Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems

TL;DR

This work presents a LOT-based, measure-valued Euler scheme on the Wasserstein manifold to predict the evolution of time-dependent probability measures using only short-time oracle data $E_h$, avoiding explicit operator learning. The authors prove a local truncation error of $O(H^2)$ and a global error of $O(H)$ under regularity when evolving smoothly, and they translate this into a practical algorithm for empirical measures $\mu_t^N$ arising from particle systems. They demonstrate the approach on three simulation studies—biological chemotaxis, a PDE surrogate (Burgers) and Langevin dynamics—showing reduced micro-scale steps while preserving accuracy and offering a macro-scale timestepper suitable for fixed-point, bifurcation, and control analyses. The results highlight the method’s capacity to extract macroscopic, slow dynamics from chaotic micro-scale motion by leveraging optimal transport geometry, thus enabling efficient forward prediction in multiscale particle systems.

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, 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.

Figures (14)

• Figure 1: Example of the difference between the optimal transport map and the particle-wise map. In the particle-wise image (left), an arrow is drawn to connect each particle's starting position to its ending position. In the optimal transport image (right), we compute the OT map between the starting and ending distributions and draw an arrow connecting each particle's starting location to its corresponding output under the OT map.
• Figure 2: Vector field approximation error $\|\mathbf{v}_t - \mathbf{v}_t^{h,N}\|_{L^2(\mu_t^N)}^2$ versus Euler step approximation error $W_2(\mu_{t+H}^N, \widetilde{\mu}_{t+H}^N)^2$ for Brownian motion with $t = 1$ and $H = 99$ ($h$ and $N$ vary)
• Figure 3: Results of vector field error computations for Brownian motion and comparison with an estimated error using the difference between consecutive approximations.
• Figure 4: Histograms of bacteria locations at various times $t$ from $t=0$ to $t=4000$ for the control simulation of chemotaxis.
• Figure 5: Histograms of bacteria locations after first two Euler steps (bottom) compared with corresponding times from control simulation (top). We run the micro-scale simulation until $t=200$ (not pictured), take an Euler step to $t=300$ (leftmost), burn in until $t=400$ (second from left), take an Euler step to $t=500$ (second from right), and burn in until $t=600$ (rightmost).

Theorems & Definitions (3)

• proof
• proof : Proof of Theorem \ref{['thm:w2-euler-global-error']}
• proof