HEI Yafang,HU Jiancheng.Numerical Solutions and Methods for Ordinary Differential Equations[J].Journal of Chengdu University of Information Technology,2024,39(04):499-511.[doi:10.16836/j.cnki.jcuit.2024.04.017]
常微分方程的数值求解与方法
- Title:
- Numerical Solutions and Methods for Ordinary Differential Equations
- 文章编号:
- 2096-1618(2024)04-0499-13
- 分类号:
- O241.81
- 文献标志码:
- A
- 摘要:
- 针对各种微分方程数值求解的方法,如有限差分法、有限元法等方法,在求解微分方程时存在计算存储量、计算时间等都随着微分方程维数的增加而剧烈增长的问题,严重制约了高维问题的求解。 神经网络因其能够无限逼近任意非线性函数的特性,为求解微分方程提供了一种新的思路。通过神经网络训练,得到微分方程的近似解是连续函数,且具有足够的精度,因此可以得到解的任意阶导数。该方法的优势在于当问题维数增大时,计算量和存储量增加相对较小,可以克服维数灾难求解高维问题; 同时,具有良好的泛化性和求解复杂区域问题的能力。提出一种求解微分方程的神经网络方法,即通过物理约束耦合神经网络的方法。通过数值算例说明,神经网络方法在求解微分方程问题上有高效率、很好的泛化等优点,能够保证优化算法的收敛性,且近似解具有足够的精度,为微分方程求解提供了一种有效的途径。
- Abstract:
- For the solution of various numerical methods of differential equations, such as finite difference and finite element methods, there exists a problem that the calculation storage and calculation time increase dramatically with the dimension of the differential equation, which seriously restricts the solution of high-dimensional problems. Neural network provides a new approach to solving differential equations, as it can infinite approximation to any nonlinear function. Through neural network training, the approximate solution of a differential equation becomes a continuous function, with enough precision to obtain any order derivative of the solution. The advantage of this method is that when the dimension of the problem increases, the increase of computation and storage is relatively small, and it can overcome the disaster of dimension and solve the high-dimension problem. At the same time, it has good generalization and the ability to solve complex regional problems. In this paper, a neural network method for solving differential equations is proposed, which is coupled by physical constraints. The numerical examples demonstrate that this method has high efficiency and good generalization, ensuring the convergence of the optimization algorithm, and the approximate solution has enough precision, which provides an effective way of solving differential equations.
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备注/Memo
收稿日期:2023-02-09
基金项目:四川省科技计划重点研发资助项目(2019YFS0143)