This paper addresses the nonparametric estimation of the drift function over a compact domain for a time-homogeneous diffusion process, based on high-frequency discrete observations from $N$ independent trajectories. We propose a neural network-based estimator and derive a non-asymptotic convergence rate, decomposed into a training error, an approximation error, and a diffusion-related term scaling as ${\log N}/{N}$. For compositional drift functions, we establish an explicit rate. In the numerical experiments, we consider a drift function with local fluctuations generated by a double-layer compositional structure featuring local oscillations, and show that the empirical convergence rate becomes independent of the input dimension $d$. Compared to the $B$-spline method, the neural network estimator achieves better convergence rates and more effectively captures local features, particularly in higher-dimensional settings.

翻译：本文研究基于高频离散观测的N条独立轨迹，在紧致域上对时间齐次扩散过程的漂移函数进行非参数估计。我们提出一种基于神经网络的估计量，并推导出其非渐近收敛速度，该速度由训练误差、逼近误差及扩散相关项（量级为${\log N}/{N}$）共同构成。针对复合型漂移函数，我们建立了显式收敛速率。数值实验中，我们考虑具有局部振荡双层复合结构的漂移函数，证明经验收敛速率与输入维度d无关。相较于B样条方法，神经网络估计量能实现更优的收敛速率，并在高维场景下更有效地捕捉局部特征。