标题： 基于 RKHS 正则化的导数估计用于从时间序列数据学习动力学 显示英文标题

标题： Derivative estimation by RKHS regularization for learning dynamics from time-series data

摘要： 从时间序列数据中学习控制方程因其从真实世界数据中提取有用动态的潜力而引起了越来越多的关注。尽管取得了显著进展，但在存在噪声的情况下变得具有挑战性，特别是当需要计算导数时。 为了减少噪声的影响，我们提出了一种同时拟合噪声时间序列数据的导数和轨迹的方法。我们的方法将导数估计表述为涉及正向模型中积分算子的反问题，并通过在向量值再生核希尔伯特空间（vRKHS）中求解正则化问题来估计导数函数。我们推导出一个积分形式的表示定理，这使得可以通过求解有限维问题来计算正则化解，并促进有效估计最优正则化参数。通过将动力学嵌入到vRKHS中并利用拟合的导数和轨迹，我们可以通过求解线性正则化问题从噪声数据中恢复潜在的动力学。进行了几个数值实验以验证我们方法的有效性和效率。 显示英文摘要

摘要： Learning the governing equations from time-series data has gained increasing attention due to its potential to extract useful dynamics from real-world data. Despite significant progress, it becomes challenging in the presence of noise, especially when derivatives need to be calculated. To reduce the effect of noise, we propose a method that simultaneously fits both the derivative and trajectory from noisy time-series data. Our approach formulates derivative estimation as an inverse problem involving integral operators within the forward model, and estimates the derivative function by solving a regularization problem in a vector-valued reproducing kernel Hilbert space (vRKHS). We derive an integral-form representer theorem, which enables the computation of the regularized solution by solving a finite-dimensional problem and facilitates efficiently estimating the optimal regularization parameter. By embedding the dynamics within a vRKHS and utilizing the fitted derivative and trajectory, we can recover the underlying dynamics from noisy data by solving a linear regularization problem. Several numerical experiments are conducted to validate the effectiveness and efficiency of our method.