RODE-Net：从数据中学习具有随机性的常微分方程
RODE-Net: Learning Ordinary Differential Equations with Randomness from Data
摘要：
随机常微分方程（RODE），即具有随机参数的常微分方程，经常被用来模拟复杂的动力学。大多数现有的方法，以确定未知的执政RODE从观测数据往往依赖于强大的先验知识。从先验知识较少的数据中提取控制方程仍然是一个巨大的挑战。在本文中，我们提出了一种称为RODE-Net的深度神经网络，通过同时拟合微分方程的符号表达式和参数分布来应对这一挑战。为了训练RODE-Net，我们首先使用符号网络\cite{long2019pde}通过求解一组基于测量数据的确定性逆问题来估计未知RODE的参数，并使用生成对抗网络（GAN）来估计RODE参数的真实分布。然后，我们使用训练好的GAN作为正则化，以进一步提高ODE参数的估计。这两个步骤交替操作。数值结果表明，所提出的RODE-Net可以很好地估计模型参数的分布，利用模拟数据，可以作出可靠的预测。值得注意的是，GAN在RODE-Net中用作数据驱动的正则化，并且比经常用于系统辨识的基于$\ell_1$的正则化更有效。
英文摘要：
Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge. Extracting the governing equations from data with less prior knowledge remains a great challenge. In this paper, we propose a deep neural network, called RODE-Net, to tackle such challenge by fitting a symbolic expression of the differential equation and the distribution of parameters simultaneously. To train the RODE-Net, we first estimate the parameters of the unknown RODE using the symbolic networks \cite{long2019pde} by solving a set of deterministic inverse problems based on the measured data, and use a generative adversarial network (GAN) to estimate the true distribution of the RODE's parameters. Then, we use the trained GAN as a regularization to further improve the estimation of the ODE's parameters. The two steps are operated alternatively. Numerical results show that the proposed RODE-Net can well estimate the distribution of model parameters using simulated data and can make reliable predictions. It is worth noting that, GAN serves as a data driven regularization in RODE-Net and is more effective than the $\ell_1$ based regularization that is often used in system identifications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
