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Random ordinary differential equations (RODEs), i.e. ODEs with randomparameters, are often used to model complex dynamics. Most existing methods toidentify unknown governing RODEs from observed data often rely on strong priorknowledge. Extracting the governing equations from data with less priorknowledge remains a great challenge. In this paper, we propose a deep neuralnetwork, called RODE-Net, to tackle such challenge by fitting a symbolicexpression of the differential equation and the distribution of parameterssimultaneously. To train the RODE-Net, we first estimate the parameters of theunknown RODE using the symbolic networks \cite{long2019pde} by solving a set ofdeterministic inverse problems based on the measured data, and use a generativeadversarial network (GAN) to estimate the true distribution of the RODE'sparameters. Then, we use the trained GAN as a regularization to further improvethe estimation of the ODE's parameters. The two steps are operatedalternatively. Numerical results show that the proposed RODE-Net can wellestimate the distribution of model parameters using simulated data and can makereliable predictions. It is worth noting that, GAN serves as a data drivenregularization in RODE-Net and is more effective than the $\ell_1$ basedregularization that is often used in system identifications.

RODE-Net: Learning Ordinary Differential Equations with Randomness from Data