Exact Inference for Continuous-Time Gaussian Process Dynamics
This work addresses a domain-specific problem for researchers and practitioners in machine learning and physics who need accurate continuous-time dynamics modeling, though it is incremental as it builds on existing GP and integrator techniques.
The paper tackles the problem of learning continuous-time Gaussian process dynamics models from discrete-time data, which is challenging with standard methods due to intractable exact inference when using higher-order numerical integrators. The result is a method that enables exact GP inference by leveraging multistep and Taylor integrators, achieving an accurate representation of the continuous-time system as demonstrated empirically and theoretically.
Physical systems can often be described via a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. This can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. Higher-order numerical integrators provide the necessary tools to address this problem by discretizing the dynamics function with arbitrary accuracy. Many higher-order integrators require dynamics evaluations at intermediate time steps making exact GP inference intractable. In previous work, this problem is often tackled by approximating the GP posterior with variational inference. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to make direct inference tractable, we propose to leverage multistep and Taylor integrators. We demonstrate how to derive flexible inference schemes for these types of integrators. Further, we derive tailored sampling schemes that allow to draw consistent dynamics functions from the learned posterior. This is crucial to sample consistent predictions from the dynamics model. We demonstrate empirically and theoretically that our approach yields an accurate representation of the continuous-time system.