发表机构
KTH Royal Institute of Technology; Digital Futures; Grenoble INP – Ensimag; Université Grenoble Alpes; University of Ljubljana（KTH皇家理工学院; 数字未来中心; 格勒诺布尔国立理工学院-格勒诺布尔国立高等计算机与应用数学学院; 格勒诺布尔阿尔卑斯大学; 卢布尔雅那大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对含公共噪声的McKean-Vlasov方程，提出条件圆柱型神经网络结合混合密度网络的两阶段方法，证明其L²通用逼近定理，数值实验显示该方法优于经验粒子插入法。
AI 中文摘要
我们引入条件圆柱型神经网络，用于逼近含公共噪声的McKean-Vlasov方程中条件律的泛函。初始律的傅里叶矩与时增公共噪声的截断特征通过混合密度网络映射为条件律的高斯混合逼近，随后圆柱型神经网络通过与该预测测度的解析积分计算目标泛函。粗糙路径适定性与稳定性提供了一个条件律映射，其在初始分布与粗糙驱动下连续，且在Itô布朗提升处几乎必然与经典条件律一致。结合该连续性、傅里叶分离性、特征唯一性、高斯混合的Wasserstein密度性及神经通用逼近性，我们证明了连续平方可积泛函的L²通用逼近定理。数值研究将所得两阶段流程应用于六个示例，包括非高斯初始律、非线性漂移、乘性公共噪声及二维状态；当无闭式律可用时，采用独立粒子参考。学习得到的条件律与泛函逼近始终优于经验粒子插入法，额外实验还检验了特征敏感性、每个公共噪声场景从一个终端观测值训练、Itô–Stratonovich一致性。
英文摘要
We introduce conditional cylindrical neural networks for approximating functionals of conditional laws in McKean-Vlasov equations with common noise. Fourier moments of the initial law and truncated signatures of the time augmented common noise are mapped by a mixture density network to a Gaussian mixture approximation of the conditional law. A cylindrical neural network then evaluates the target functional through analytic integrals against this predicted measure. Rough path well posedness and stability provide a conditional law map that is continuous in the initial distribution and the rough driver and agrees almost surely with the classical conditional law at the Itô Brownian lift. Combining this continuity with Fourier separation, signature uniqueness, Wasserstein density of Gaussian mixtures, and neural universal approximation, we prove an $L^2$ universal approximation theorem for continuous square integrable functionals. The numerical study implements the resulting two stage procedure on six examples, including non Gaussian initial laws, nonlinear drift, multiplicative common noise, and a two dimensional state. Independent particle references are used when no closed form law is available. The learned conditional law and functional approximations consistently improve on the empirical particle plug in, and additional experiments examine feature sensitivity, training from one terminal observation per common noise scenario, and Itô--Stratonovich consistency.
Comments32 pages, 7 figures, 4 tables. Code available at https://github.com/HmiouiReda/cylindrical-mckean-vlasov
