We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.

翻译：我们提出了一种新颖的非参数学习范式，用于识别多维非线性随机微分方程的漂移系数和扩散系数，该方法依赖于对系统状态的离散时间观测。其核心思想本质上是通过拟合基于再生核希尔伯特空间（RKHS）的相应福克-普朗克方程近似来匹配观测数据，从而获得非渐近学习速率的理论估计。与以往研究不同，当未知漂移系数和扩散系数的正则性越高时，我们的估计会变得越精确。由于本方法基于核函数，可通过离线预处理有效实现高效数值计算，在精度与计算复杂度之间实现了优异平衡。