Random Batch Methods for Large Interacting Particle Systems

We develop Random Batch Methods (RBM) for large interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from \(O(N^{2})\) per time step to \(O(N)\), for a system with \(N\) particles with binary interactions. The algorithms are motivated by the mini-batch idea in machine learning. For interacting particle systems with bounded Lipschtiz interaction kernel and a strong convex confinement potential, we provide error estimate of the first marginal under Wasserstein distance uniformly in particle number and in time, a result of the law of large numbers in time. We apply RBM to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.