Some Mathematical Aspects of Constructing Artificial Neural Networks

Искусственные нейронные сети (ИНС) в настоящее время являются полем интенсивных исследований. Они зарекомендовали себя при решении задач распознавания образов, аудио и текстовой информации. Планируется их применение в медицине, в беспилотных автомобилях и летательных аппаратах. Однако крайне мало научных работ посвящено обсуждению возможности построения искусственного интеллекта (ИИ), способного эффективно решать очерченный круг задач. Отсутствует гарантия штатного функционирования ИИ в любой реальной, а не специально созданной ситуации. В данной работе предпринимается попытка обоснования ненадежности функционирования современных искусственных нейронных сетей. Показывается, что задача построения интерполяционных многочленов является прообразом проблем, возникающих при создании ИНС. Известны примеры К.Д.Т. Рунге, С.Н. Бернштейна и общая теорема Фабера о том, что для любого наперед заданного натурального числа, соответствующего количеству узлов в интерполяционной таблице, найдется точка из области интерполяции и непрерывная функция, что интерполяционный многочлен не сходится к значению функции в этой точке при неограниченном росте числа узлов. Отсюда следует невозможность обеспечения эффективной работы ИИ лишь за счет неограниченного роста числа нейронов и объемов данных (Big Data), используемых в качестве обучающих выборок. Artificial neural networks (ANN) are currently a field of intensive research. They are a proven pattern/audio/text recognition tool. ANNs will be used in medicine, autonomous vehicles, and drones. Still, very few works discuss building artificial intelligence (AI) that can effectively solve the mentioned problems. There is no guarantee that AI will operate properly in any reallife, not simulated situation. In this work, an attempt is made to prove the unreliability of modern artificial neural networks. It is shown that constructing interpolation polynomials is a prototype of the problems associated with the ANN generation. There are examples by C.D.T. Runge, S.N. Bernstein, and the general Faber theorem stating that for any predetermined natural number corresponding to the number of nodes in the lookup table there is a point from the interpolation region and a continuous function that the interpolation polynomial does not converge to the value of the function at this point as the number of nodes increases indefinitely. This means the impossibility of ensuring efficient AI operation only by an unlimited increase in the number of neurons and data volumes (Big Data) used as training datasets.

Some aspects of extrapolation based on interpolation polynomials

The problem of extrapolation on the basis of interpolation polynomials is considered in the paper. A simple computational procedure is proposed to find the predicted value for a polynomial of any degree under conditions of a uniform grid. An algorithm for determining the best polynomial for extrapolation is proposed. To construction of integral transformation for operator of equation of convective diffusion under mixed boundary conditions.

Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

The focus of this paper is on the use of Polynomial Chaos for developing surrogate models for Differential Algebraic Equations with time-invariant uncertainties. Intrusive and non-intrusive approaches to synthesize Polynomial Chaos surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations, if the algebraic constraints are a function of the stochastic variable, some initial conditions of the differential algebraic equations are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a Polynomial Chaos based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the Polynomial Chaos models are validated using Monte Carlo simulations in addition to estimating the evolving PDFs of the states of the pendulum.

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ℒ m AG f has the property of m m ∈ ℤ , m > 0 degree polynomial reproducing and converges up to a rate of m + 1 . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.

Modelling Development of Voluntary Pension Fund Using Mathematical Model of Approximation with Lagrange Interpolation Polynomials

Corporate social responsibility (CSR), as a concept that tackles economic, The introduction of private pension funds is the essence of the reform of the pension system in Serbia. Private pension funds in Serbia are based on voluntary benefits. Thus, the functioning of the pension system takes place in three interconnected processes: payments to a voluntary pension fund, investment of free funds, and ultimately programmed payments – pensions. The stability in the voluntary pension funds and the predictability of payments allow the quality of investment portfolio to be formed and achieve a long-term yield of investment. In this paper, we implement a well-known approximation method of Lagrange polynomial interpolation. We use it in order to find appropriate mathematical model for prediction of the number of fund members and the average salary in Serbia. This calculation is based on data (average salaries and fund member) from the last five years, i.e. from the period 2015-2019. We calculated the exact mathematical formula, then we compared the results and predictions obtained with that formula and with the formula from one of our previous works. In keeping with that, the appropriate conclusions were given..

Asymptotics of Polynomial Interpolation and the Bernstein Constants

It is well known that the interpolation error for $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in $$L_{\infty }\left[ -1,1\right] $$ L ∞ - 1 , 1 by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when $$\alpha $$ α tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in the $$L_{\infty }$$ L ∞ norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

Grouped Secret Sharing Schemes Based on Lagrange Interpolation Polynomials and Chinese Remainder Theorem

In a t , n threshold secret sharing (SS) scheme, whether or not a shareholder set is an authorized set totally depends on the number of shareholders in the set. When the access structure is not threshold, (t,n) threshold SS is not suitable. This paper proposes a new kind of SS named grouped secret sharing (GSS), which is specific multipartite SS. Moreover, in order to implement GSS, we utilize both Lagrange interpolation polynomials and Chinese remainder theorem to design two GSS schemes, respectively. Detailed analysis shows that both GSS schemes are correct and perfect, which means any authorized set can recover the secret while an unauthorized set cannot get any information about the secret.