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.

将分布依赖显式地包含在随机微分方程的参数化中对于建模具有交互性的时间数据是有效的，并在维持标准 Itô-SDE 的优越性能的同时，通过 MV-SDE 关联的更丰富的概率流类来进行时间数据的建模。

神经麦克林 - 瓦萨夫过程：扩散过程中的分布依赖

Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion  Processes

In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasov
stochastic differential equations (SDEs) assuming only a standard monotonicity
condition on the drift and diffusion coefficients but no global Lipschitz
continuity in the state variable for either, while global Lipschitz continuity
is required for the measure component only. We prove moment stability of the
discretised processes and a strong convergence rate of $1/2$. Several numerical
examples, centred around a mean-field model for FitzHugh-Nagumo neurons,
illustrate that the standard uniform scheme fails and that the adaptive
approach shows in most cases superior performance to tamed approximation
schemes. In addition, we introduce and analyse an adaptive Milstein scheme for
a certain sub-class of McKean-Vlasov SDEs with linear measure-dependence of the
drift.

本文提出使用调整的 Euler-Maruyama 方案来处理 McKean-Vlasov 随机微分方程，该方案仅假设漂移和扩散系数具有标准单调性条件但状态变量没有全局 Lipschitz 连续性，措施项组件仅需要全局 Lipschitz 连续性，针对 FitzHugh-Nagumo 神经元的平均场模型，数值结果表明该调整方案在大多数情况下优于经过调整的逼近方案，同时还介绍并分析了一个针对具有线性措施依赖性漂移的某些子类 McKean-Vlasov SDE 的自适应 Milstein 方案。

具有超线性增长的 McKean-Vlasov SDEs 的自适应 Euler-Maruyama 方案及其在平均场 FitzHugh-Nagumo 模型中的应用

An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with  super-linear growth and application to the mean-field FitzHugh-Nagumo model

We present two fully probabilistic Euler schemes, one explicit and one
implicit, for the simulation of McKean-Vlasov Stochastic Differential Equations
(MV-SDEs) with drifts of super-linear growth and random initial condition.
We provide a pathwise propagation of chaos result and show strong convergence
for both schemes on the consequent particle system. The explicit scheme attains
the standard $1/2$ rate in stepsize. From a technical point of view, we
successfully use stopping times to prove the convergence of the implicit method
although we avoid them altogether for the explicit one. The combination of
particle interactions and random initial condition makes the proofs technically
more involved.
Numerical tests recover the theoretical convergence rates and illustrate a
computational complexity advantage of the explicit over the implicit scheme. A
comparative analysis is carried out on a stylized non-Lipschitz MV-SDE and a
mean-field model for FitzHugh-Nagumo neurons. We provide numerical tests
illustrating "particle corruption" effect where one single particle diverging
can "corrupt" the whole particle system. Moreover, the more particles in the
system the more likely this divergence is to occur.

本研究提出了两种完全概率欧拉方案，一种是显式的，一种是隐式的，用于模拟随机初始条件下具有超线性增长漂移的麦凯恩 - 弗拉索夫随机微分方程，并证明了在其后的粒子系统中两种方案都有强收敛性。经过数值测试发现，显式方案具有计算复杂度优势，但也存在 “粒子污染” 效应，需要更多的理论分析。