The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero is provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

翻译：本文提出了一种从数据中学习由欧拉-拉格朗日方程支配的动力学系统的方法。该方法基于高斯过程回归，能够识别连续或离散拉格朗日量，因此本质上具有结构保持性。我们严格证明了当观测数据点间距收敛到零时方法的收敛性。除收敛保证外，该方法还允许量化模型不确定性，这可为自适应采样技术提供理论基础。我们实现了对任何关于拉格朗日量为线性的可观测量（包括哈密顿函数（能量）与辛结构）的高效不确定性量化，这在系统辨识领域具有重要价值。通过精心设计几何正则化策略并利用再生核希尔伯特空间中凸最小化问题的关联性，本文克服了（离散）拉格朗日量辨识任务不适定性所带来的重大实践与理论困难。