Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems
This work provides a macro-scale timestepper for computational modeling in fields like biology and physics, though it is incremental as it builds on existing optimal transport and Euler methods.
The authors tackled the problem of predicting the evolution of stochastic particle systems without explicitly learning governing operators, by developing an Euler-type method using linearized optimal transport, which accurately predicts collective behavior over large time steps, reducing micro-scale simulation steps.
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, thus reducing the number of micro-scale steps required to step forward in time. In this way, our algorithm provides a ``macro-scale timestepper'' that requires less micro-scale data to still maintain accuracy, which we demonstrate with three illustrative examples: a biological agent-based model, a model of a PDE, and a model of Langevin dynamics.