Refining Mean-field Approximations by Dynamic State Truncation

Abstract

Mean-field models are an established method to analyze large stochastic systems with N interacting objects by means of simple deterministic equations that are asymptotically correct when N tends to infinity. For finite N, mean-field equations provide an approximation whose accuracy is model- and parameter-dependent. Recent research has focused on refining the approximation by computing suitable quantities associated with expansions of order $1/N$ and $1/N^2$ to the mean-field equation. In this paper we present a new method for refining mean-field approximations. It couples the master equation governing the evolution of the probability distribution of a truncation of the original state space with a mean-field approximation of a time-inhomogeneous population process that dynamically shifts the truncation across the whole state space. We provide a result of asymptotic correctness in the limit when the truncation covers the state space; for finite truncations, the equations give a correction of the mean-field approximation. We apply our method to examples from the literature to show that, even with modest truncations, it is effective in models that cannot be refined using existing techniques due to non-differentiable drifts, and that it can outperform the state of the art in challenging models that cause instability due orbit cycles in their mean-field equations.