  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.
