几类微分-代数方程的神经网络求解法

On Solutions to Several Classes of Differential-Algebraic Equations Based on Artificial Neural Networks

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.

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 differential-algebraic 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 Euler-Lagrange 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 Runge-Kutta method.High accuracy of the proposed method was demonstrated.