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In this paper, a numerical method to solve second order nonlinear ordinary differential equations with singular initial value problems was proposed. Here, block Legendre basis neural network (B-LBNN) was developed to create approximate solution and its derivatives. We use Legendre polynomials to expand the input pattern and eliminate hidden layer. By constructing block subnetwork and transforming variables, the problem of Emden–Fowler type equations was converted to solving the algebraic equation systems. The improved extreme learning machine algorithm was used to train network weights. A series of homogeneous or nonhomogeneous Emden–fowler equations were test to validate the accuracy and efficiency of B-LBNN model. Experimental results showed that the proposed algorithm provides a new tool for this equations and can outperform other approaches in literature in terms of accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we study numerical solution of ruin probability of continuous time model. We develop an effective optimal adaptive particle swarm optimization-triangular neural network (PSO-TNN), which consists of three parts: particle swarm optimization algorithm (PSO) improved trigonometric function, extreme learning machine algorithm with initial conditions (IELM) and improved reduction algorithm. The results obtained that PSO-TNN is superior to triangular neural network (TNN) and physics-informed neural networks (PINNs), and PSO is superior to Aquila Optimizer (AO), Smell Agent Optimization (SAO), African vultures optimization algorithm (AVOA), Arithmetic optimization algorithm (AOA) in the optimization of neural network. Because the relationship between the number of neural networks and the mean square error is uncertain, we propose the adaptive reduction algorithm (AR). Through the comparison of numerical solutions with the analytical solutions and traditional numerical solutions, the PSO-TNN algorithm clearly reduced the mean square error and relative error. The PSO-TNN algorithm shows a clear improvement in terms of accuracy and overall efficiency. [ABSTRACT FROM AUTHOR]

In this paper, block neural network (BNN) method is proposed to solve several kinds of differential equations. BNN is used to construct approximating functions and its derivatives, the improved extreme learning machine (IELM) algorithm are designed to train network weights. To evaluate the performance of the proposed method, numerical examples are performed by the presented method. Comparison of the numerical results with exact solutions validate the feasibility of the proposed method in accuracy. Results compared with other recent research works also validate the superiority of the proposed approach. Numerical results show that the proposed BNN with IELM algorithm perform well in accuracy and requires less hidden neurons. [ABSTRACT FROM AUTHOR]

To provide a new numerical algorithm for solving elliptic partial differential equations (PDEs), the Legendre neural network (LNN) and improved extreme learning machine (IELM) algorithm are introduced to propose a Legendre improved extreme learning machine (L-IELM) method, which is applied to solving elliptic PDEs in this paper. The product of two Legendre polynomials is chosen as basis functions of hidden neurons. Single hidden layer LNN is used to construct approximate solutions and its derivatives of differential equations. IELM algorithm is used for network weights training, and the algorithm steps of the proposed L-IELM method are summarized. Finally, in order to evaluate the present algorithm, various test examples are selected and solved by the proposed approach to validate the calculation accuracy. Comparative study with the earlier methods in literature is described to verify the superiority of the presented L-IELM method. Experiment results show that the proposed L-IELM algorithm can perform well in terms of accuracy and execution time, which in addition provides a new algorithm for solving elliptic PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method based on Bernstein Neural Network model (BeNN) and extreme learning machine algorithm to solve the differential equation. In the proposed method, we develop a single-layer functional link BeNN, the hidden layer is eliminated by expanding the input pattern by Bernstein polynomials. The network parameters are obtained by solving a system of linear equations using the extreme learning machine algorithm. Finally, the numerical experiment is carried out by MATLAB, results obtained are compared with the existing method, which proves the feasibility and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden-Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time. [ABSTRACT FROM AUTHOR]