Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems

1995 ◽  
Vol 6 (4) ◽  
pp. 911-917 ◽  
Author(s):  
Tianping Chen ◽  
Hong Chen
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Universal Approximation ◽  
Activation Functions ◽  
Nonlinear Operators

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 Interval universal approximation for neural networks

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-29
Author(s):  
Zi Wang ◽  
Aws Albarghouthi ◽  
Gautam Prakriya ◽  
Somesh Jha
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Approximation Problem ◽  
Activation Function ◽  
Constructive Proof ◽  
Universal Approximation ◽  
Activation Functions ◽  
Complete Problems ◽  
Robust Network ◽  
Interval Approximation

To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation , primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable functions . Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ ∞ -norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ NP . As a result, IUA is an inherently hard problem : No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.

RECURRENT NEURAL NETWORKS ARE UNIVERSAL APPROXIMATORS

2007 ◽  
Vol 17 (04) ◽  
pp. 253-263 ◽  
Author(s):  
ANTON MAXIMILIAN SCHÄFER ◽  
HANS-GEORG ZIMMERMANN
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Dynamical Systems ◽  
Error Correction ◽  
Recurrent Neural Networks ◽  
State Space Model ◽  
Universal Approximation ◽  
Space Model ◽  
Open Dynamical System ◽  
Universal Approximators

Recurrent Neural Networks (RNN) have been developed for a better understanding and analysis of open dynamical systems. Still the question often arises if RNN are able to map every open dynamical system, which would be desirable for a broad spectrum of applications. In this article we give a proof for the universal approximation ability of RNN in state space model form and even extend it to Error Correction and Normalized Recurrent Neural Networks.

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.

Application of neural networks with orthogonal activation functions in control of dynamical systems

2015 ◽  
Vol 103 (4) ◽  
pp. 667-685 ◽  
Author(s):  
Saša S. Nikolić ◽  
Dragan S. Antić ◽  
Marko T. Milojković ◽  
Miroslav B. Milovanović ◽  
Staniša Lj. Perić ◽  
...  
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Activation Functions

Analysis of Non-Linear Activation Functions for Classification Tasks Using Convolutional Neural Networks

2019 ◽  
Vol 12 (3) ◽  
pp. 156-161 ◽  
Author(s):  
Aman Dureja ◽  
Payal Pahwa
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Activation Function ◽  
Primary Objective ◽  
Experimental Comparison ◽  
Activation Functions ◽  
Practical Applications ◽  
Network Activation ◽  
Non Linear ◽  
Hidden Layer

Background: In making the deep neural network, activation functions play an important role. But the choice of activation functions also affects the network in term of optimization and to retrieve the better results. Several activation functions have been introduced in machine learning for many practical applications. But which activation function should use at hidden layer of deep neural networks was not identified. Objective: The primary objective of this analysis was to describe which activation function must be used at hidden layers for deep neural networks to solve complex non-linear problems. Methods: The configuration for this comparative model was used by using the datasets of 2 classes (Cat/Dog). The number of Convolutional layer used in this network was 3 and the pooling layer was also introduced after each layer of CNN layer. The total of the dataset was divided into the two parts. The first 8000 images were mainly used for training the network and the next 2000 images were used for testing the network. Results: The experimental comparison was done by analyzing the network by taking different activation functions on each layer of CNN network. The validation error and accuracy on Cat/Dog dataset were analyzed using activation functions (ReLU, Tanh, Selu, PRelu, Elu) at number of hidden layers. Overall the Relu gave best performance with the validation loss at 25th Epoch 0.3912 and validation accuracy at 25th Epoch 0.8320. Conclusion: It is found that a CNN model with ReLU hidden layers (3 hidden layers here) gives best results and improve overall performance better in term of accuracy and speed. These advantages of ReLU in CNN at number of hidden layers are helpful to effectively and fast retrieval of images from the databases.

scholarly journals Hardware implementation of radial-basis neural networks with Gaussian activation functions on FPGA

2021 ◽  
Author(s):  
Volodymyr Shymkovych ◽  
Sergii Telenyk ◽  
Petro Kravets
Keyword(s):  
Neural Networks ◽  
Hardware Implementation ◽  
Gaussian Function ◽  
Activation Function ◽  
Rbf Neural Networks ◽  
Activation Functions ◽  
Rbf Network ◽  
Combination Scheme ◽  
Radial Basis ◽  
Hidden Layer

AbstractThis article introduces a method for realizing the Gaussian activation function of radial-basis (RBF) neural networks with their hardware implementation on field-programmable gaits area (FPGAs). The results of modeling of the Gaussian function on FPGA chips of different families have been presented. RBF neural networks of various topologies have been synthesized and investigated. The hardware component implemented by this algorithm is an RBF neural network with four neurons of the latent layer and one neuron with a sigmoid activation function on an FPGA using 16-bit numbers with a fixed point, which took 1193 logic matrix gate (LUTs—LookUpTable). Each hidden layer neuron of the RBF network is designed on an FPGA as a separate computing unit. The speed as a total delay of the combination scheme of the block RBF network was 101.579 ns. The implementation of the Gaussian activation functions of the hidden layer of the RBF network occupies 106 LUTs, and the speed of the Gaussian activation functions is 29.33 ns. The absolute error is ± 0.005. The Spartan 3 family of chips for modeling has been used to get these results. Modeling on chips of other series has been also introduced in the article. RBF neural networks of various topologies have been synthesized and investigated. Hardware implementation of RBF neural networks with such speed allows them to be used in real-time control systems for high-speed objects.

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