Artificial neural networks for solving ordinary and partial differential equations
Artificial neural networks for solving ordinary and partial differential equations
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DOI：
10.1109/72.712178
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发表时间：
1998-09-01
影响因子：
--
通讯作者：
Fotiadis, DI
中科院分区：
文献类型：
--
作者：
Lagaris, IE;Likas, A;Fotiadis, DI
提出了一种利用人工神经网络求解初值和边值问题的方法。微分方程的试解写成两部分的和。第一部分满足初始/边界条件，不包含可调参数。第二部分是为了不影响初始/边界条件而构造的。这部分涉及一个前馈神经网络，其中包含可调参数（权重）。因此，通过构造可以满足初始/边界条件，并且可以训练网络满足微分方程。这种方法的适用范围从单一的常微分方程到耦合的常微分方程系统，以及偏微分方程。在本文中，我们通过求解各种模型问题来说明该方法，并对几种偏微分方程的解与Galekrkin有限元法的解进行了比较。随着神经处理器和数字信号处理器的出现，由于预期的执行速度的基本增益，该方法变得特别有趣。
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural net work containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of coupled ODE's and also to partial differential equations (PDE's). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.