On Solutions to Several Classes of Differential-Algebraic Equations Based on Artificial Neural Networks
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摘要: 在非线性科学中,寻求微分方程的近似解析解一直是重要的研究课题和研究热点.利用人工神经网络原理,结合最优化方法,研究了几类微分-代数方程的近似解析解,包括指标1,2,3型Hessenberg方程及指标3型Euler-Lagrange方程,得到了方程近似解析解的表达式.通过与精确解或Runge-Kutta(龙格-库塔)数值计算结果对比,表明神经网络方法的结果有很高的精度.Abstract: In nonlinear science, it is always an important subject and research focus to find the approximate analytical solutions to differential equations. The artificial neural network and the optimization method were combined to solve 2 special classes of differentialalgebraic equations (DAEs). The 1st 3 numerical examples, namely, the Hessenberg DAEs with indices 1, 2, 3, fell into a category of pure mathematical problems. Then the 2nd example related to EulerLagrange DAEs with indices 3, i.e. a pendulum without external force, arising from the background of nonholonomic mechanics. The approximate analytical solutions to the above 4 examples were obtained and compared with the exact solutions and the results from the RungeKutta method. High accuracy of the proposed method was demonstrated.
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