Neural networks with radial-basis functions for forming of soil profile in pipeline’s cathodic protection system

2021 ◽  
Author(s):  
Sergei Nikulin ◽  
Dmitriy Vaganov
Keyword(s):  
Neural Networks ◽  
Radial Basis Functions ◽  
Cathodic Protection ◽  
Soil Profile ◽  
Basis Functions ◽  
Protection System ◽  
Radial Basis ◽  
Cathodic Protection System

Regularization Theory and Neural Networks Architectures

1995 ◽  
Vol 7 (2) ◽  
pp. 219-269 ◽  
Author(s):  
Federico Girosi ◽  
Michael Jones ◽  
Tomaso Poggio
Keyword(s):  
Neural Networks ◽  
Tensor Product ◽  
Radial Basis Functions ◽  
Basis Functions ◽  
Additive Models ◽  
Approximation Schemes ◽  
Tensor Product Splines ◽  
Radial Basis ◽  
Regularization Networks ◽  
Hidden Layer

We had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units, called regularization networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known radial basis functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends radial basis functions (RBF) to hyper basis functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, some forms of projection pursuit regression, and several types of neural networks. We propose to use the term generalized regularization networks for this broad class of approximation schemes that follow from an extension of regularization. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In summary, different multilayer networks with one hidden layer, which we collectively call generalized regularization networks, correspond to different classes of priors and associated smoothness functionals in a classical regularization principle. Three broad classes are (1) radial basis functions that can be generalized to hyper basis functions, (2) some tensor product splines, and (3) additive splines that can be generalized to schemes of the type of ridge approximation, hinge functions, and several perceptron-like neural networks with one hidden layer.

scholarly journals Feedback Analysis of Radial Basis Functions Neural Networks via Small Gain Theorem

2008 ◽  
Vol 41 (2) ◽  
pp. 7463-7467
Author(s):  
Ali S. Saad Azhar ◽  
Muhammad Shafiq ◽  
Jamil M. Bakhashwain ◽  
Fouad M. AL-Sunni
Keyword(s):  
Neural Networks ◽  
Radial Basis Functions ◽  
Basis Functions ◽  
Feedback Analysis ◽  
Small Gain Theorem ◽  
Radial Basis ◽  
Small Gain

On the Problem in Model Selection of Neural Network Regression in Overrealizable Scenario

2002 ◽  
Vol 14 (8) ◽  
pp. 1979-2002 ◽  
Author(s):  
Katsuyuki Hagiwara
Keyword(s):  
Neural Networks ◽  
Lower Bound ◽  
Model Selection ◽  
Radial Basis Functions ◽  
Selection Criterion ◽  
Basis Functions ◽  
Generalization Error ◽  
Radial Basis ◽  
Training Error ◽  
Selection Of

In considering a statistical model selection of neural networks and radial basis functions under an overrealizable case, the problem of unidentifiability emerges. Because the model selection criterion is an unbiased estimator of the generalization error based on the training error, this article analyzes the expected training error and the expected generalization error of neural networks and radial basis functions in overrealizable cases and clarifies the difference from regular models, for which identifiability holds. As a special case of an overrealizable scenario, we assumed a gaussian noise sequence as training data. In the least-squares estimation under this assumption, we first formulated the problem, in which the calculation of the expected errors of unidentifiable networks is reduced to the calculation of the expectation of the supremum of thex2 process. Under this formulation, we gave an upper bound of the expected training error and a lower bound of the expected generalization error, where the generalization is measured at a set of training inputs. Furthermore, we gave stochastic bounds on the training error and the generalization error. The obtained upper bound of the expected training error is smaller than in regular models, and the lower bound of the expected generalization error is larger than in regular models. The result tells us that the degree of overfitting in neural networks and radial basis functions is higher than in regular models. Correspondingly, it also tells us that the generalization capability is worse than in the case of regular models. The article may be enough to show a difference between neural networks and regular models in the context of the least-squares estimation in a simple situation. This is a first step in constructing a model selection criterion in an overrealizable case. Further important problems in this direction are also included in this article.

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