DeepParticle: Learning Invariant Measure by a Deep Neural Network Minimizing Wasserstein Distance on Data Generated from an Interacting Particle Method

Abstract

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.