Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion
  Processes
Haoming Yang Ali Hasan Yuting Ng Vahid Tarokh
McKean-Vlasov stochastic differential equations (MV-SDEs) provide a
mathematical description of the behavior of an infinite number of interacting
particles by imposing a dependence on the particle density. As such, we study
the influence of explicitly including distributional information in the
parameterization of the SDE. We propose a series of semi-parametric methods for
representing MV-SDEs, and corresponding estimators for inferring parameters
from data based on the properties of the MV-SDE. We analyze the characteristics
of the different architectures and estimators, and consider their applicability
in relevant machine learning problems. We empirically compare the performance
of the different architectures and estimators on real and synthetic datasets
for time series and probabilistic modeling. The results suggest that explicitly
including distributional dependence in the parameterization of the SDE is
effective in modeling temporal data with interaction under an exchangeability
assumption while maintaining strong performance for standard It\^o-SDEs due to
the richer class of probability flows associated with MV-SDEs.