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Explore an advanced mathematical approach to predicting the evolution of stochastic particle systems through linearized optimal transport theory in this 41-minute conference talk. Learn how to approximate the time evolution of probability measures without explicitly learning governing operators, particularly for discrete measures arising from particle systems where individual particles move chaotically on short time scales but bulk distributions evolve smoothly. Discover the development of an Euler-like scheme that uses "tangent vector fields" computed as limits of optimal transport maps to predict evolution when measures are known on specific time intervals. Examine the theoretical foundations including conditions for first-order accuracy, stability over multiple Euler steps, and computable conditions for maximum step sizes. Understand how this approach extends to discrete measure evolution and see practical demonstrations through two illustrative examples: Gaussian diffusion and a cell chemotaxis model, where the method successfully predicts bulk behavior over large steps while significantly reducing required microscale simulation steps to reach steady state distributions.
