One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in a poor estimation when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to two typical scenarios for dynamical systems: parameter identification and estimation with an affine structure of the system, and nonlinear parametric approximation without prior knowledge.

翻译：科学机器学习的关键任务之一是从时间序列数据中表征潜在的动力系统。许多此类动力学学习方法明确要求状态数据的导数，这些导数无法直接获取，通常可通过有限差分进行近似。然而，当状态数据稀疏和/或受噪声污染时，时间导数的离散近似可能导致估计效果不佳，从而削弱所学动力学模型的预测能力。为克服这一技术难题，我们提出了一种新方法，通过贝叶斯推断表征模型参数来学习非线性动力学。该方法利用状态的高斯过程表示，通过状态数据与其导数之间的相关性构建似然函数，同时避免显式计算时间导数。通过贝叶斯框架，模型参数的概率估计由后验分布给出，从而便于量化噪声状态数据和学习过程中的不确定性。具体而言，我们将讨论所提方法在两类典型动力系统场景中的适用性：具有仿射结构系统的参数辨识与估计，以及无先验知识的非线性参数逼近。