functionals defined on some compact set of a Banach space and nonlinear operators has been shown; and 4) the possibility by neural computation to approximate the output as a whole (not at a fixed point) of a dynamical system, thus identifying the system.
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DEM is not simply a nonlinear function approximation device because the network connections parameterise a generative model as opposed to its inverse.
“However, unlike back propagation of errors or universal approximation in neural networks [22], DEM is not simply a nonlinear function approximation device. This is because the network connections parameterise a generative model as opposed to its inverse; h: y→η (i.e., recognition model).”
From Hierarchical Models in the Brain (2008) · cites this paper as [22]
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Forecasting with artificial neural networks:
Zhang, G., Eddy Patuwo, B., & Y. Hu, M. (1998) · International Journal of Forecasting · doi:10.1016/s0169-2070(97)00044-7 · Cited by 4,229 · see citing works
Hierarchical Models in the Brain
Friston, K. (2008) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1000211 · Cited by 952 · see citing works
“However, unlike back propagation of errors or universal approximation in neural networks [22], DEM is not simply a nonlinear function approximation device.”
BackgroundDeep learning in spatial transcriptomics: Learning from the next next-generation sequencing
Heydari, A. A., & Sindi, S. S. (2023) · Biophysics Reviews · doi:10.1063/5.0091135 · Cited by 44 · see citing works
Nonlocality and Nonlinearity Implies Universality in Operator Learning
Lanthaler, S., Li, Z., & Stuart, A. M. (2023) · arXiv · doi:10.48550/arxiv.2304.13221 · Cited by 25 · see citing works
Toms, B. A., Kashinath, K., Prabhat, & Yang, D. (2021) · Geoscientific Model Development · doi:10.5194/gmd-14-4495-2021 · Cited by 20 · see citing works
Bhan, L., Krstic, M., & Shi, Y. (2025) · arXiv · doi:10.48550/arxiv.2509.26443
“Neural operators were first introduced in [21] and then more recently popularized for learning PDE solutions in [22] and [23].”
- Authors
- Chen, T., & Chen, H.
- Affiliations
- Fudan University (CN)
- Venue
- IEEE Transactions on Neural Networks
- Published
- 1995-07
- Volume
- 6
- Issue
- 4
- Pages
- 911-917
- ISSN
- 1045-9227 (ISSN-L)
- OpenAlex
- W2139923370
- Cited by
- 1,230
- References
- 0
- Citations by year
- 2026: 195 · 2025: 210 · 2024: 218 · 2023: 127 · 2022: 84 · 2021: 78 · 2020: 30 · 2019: 22 · 2018: 14 · 2017: 7 · 2016: 6 · 2015: 14 · 2014: 13 · 2013: 17 · 2012: 19
- FWCI
- 10.93
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Neural Networks (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Machine Learning and ELM
- Physical Sciences › Mathematics › Statistics and Probability › Approximation Theory and Sequence Spaces
- Keywords
- universal approximation, artificial neural networks, activation functions, nonlinear operators, nonlinear functionals, dynamical systems, system identification, hidden layers, Banach spaces
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