基于深度学习的伊藤扩散稳态密度估计
Stationary Density Estimation of Itô Diffusions Using Deep Learning
摘要：
本文考虑了离散时间序列中遍历Itô扩散平稳测度的密度估计问题，该离散时间序列逼近随机微分方程的解.为了利用通过抛物型福克-普朗克偏微分方程的定态解表征密度函数，我们进行如下。首先，我们使用深度神经网络通过解决适当的监督学习任务来近似估计的漂移和扩散项。随后，我们解决了稳态福克普朗克方程与估计的漂移和扩散系数与基于神经网络的最小二乘法。我们建立了适当的数学假设下，该计划的收敛性，考虑到回归的漂移和扩散系数，和偏微分方程求解器引起的泛化误差。本理论研究依赖于最近的扰动理论的马尔可夫链的结果，显示了线性依赖的密度估计的误差估计的漂移项，和推广的误差结果的非参数回归和偏微分方程回归解与神经网络模型。二维Student t分布和20维Langevin动力学的数值模拟表明了该方法的有效性。
英文摘要：
In this paper, we consider the density estimation problem associated with the stationary measure of ergodic Itô diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Student's t distribution and a 20-dimensional Langevin dynamics.
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