  This paper presents a machine learning framework for Bayesian systems
identification from noisy, sparse and irregular observations of nonlinear
dynamical systems. The proposed method takes advantage of recent developments
in differentiable programming to propagate gradient information through
ordinary differential equation solvers and perform Bayesian inference with
respect to unknown model parameters using Hamiltonian Monte Carlo. This allows
us to efficiently infer posterior distributions over plausible models with
quantified uncertainty, while the use of sparsity-promoting priors enables the
discovery of interpretable and parsimonious representations for the underlying
latent dynamics. A series of numerical studies is presented to demonstrate the
effectiveness of the proposed methods including nonlinear oscillators,
predator-prey systems, chaotic dynamics and systems biology. Taken all
together, our findings put forth a novel, flexible and robust workflow for
data-driven model discovery under uncertainty.
