Gradient flows on graphons: existence, convergence, continuity equations

Understand

Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems.

• They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential.
• However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable.
• Such large graphs are known to converge to continuum limits called graphons as their size grow to infinity.