Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while providing strong learning guarantees for safe and reliable performance. However, existing approaches often focus on simplified scenarios, such as deterministic models, known diffusion, discrete systems, one-dimensional dynamics, or systems constrained by strong structural assumptions such as linearity. This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled stochastic differential equations with non-uniform diffusion. We assume regularity of the coefficients within a Sobolev space, allowing for broad applicability to various dynamical systems in robotics, finance, climate modeling, and biology. Leveraging the Fokker-Planck equation, we split the estimation into two tasks: (a) estimating system dynamics for a finite set of controls, and (b) estimating coefficients that govern those dynamics. We provide strong theoretical guarantees, including finite-sample bounds for \(L^2\), \(L^\infty\), and risk metrics, with learning rates adaptive to coefficients' regularity, similar to those in nonparametric least-squares regression literature. The practical effectiveness of our approach is demonstrated through extensive numerical experiments. Our method is available as an open-source Python library.

本研究解决了现有方法在处理复杂非线性动态系统时的局限性，尤其是在控制、预测和优化等应用领域。我们提出了一种新方法，能够高效估计多维非线性受控随机微分方程的漂移和扩散系数，并提供了强有力的理论保证和实际有效性，极大地推动了该领域的研究进展。

学习受控随机微分方程