A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

Abstract

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.

Solving two-dimensional linear partial differential equations based on Chebyshev neural network with extreme learning machine algorithm

“…Figure 7 depicts the deviation error of this example in the case of random sampling. When Legendre improved ELM method (L-IELM) (Yang et al, 2018) with 15 neurons and 30 Â 30 training points is used to solve this problem, the MAE is 1.7647 Â 10 À6 . Table 2 shows the execution time with uniform sampling and the errors between the output of the model and the exact solution under different hidden neurons.…”

“…With the development of computer technology and the improvement of neural network theory, the neural network has been applied to various fields, such as approximating (Park and Sandberg, 1991;Hou et al, 2009;Hou and Han, 2010, forecasting , image classification (Yan et al, 2014), information processing (Kötter and Stephan, 2003), face recognition (Kim et al, 2017), finite deformation hyperelasticity and solving Des . At present, there are many neural networks for solving DEs, such as wavelet neural network (Li, 2010), cellular neural networks (Aein and Talebi, 2009;Klinkenbusch et al, 2011), finite element neural network (Beltzer et al, 2003), Chebyshev neural network (ChNN) , Legendre neural network (Yang et al, 2018), radial basis function neural network (Rizaner and Rizaner, 2018), Bernstein neural network (BeNN) (Sun et al, 2018) and hybrid neural network (Malek and Beidokhti, 2006). Cosmin et al (2019) proposes a partial differential equation (PDE) solution algorithm based on neural network, which adopts the adaptive configuration strategy, especially for non-smooth solutions, which saves a lot of computation and also improves the robustness of neural network approximation.…”

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

“…System (23), which consists of 3N -2M + 1 equations with unknowns (α, η, β, b, y), is solved by Newton's method. The LS-SVM model in the dual form becomeŝ…”

“…Neural network, which is one of machine intelligence techniques, has universal function approximation capabilities [20][21][22], and the solution obtained from the neural network is differentiable and in closed analytic form. Neural network has been widely used for solving ordinary differential equations [23,24], partial differential equations [25][26][27], fractional differential equations [28][29][30], and integro-differential equations [31,32]. Chakraverty and Mall [33] analyzed a regression-based neural network algorithm to solve two-point boundary value problems of fourth-order linear ordinary differential equations.…”

Designing optimal models of nonlinear MIMO systems based on orthogonal polynomial neural networks

“…Paper [9] is also interesting because it considers training methods for Chebyshev neural network utilized for fault classification in series compensated transmission line. Neural networks with Legendre basis in hidden neurons were also used in various applications like solving systems of ordinary differential equations [10] or prediction of chaotic time series [11]. However, it should be emphasized that classical, plain Legendre basis used in previous papers is different than quasi-orthogonal variant with adaptive capabilities, for the first time considered in our paper and applied in neural networks.…”