Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems (2408.01857v3)

Abstract: We develop an Euler-type method to predict the evolution of a time-dependent probability measure without explicitly learning an operator that governs its evolution. We use linearized optimal transport theory to prove that the measure-valued analog of Euler's method is first-order accurate when the measure evolves ``smoothly.'' In applications of interest, however, the measure is an empirical distribution of a system of stochastic particles whose behavior is only accessible through an agent-based micro-scale simulation. In such cases, this empirical measure does not evolve smoothly because the individual particles move chaotically on short time scales. However, we can still perform our Euler-type method, and when the particles' collective distribution approximates a measure that \emph{does} evolve smoothly, we observe that the algorithm still accurately predicts this collective behavior over relatively large Euler steps. We specifically demonstrate the efficacy of our approach by showing that our algorithm vastly reduces the number of micro-scale steps needed to correctly approximate long-term behavior in two illustrative examples, reflected Brownian motion and a model of bacterial chemotaxis.