Parameter identification and comparison of dynamical systems is a challenging task in many fields. Bayesian approaches based on Gaussian process regression over time-series data have been successfully applied to infer the parameters of a dynamical system without explicitly solving it. While the benefits in computational cost are well established, the theoretical foundation has been criticized in the past. We offer a novel interpretation which leads to a better understanding, improvements in state-of-the-art performance in terms of accuracy and robustness and a decrease in run time due to a more efficient setup for general nonlinear dynamical systems.

Many real world dynamical systems are described by stochastic differential equations. Thus parameter inference is a challenging and important problem in many disciplines. We provide a grid free and flexible algorithm offering parameter and state inference for stochastic systems and compare our approch based on variational approximations to state of the art methods showing significant advantages both in runtime and accuracy.

Gradient matching is a promising tool for learning parameters and state dynamics of ordinary differential equations. It is a grid free inference approach, which, for fully observable systems is at times competitive with numerical integration. However, for many real-world applications, only sparse observations are available or even unobserved variables are included in the model description. In these cases most gradient matching methods are difficult to apply or simply do not provide satisfactory results. That is why, despite the high computational cost, numerical integration is still the gold standard in many applications. Using an existing gradient matching approach, we propose a scalable variational inference framework which can infer states and parameters simultaneously, offers computational speedups, improved accuracy and works well even under model misspecifications in a partially observable system.