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.

翻译：非线性动力系统的辨识在多个领域至关重要，它有助于实现控制、预测、优化和故障检测等任务。许多应用需要能够处理复杂系统，同时为安全可靠性能提供强有力学习保证的方法。然而，现有方法通常侧重于简化场景，例如确定性模型、已知扩散、离散系统、一维动力学，或受线性等强结构假设约束的系统。本文提出了一种新颖的方法，用于估计具有非均匀扩散的连续、多维、非线性受控随机微分方程的漂移系数和扩散系数。我们假设系数在Sobolev空间内具有正则性，这使得该方法可广泛应用于机器人学、金融、气候建模和生物学中的各种动力系统。利用Fokker-Planck方程，我们将估计分为两个任务：(a) 针对有限控制集估计系统动力学，以及 (b) 估计支配这些动力学的系数。我们提供了强有力的理论保证，包括针对 \(L^2\)、\(L^\infty\) 和风险度量的有限样本界，其学习率可自适应于系数的正则性，类似于非参数最小二乘回归文献中的结果。我们通过大量数值实验证明了所提方法的实际有效性。我们的方法已作为一个开源Python库提供。