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.

翻译：从时间序列数据中学习控制方程因其从现实世界数据中提取有用动力学的潜力而受到越来越多的关注。尽管取得了显著进展，但在存在噪声的情况下，尤其是在需要计算导数时，这一任务变得具有挑战性。为了降低噪声的影响，我们提出了一种从含噪时间序列数据中同时拟合导数和轨迹的方法。我们的方法将导数估计表述为一个在前向模型中涉及积分算子的逆问题，并通过在向量值再生核希尔伯特空间（vRKHS）中求解一个正则化问题来估计导数函数。我们推导了一个积分形式的表示定理，该定理使得可以通过求解一个有限维问题来计算正则化解，并有助于高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，我们可以通过求解一个线性正则化问题从含噪数据中恢复潜在的动力学。我们进行了若干数值实验以验证所提方法的有效性和效率。