Evolutionary Integrated Heuristic with Gudermannian Neural Networks for Second Kind of Lane–Emden Nonlinear Singular Models

In this work, a new heuristic computing design is presented with an artificial intelligence approach to exploit the models with feed-forward (FF) Gudermannian neural networks (GNN) accomplished with global search capability of genetic algorithms (GA) combined with local convergence aptitude of active-set method (ASM), i.e., FF-GNN-GAASM to solve the second kind of Lane–Emden nonlinear singular models (LE-NSM). The proposed method based on the computing intelligent Gudermannian kernel is incorporated with the hidden layer configuration of FF-GNN models of differential operatives of the LE-NSM, which are arbitrarily associated with presenting an error-based objective function that is used to optimize by the hybrid heuristics of GAASM. Three LE-NSM-based examples are numerically solved to authenticate the effectiveness, accurateness, and efficiency of the suggested FF-GNN-GAASM. The reliability of the scheme via statistical valuations is verified in order to authenticate the stability, accuracy, and convergence

Integrated computational intelligent paradigm for nonlinear electric circuit models using neural networks, genetic algorithms and sequential quadratic programming

© 2019, Springer-Verlag London Ltd., part of Springer Nature. In this paper, a novel application of biologically inspired computing paradigm is presented for solving initial value problem (IVP) of electric circuits based on nonlinear RL model by exploiting the competency of accurate modeling with feed forward artificial neural network (FF-ANN), global search efficacy of genetic algorithms (GA) and rapid local search with sequential quadratic programming (SQP). The fitness function for IVP of associated nonlinear RL circuit is developed by exploiting the approximation theory in mean squared error sense using an approximate FF-ANN model. Training of the networks is conducted by integrated computational heuristic based on GA-aided with SQP, i.e., GA-SQP. The designed methodology is evaluated to variants of nonlinear RL systems based on both AC and DC excitations for number of scenarios with different voltages, resistances and inductance parameters. The comparative studies of the proposed results with Adam’s numerical solutions in terms of various performance measures verify the accuracy of the scheme. Results of statistics based on Monte-Carlo simulations validate the accuracy, convergence, stability and robustness of the designed scheme for solving problem in nonlinear circuit theory

Connectionist Learning Based Numerical Solution of Differential Equations

It is well known that the differential equations are back bone of different physical systems. Many real world problems of science and engineering may be modeled by various ordinary or partial differential equations. These differential equations may be solved by different approximate methods such as Euler, Runge-Kutta, predictor-corrector, finite difference, finite element, boundary element and other numerical techniques when the problems cannot be solved by exact/analytical methods. Although these methods provide good approximations to the solution, they require a discretization of the domain via meshing, which may be challenging in two or higher dimension problems. These procedures provide solutions at the pre-defined points and computational complexity increases with the number of sampling points.In recent decades, various machine intelligence methods in particular connectionist learning or Artificial Neural Network (ANN) models are being used to solve a variety of real-world problems because of its excellent learning capacity. Recently, a lot of attention has been given to use ANN for solving differential equations. The approximate solution of differential equations by ANN is found to be advantageous but it depends upon the ANN model that one considers. Here our target is to solve ordinary as well as partial differential equations using ANN. The approximate solution of differential equations by ANN method has various inherent benefits in comparison with other numerical methods such as (i) the approximate solution is differentiable in the given domain, (ii) computational complexity does not increase considerably with the increase in number of sampling points and dimension of the problem, (iii) it can be applied to solve linear as well as non linear Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). Moreover, the traditional numerical methods are usually iterative in nature, where we fix the step size before the start of the computation. After the solution is obtained, if we want to know the solution in between steps then again the procedure is to be repeated from initial stage. ANN may be one of the ways where we may overcome this repetition of iterations. Also, we may use it as a black box to get numerical results at any arbitrary point in the domain after training of the model.Few authors have solved ordinary and partial differential equations by combining the feed forward neural network and optimization technique. As said above that the objective of this thesis is to solve various types of ODEs and PDEs using efficient neural network. Algorithms are developed where no desired values are known and the output of the model can be generated by training only. The architectures of the existing neural models are usually problem dependent and the number of nodes etc. are taken by trial and error method. Also, the training depends upon the weights of the connecting nodes. In general, these weights are taken as random number which dictates the training. In this investigation, firstly a new method viz. Regression Based Neural Network (RBNN) has been developed to handle differential equations. In RBNN model, the number of nodes in hidden layer may be fixed by using the regression method. For this, the input and output data are fitted first with various degree polynomials using regression analysis and the coefficients involved are taken as initial weights to start with the neural training. Fixing of the hidden nodes depends upon the degree of the polynomial.We have considered RBNN model for solving different ODEs with initial/boundary conditions. Feed forward neural model and unsupervised error back propagation algorithm have been used for minimizing the error function and modification of the parameters (weights and biases) without use of any optimization technique. Next, single layer Functional Link Artificial Neural Network (FLANN) architecture has been developed for solving differential equations for the first time. In FLANN, the hidden layer is replaced by a functional expansion block for enhancement of the input patterns using orthogonal polynomials such as Chebyshev, Legendre, Hermite, etc. The computations become efficient because the procedure does not need to have hidden layer. Thus, the numbers of network parameters are less than the traditional ANN model. Varieties of differential equations are solved here using the above mentioned methods to show the reliability, powerfulness, and easy computer implementation of the methods. As such singular nonlinear initial value problems such as Lane-Emden and Emden-Fowler type equations have been solved using Chebyshev Neural Network (ChNN) model. Single layer Legendre Neural Network (LeNN) model has also been developed to handle Lane-Emden equation, Boundary Value Problem (BVP) and system of coupled ordinary differential equations. Unforced Duffing oscillator and unforced Van der Pol-Duffing oscillator equations are solved by developing Simple Orthogonal Polynomial based Neural Network (SOPNN) model. Further, Hermite Neural Network (HeNN) model is proposed to handle the Van der Pol-Duffing oscillator equation. Finally, a single layer Chebyshev Neural Network (ChNN) model has also been implemented to solve partial differential equations

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery–Hamel flow

© 2018 Taiwan Institute of Chemical Engineers In this paper, a neuro-heuristic technique by incorporating artificial neural network models (NNMs) optimized with sequential quadratic programming (SQP) is proposed to solve the dynamics of nanofluidics system based on magneto-hydrodynamic (MHD) Jeffery–Hamel (JHF) problem involving nano-meterials. Original partial differential equations associated with MHD–JHF are transformed into third order ordinary differential equations based model. Furthermore, the transformed system has been implemented by the differential equation NNMs (DE-NNMs) which are constructed by a defined error function using log-sigmoid, radial basis and tan-sigmoid windowing kernels. The parameters of DE-NNM of nanofluidics system are optimized with SQP algorithm. To illustrate the performance of the proposed system, MHD–JHF models with base-fluid water mixed with alumina, silver and copper nanoparticles for different Hartman numbers, Reynolds numbers, angles of the channel and volume fractions with three different proposed DE-NNMs are designed to evaluate. For comparison purpose, the proposed results with reference numerical solutions of Adams solver illustrate their worth. Statistical inferences through different performance indices are given to demostrate the accuracy, stability and robustness of the stochastic solvers

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”