Working with differential, functional and difference equations using functional networks

Abstract

In this paper we first analyze the problem of equivalence of differential, functional and difference equations and give methods to move between them. We also introduce functional networks, a powerful alternative to neural networks, which allow neural functions to be different, multidimensional, multi-argument and constrained by link connections, and use them for predicting values of magnitudes satisfying differential, functional and/or difference equations, and for obtaining the difference and differential equation associated with a set of data. The estimation of the differential or difference equation coefficients is done by simply solving systems of linear equations, in the cases of equally or unequally spaced or missing data points. Some examples of applications are given to illustrate the method.

Prediction of Uplift Capacity of Suction Caisson in Clay Using Functional Network and Multivariate Adaptive Regression Spline

“…Castillo et al introduced Functional Network (FN) as a powerful alternative tool to deal with the limitations of standard ANN [46][47][48]. FN is termed as a novel generalization of ANN due to its ability to take into account both data as well as properties of functions being modelled (domain knowledge) to estimate the unknown neuron functions.…”

“…According to Castillo et al, they require domain knowledge for deriving the functional equations and making assumptions about the form the unknown functions should take [46]. Though FN is similar to ANN, few di erences in features make it more powerful and exible compared to ANN [46][47][48]. Figure 3 shows a typical neural network and its corresponding functional network.…”

“…The knowledge of functional equations is used to work e ciently with functional networks, [47]. A functional equation is an equation in which the unknowns …”

“…Most recently, FN has been introduced by Castillo et al to overcome the drawbacks of neural networks [46][47][48]. Castillo et al used FNs to solve structural engineering problems such as predicting shear, moment, slope, and de ection of a beam [49].…”

Functional Network Design using Parallel Cultural Algorithm

“…Unlike neural networks, it deals with general functional models instead of sigmoid-like ones, in these networks there are no weights associated with the links connecting neurons, and the neural functions are unknown from given families to be estimated during the learning process. We can select appropriate families for each specific problem, such as polynomials, Fourier expansions and trigonometric functions, etc.. At present, the functional network is a very useful general framework for solving a wide range of problems: The solving of differential functional and difference equation [2], nonlinear time series and prediction modeling [3], factorization model of multivariate polynomials [4], the identification of nonlinear system [5], CAD, linear and nonlinear regression [4], etc. The functional networks have shown excellent performance in the above-mentioned problems.…”