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Solve Systems of Ordinary Differential Equations Using Deep Neural Networks

机译：使用深神经网络解决常微分方程的系统

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摘要

The systems of ordinary differential equations have been ubiquitously investigated and had many applications for various areas in real life. This paper investigates a deep learning method to solve the systems of ordinary differential equations (ODEs). We formulate the original problem with the initial conditions as an optimization problem. By minimizing a loss function associated with the optimization problem, we can construct an appropriate neural network to estimate the exact solutions of the systems of equations. We do experiments by considering two types of ODEs, Lotka-Volterra and Biochemical Oscillator equations. The experimental results show that we can obtain accurate results in solving these two systems of ODEs, where the numerical errors (mean square errors) vary from 10−6 to 10−11 for different neural networks, compared to the traditional approaches.
机译：常微分方程的系统已经无所不在调查和有在现实生活中的各个领域有许多应用。本文研究了深刻的学习方法来解决常微分方程（常微分方程）的系统。我们制定了原来的问题与初始条件的优化问题。通过最小化与优化问题相关的损失函数，我们可以构建一个合适的神经网络来估计方程组的精确解。我们考虑两种类型的常微分方程，洛特卡 - 沃尔泰拉生化振荡器方程做实验。实验结果表明，我们可以在解决常微分方程，这两个系统在数值误差（均方误差）从10变化获得准确的结果 -6 到10. -11 针对不同的神经网络，相比传统方法。

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