  We present a deep learning model for data-driven simulations of random
dynamical systems without a distributional assumption. The deep learning model
consists of a recurrent neural network, which aims to learn the time marching
structure, and a generative adversarial network (GAN) to learn and sample from
the probability distribution of the random dynamical system. Although GANs
provide a powerful tool to model a complex probability distribution, the
training often fails without a proper regularization. Here, we propose a
regularization strategy for a GAN based on consistency conditions for the
sequential inference problems. First, the maximum mean discrepancy (MMD) is
used to enforce the consistency between conditional and marginal distributions
of a stochastic process. Then, the marginal distributions of the multiple-step
predictions are regularized by using MMD or from multiple discriminators. The
behavior of the proposed model is studied by using three stochastic processes
with complex noise structures.
