  Recently, Gaussian processes have been used to model the vector field of
continuous dynamical systems, referred to as GPODEs, which are characterized by
a probabilistic ODE equation. Bayesian inference for these models has been
extensively studied and applied in tasks such as time series prediction.
However, the use of standard GPs with basic kernels like squared exponential
kernels has been common in GPODE research, limiting the model's ability to
represent complex scenarios. To address this limitation, we introduce
normalizing flows to reparameterize the ODE vector field, resulting in a
data-driven prior distribution, thereby increasing flexibility and expressive
power. We develop a data-driven variational learning algorithm that utilizes
analytically tractable probability density functions of normalizing flows,
enabling simultaneous learning and inference of unknown continuous dynamics.
Additionally, we also apply normalizing flows to the posterior inference of GP
ODEs to resolve the issue of strong mean-field assumptions in posterior
inference. By applying normalizing flows in both these ways, our model improves
accuracy and uncertainty estimates for Bayesian Gaussian Process ODEs. We
validate the effectiveness of our approach on simulated dynamical systems and
real-world human motion data, including time series prediction and missing data
recovery tasks. Experimental results show that our proposed method effectively
captures model uncertainty while improving accuracy.
