Solving ordinary differential equations using an optimization technique based on training improved artificial neural net
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METHODOLOGIES AND APPLICATION
Solving ordinary differential equations using an optimization technique based on training improved artificial neural networks Shangjie Li1 • Xingang Wang2
Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The solution of ordinary differential equations (ODEs) arises in a wide variety of engineering problems. This paper presents a novel method for the numerical solution of ODEs using improved artificial neural networks (IANNs). In the first step, we derive an approximate solution of ODEs by artificial neural networks (ANNs). Then, we construct a joint cost function of network system, it consists of several error functions corresponding to different sample points, and we reformulate Levenberg–Marquardt (RLM) algorithm to adjust the network parameters. The advantages of this method are high calculation accuracy and fast convergence speed compared with other existed methods, also increasing the simulation stability of ANNs method. The performance of the new proposed method in terms of calculation accuracy and convergence speed is analyzed for several different types of nonlinear ODEs. Keywords Differential equations Improved artificial neural networks Joint cost function Reformulate Levenberg– Marquardt algorithm
1 Introduction Differential equations are the basic structures in representation of engineering problems; many problems in science and engineering can be reduced to a set of differential equations through a process of mathematical modeling (Sneddon 2006; Ricardo 2009). Problems involving ordinary differential equations (ODEs) can be divided into two main categories, namely initial value problems (IVPs) and boundary value problems (BVPs). Analytic solutions for these problems are not generally available, especially for nonlinear ODEs; hence, the numerical methods must be applied. So various numerical methods such as predictor– corrector (Douglas and Jones 1963) and Runge–Kutta (Wambecq 1978) have been developed to solve these equations. These numerical methods require the
Communicated by V. Loia. & Xingang Wang [email protected] 1
School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
2
Research Center of Mechanical Kinetics and Reliability, Northeastern University, Qinhuangdao 066004, China
discretization of domain into the number of finite domains or points where the function is approximated locally. Various machine intelligence methods, in particular artificial neural networks (ANNs), have been used to solve differential equations; the ANNs method in comparison with other numerical methods has more advantages. Most other numerical methods are usually iterative in nature, where we fix the step size before initiating 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. The ANNs method may be one of the reliefs where we may overcome this repetition o
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