  We propose a recurrent neural network for a "model-free" simulation of a
dynamical system with unknown parameters without prior knowledge. The deep
learning model aims to jointly learn the nonlinear time marching operator and
the effects of the unknown parameters from a time series dataset. We assume
that the time series data set consists of an ensemble of trajectories for a
range of the parameters. The learning task is formulated as a statistical
inference problem by considering the unknown parameters as random variables. A
latent variable is introduced to model the effects of the unknown parameters,
and a variational inference method is employed to simultaneously train
probabilistic models for the time marching operator and an approximate
posterior distribution for the latent variable. Unlike the classical
variational inference, where a factorized distribution is used to approximate
the posterior, we employ a feedforward neural network supplemented by an
encoder recurrent neural network to develop a more flexible probabilistic
model. The approximate posterior distribution makes an inference on a
trajectory to identify the effects of the unknown parameters. The time marching
operator is approximated by a recurrent neural network, which takes a latent
state sampled from the approximate posterior distribution as one of the input
variables, to compute the time evolution of the probability distribution
conditioned on the latent variable. In the numerical experiments, it is shown
that the proposed variational inference model makes a more accurate simulation
compared to the standard recurrent neural networks. It is found that the
proposed deep learning model is capable of correctly identifying the dimensions
of the random parameters and learning a representation of complex time series
data.
