Diffusion PDEs, such as the heat equation, porous medium equation, and Fokker-Planck equation, can be obtained as gradient flows of energy functionals with respect to the Wasserstein-2 metric on the space of probability measures. These equations are often highly non linear, and their underlying geometric structure necessitates that numerical schemes must produce positive, mass-conserving, entropy stable solutions. In this seminar, I begin in the classical regime by presenting two first-order, structure-preserving finite-element schemes tailored to the porous medium equation. I then demonstrate that these PDEs admit a Mean-Field Control reformulation that seeks solutions by solving generalized Optimal Transport type problems. The MFC framework enables the computation of generalized barycenters in the Wasserstein-2 metric space, incorporating nonlinear reaction-diffusion dynamics. Finally, I outline an application of MFC to develop arbitrarily high-order finite-element schemes capable of solving general Wasserstein gradient flows, showcasing preliminary results.

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