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We introduce the so called DeepParticle method to learn and generateinvariant measures of stochastic dynamical systems with physical parametersbased on data computed from an interacting particle method (IPM). We utilizethe expressiveness of deep neural networks (DNNs) to represent the transform ofsamples from a given input (source) distribution to an arbitrary targetdistribution, neither assuming distribution functions in closed form nor afinite state space for the samples. In training, we update the network weightsto minimize a discrete Wasserstein distance between the input and targetsamples. To reduce computational cost, we propose an iterativedivide-and-conquer (a mini-batch interior point) algorithm, to find the optimaltransition matrix in the Wasserstein distance. We present numerical results todemonstrate the performance of our method for accelerating IPM computation ofinvariant measures of stochastic dynamical systems arising in computingreaction-diffusion front speeds through chaotic flows. The physical parameteris a large Pecl\'et number reflecting the advection dominated regime of ourinterest.

DeepParticle: learning invariant measure by a deep neural network  minimizing Wasserstein distance on data generated from an interacting  particle method