  Nonlinear dynamics are ubiquitous in science and engineering applications,
but the physics of most complex systems is far from being fully understood.
Discovering interpretable governing equations from measurement data can help us
understand and predict the behavior of complex dynamic systems. Although
extensive work has recently been done in this field, robustly distilling
explicit model forms from very sparse data with considerable noise remains
intractable. Moreover, quantifying and propagating the uncertainty of the
identified system from noisy data is challenging, and relevant literature is
still limited. To bridge this gap, we develop a novel Bayesian spline learning
framework to identify parsimonious governing equations of nonlinear
(spatio)temporal dynamics from sparse, noisy data with quantified uncertainty.
The proposed method utilizes spline basis to handle the data scarcity and
measurement noise, upon which a group of derivatives can be accurately computed
to form a library of candidate model terms. The equation residuals are used to
inform the spline learning in a Bayesian manner, where approximate Bayesian
uncertainty calibration techniques are employed to approximate posterior
distributions of the trainable parameters. To promote the sparsity, an
iterative sequential-threshold Bayesian learning approach is developed, using
the alternative direction optimization strategy to systematically approximate
L0 sparsity constraints. The proposed algorithm is evaluated on multiple
nonlinear dynamical systems governed by canonical ordinary and partial
differential equations, and the merit/superiority of the proposed method is
demonstrated by comparison with state-of-the-art methods.
