scholarly journals Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

2021 ◽  
Vol 3 (3) ◽  
pp. 218-229 ◽  
Author(s):  
Lu Lu ◽  
Pengzhan Jin ◽  
Guofei Pang ◽  
Zhongqiang Zhang ◽  
George Em Karniadakis
Keyword(s):  
Approximation Theorem ◽  
Universal Approximation ◽  
Nonlinear Operators

scholarly journals A Universal Approximation Theorem for Mixture-of-Experts Models

2016 ◽  
Vol 28 (12) ◽  
pp. 2585-2593 ◽  
Author(s):  
Hien D. Nguyen ◽  
Luke R. Lloyd-Jones ◽  
Geoffrey J. McLachlan
Keyword(s):  
Network Architecture ◽  
Approximation Theorem ◽  
Target Function ◽  
Continuous Functions ◽  
Universal Approximation ◽  
Mixture Of Experts ◽  
Neural Network Architecture ◽  
Alternative Result ◽  
Unit Hypercube ◽  
Uniformly Convergent

The mixture-of-experts (MoE) model is a popular neural network architecture for nonlinear regression and classification. The class of MoE mean functions is known to be uniformly convergent to any unknown target function, assuming that the target function is from a Sobolev space that is sufficiently differentiable and that the domain of estimation is a compact unit hypercube. We provide an alternative result, which shows that the class of MoE mean functions is dense in the class of all continuous functions over arbitrary compact domains of estimation. Our result can be viewed as a universal approximation theorem for MoE models. The theorem we present allows MoE users to be confident in applying such models for estimation when data arise from nonlinear and nondifferentiable generative processes.

Universal Approximation of Multiple Nonlinear Operators by Neural Networks

2002 ◽  
Vol 14 (11) ◽  
pp. 2561-2566 ◽  
Author(s):  
Andrew D. Back ◽  
Tianping Chen
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Universal Approximation ◽  
Nonlinear Operators ◽  
Nonlinear Dynamical ◽  
Open Questions

Recently, there has been interest in the observed capabilities of some classes of neural networks with fixed weights to model multiple nonlinear dynamical systems. While this property has been observed in simulations, open questions exist as to how this property can arise. In this article, we propose a theory that provides a possible mechanism by which this multiple modeling phenomenon can occur.

scholarly journals Universal approximation theorem for Dirichlet series

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
O. Demanze ◽  
A. Mouze
Keyword(s):  
Dirichlet Series ◽  
Simultaneous Approximation ◽  
Shift Operator ◽  
Approximation Theorem ◽  
Extension Theorem ◽  
Universal Approximation ◽  
Derivation Operator ◽  
Backward Shift ◽  
The Right ◽  
Density Results

The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that finally lead to the main theorem on simultaneous approximation.

scholarly journals Universal Approximation Theorem for Tessarine-Valued Neural Networks

2021 ◽  
Author(s):  
Rafael A. F. Carniello ◽  
Wington L. Vital ◽  
Marcos Eduardo Valle
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Approximation Theorem ◽  
Superior Performance ◽  
Universal Approximation ◽  
Activation Functions ◽  
Hidden Layer ◽  
Arbitrary Precision ◽  
Vector Valued Function ◽  
Vector Valued

The universal approximation theorem ensures that any continuous real-valued function defined on a compact subset can be approximated with arbitrary precision by a single hidden layer neural network. In this paper, we show that the universal approximation theorem also holds for tessarine-valued neural networks. Precisely, any continuous tessarine-valued function can be approximated with arbitrary precision by a single hidden layer tessarine-valued neural network with split activation functions in the hidden layer. A simple numerical example, confirming the theoretical result and revealing the superior performance of a tessarine-valued neural network over a real-valued model for interpolating a vector-valued function, is presented in the paper.

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