In this paper, we consider the density estimation problem associated with the stationary measure of ergodic It\^o 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.
翻译:在本文中,我们考虑与一个离散时间序列的静态测量 ERgodic It ⁇ o 扩散相联的密度估计问题,这个时间序列与随机偏差方程式的解决方案相近。为了利用通过抛物线型Fokker-Planck PDE的固定解决方案对密度函数的定性,我们接下来进行如下演进。首先,我们采用深神经网络,通过解决适当的受监督的学习任务,来接近SDE的漂移和扩散条件。随后,我们用一个基于神经网络的最小方程式方法,解决与估计漂移和扩散系数相关的稳定状态Fokker-Plank方程式。我们根据适当的数学假设,在计算漂移和扩散系数的回归假设和PDE解算法的固定方法下,确定拟议方案对密度函数的趋同性。这一理论研究依赖于最近对Markov 链结果的扰动理论,该理论显示在估算漂移术语时对密度估计的偏差具有直线性依赖性依赖性估计,以及非对准回归和PDE回归性解决方案的概括性结果,而以20度模型模拟的模型反映了20度的数学模型的数学模型的模型。