Georgia Institute of Technology College of Sciences

Many evolution equations arising from interacting particle systems, particularly the nonlinear Fokker–Planck (McKean–Vlasov) equation, are Wasserstein gradient flows. This talk looks at two complementary questions.
First, when the medium is spatially inhomogeneous and oscillates, does the $\varepsilon$-gradient flow homogenize as $\varepsilon$ goes to 0? We show evolutionary Gamma-convergence of the flow in energy-dissipation-inequality form to an effective gradient flow, and identify the resulting effective Wasserstein metric. Notably, it is strictly larger than the "naive" Gromov–Hausdorff limit of the $\varepsilon$-metrics themselves, showing that the gradient-flow structure encodes genuinely more information than the metric space alone.
Second, given only noisy observations of the flow, can we learn the energy — the interaction kernel, confining potential, and diffusion — that generates it? We introduce self-test loss functions, a derivative-free, high-dimension-friendly estimator built from the weak form of the equation. We give identification analysis when the resulting inverse problem is well- or ill-posed.