| 几类微分-代数方程的神经网络求解法 |
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| 引用本文： | 杨钊,兰钧,吴勇军. 几类微分-代数方程的神经网络求解法[J]. 应用数学和力学, 2019, 40(2): 115-126. DOI: 10.21656/1000-0887.390122 |
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| 作者姓名： | 杨钊 兰钧 吴勇军 |
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| 作者单位： | 1上海交通大学 工程力学系, 上海 200240;2卫宁健康科技集团股份有限公司, 上海 200072 |
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| 基金项目： | 国家自然科学基金（11772293；11272201） |
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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.
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| 关 键 词： | 人工神经网络 微分-代数方程 近似解析解 最优化方法 |
| 收稿时间： | 2018-04-18 |
On Solutions to Several Classes of Differential-Algebraic Equations Based on Artificial Neural Networks |
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| YANG Zhao,LAN Jun,WU Yongjun. On Solutions to Several Classes of Differential-Algebraic Equations Based on Artificial Neural Networks[J]. Applied Mathematics and Mechanics, 2019, 40(2): 115-126. DOI: 10.21656/1000-0887.390122 |
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| Authors: | YANG Zhao LAN Jun WU Yongjun |
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| Affiliation: | 1Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, P.R.China;2Winning Health Technology Group Co., Ltd.,Shanghai 200072, P.R.China |
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| 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 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. |
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| Keywords: | artificial neural network differential-algebraic equation approximate analytical solution optimization method |
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