Optimal Transport and PDE: Gradient Flows in the Wasserstein Metric
summary
Survey of specific PDEs that are Wasserstein gradient flows, the particle schemes used, and when they provably match the continuum: continuity equation with ODE particle limit; aggregation PDE with singular interaction potentials and empirical measures converging in Wasserstein metric; Fokker–Planck as gradient flow of V+entropy with stochastic particle methods converging almost surely; mean‑field two‑layer NN training is the gradient flow of the quadratic loss; chi‑squared flow gives a porous‑medium type PDE and is discretized by regularize with convolution to restore Lipschitz, then send regularization→0 and particles→∞.
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chapters (17)
- 0:00Introduction
- 0:55Motivation
- 2:03Continuity Equation
- 5:40PDE Properties
- 7:20Order of Convergence
- 9:11Aggregation Equation
- 10:35Dynamics
- 13:15Why PDE
- 15:07Grading flow
- 21:33Twolayer neural networks
- 25:43Chisquared divergence
- 29:05The plan
- 32:05What is perpendicular mean
- 33:44When do solutions exist
- 34:25Uniqueness
- 44:50Intuition
- 51:05Existence
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