Comparison of three unsupervised neural network models for first Painlevé Transcendent

Abstract

In this paper, a reliable soft computing framework is presented for the approximate solution of initial value problem (IVP) of first Painlevé equation using three unsupervised neural network models optimized with sequential quadratic programming (SQP). These mathematical models are constructed in the form of feed-forward architecture including log-sigmoid, radial base and tansigmoid activation functions in the hidden layers. The optimization of designed parameters for each model is performed with SQP, an efficient constraint optimization problem-solving algorithm. The designed methodology is tested on the IVP, and comparative study is carried out with standard solution based on numerical and analytical solvers. The accuracy, convergence and effectiveness of the schemes are validated on the given benchmark problem by large number of simulations and their comprehensive analysis.

Approximate Solutions for Solving Fractional-order Painlevé Equations

“…For comparison, the following numerical methods are used. Theses include varational iteration method (VIM) [12] , homotopy perturbation method (HPM) [12] , homotopy analysis method (HAM) [12] , particle swarm optimization algorithm (PSOA) [21] , neural networks algorithm (NNA) [22] , and reproducing kernel algorithm (RKA) [3] . The numerical results obtained by (23) while using N = 15 and N = 20 are shown in Table 2.…”

“…Contrary to the first Painlevé equation, in this case also we need a higher number of basis functions to get reasonable solutions. Let us consider the approximate solutions Y 25,5 (t) obtained via (22) of the model (2) In the next experiments, we fix ν = 2 and use r = 5 iterations. We employ different number of basis functions N = 20, 25, 30,35 as well as various values for the parameter µ2 = 0, 1, 2.…”

“…The most significat analytical schemes include Adomian's decomposition (ADM) and homotopy perturbation methods (HPM) [7] , variational iteration method (VIM) and HPM [12] , homotopy analysis method [8,11] , sinc collocation method and VIM [23] , optimal homotopy asymptotic method [16] , to name but a few. On the other hand, numerical techniques such as Chebyshev series [6,13] , computational intelligence technique based on neural networks and particle swarm optimization [21,22] , and reproducing kernel Hilbert space algorithms [3] have been developed in the past to solve the nonlinear equation (1)- (2).…”

A Soft Computing Approach Based on Fractional Order DPSO Algorithm Designed to Solve the Corneal Model for Eye Surgery

“…These artificial intelligence schemes are suitable to calculate solutions for the CSM. Many researchers had worked on the said scheme which includes solvers are in electromagnetic [23], circuit theory [24], fuel ignition model [25], Thomas-Fermi model [26], induction of the motor models [27], doubly singular nonlinear systems [27], nanofluidics [28], nanotechnology [29], nonlinear prey-predator models [30], Troesch's problem [31], nonlinear equations [32], optimal control [33], mathematical modeling and control theory of particle accelerators [34], signal processing [35], linear and nonlinear fractional order model [36], financial mathematics [37], physical models signified nonlinear system of equations [38] and powerfully nonlinear differential equations with many singularities of Painleve equations [39]. Recently, the design analysis of porous fins is studied with the help of a combined procedure CS-ANN in [40].…”

Optimal Type-3 Fuzzy System for Solving Singular Multi-Pantograph Equations

“…In [20], the Lagaris method is developed to estimate the real solution of PDEs by a simple NN. In [21], NNs are used to solve Painlevé equations and the effectiveness of NNs is shown. In [22], similar to above mentioned approaches, the solving of a class of differential equations by NN is investigated and the genetic algorithm is used to optimize NNs.…”