Double Normalizing Flows: Flexible Bayesian Gaussian Process ODEs Learning

Recently, Gaussian processes have been utilized to model the vector field of continuous dynamical systems. Bayesian inference for such models \cite{hegde2022variational} has been extensively studied and has been applied in tasks such as time series prediction, providing uncertain estimates. However, previous Gaussian Process Ordinary Differential Equation (ODE) models may underperform on datasets with non-Gaussian process priors, as their constrained priors and mean-field posteriors may lack flexibility. To address this limitation, we incorporate normalizing flows to reparameterize the vector field of ODEs, resulting in a more flexible and expressive prior distribution. Additionally, due to the analytically tractable probability density functions of normalizing flows, we apply them to the posterior inference of GP ODEs, generating a non-Gaussian posterior. Through these dual applications of normalizing flows, our model improves accuracy and uncertainty estimates for Bayesian Gaussian Process ODEs. The effectiveness of our approach is demonstrated on simulated dynamical systems and real-world human motion data, including tasks such as time series prediction and missing data recovery. Experimental results indicate that our proposed method effectively captures model uncertainty while improving accuracy.