Variational inference formulation for a model-free simulation of a dynamical system with unknown parameters by a recurrent neural network

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 variational inference method is employed to train a recurrent neural network jointly with a feedforward neural network for an approximately posterior distribution. The approximate posterior distribution makes an inference on a trajectory to identify the effects of the unknown parameters and a recurrent neural network makes a prediction by using the outcome of the inference. 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.