  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 poor estimations 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 several typical
scenarios for dynamical systems: identification and estimation with an affine
parametrization, nonlinear parametric approximation without prior knowledge,
and general parameter estimation for a given dynamical system.
