  We introduce the so called DeepParticle method to learn and generate
invariant measures of stochastic dynamical systems with physical parameters
based on data computed from an interacting particle method (IPM). We utilize
the expressiveness of deep neural networks (DNNs) to represent the transform of
samples from a given input (source) distribution to an arbitrary target
distribution, neither assuming distribution functions in closed form nor a
finite state space for the samples. In training, we update the network weights
to minimize a discrete Wasserstein distance between the input and target
samples. To reduce computational cost, we propose an iterative
divide-and-conquer (a mini-batch interior point) algorithm, to find the optimal
transition matrix in the Wasserstein distance. We present numerical results to
demonstrate the performance of our method for accelerating IPM computation of
invariant measures of stochastic dynamical systems arising in computing
reaction-diffusion front speeds through chaotic flows. The physical parameter
is a large Pecl\'et number reflecting the advection dominated regime of our
interest.
