  Gaussian ODE filtering is a probabilistic numerical method to solve ordinary
differential equations (ODEs). It computes a Bayesian posterior over the
solution from evaluations of the vector field defining the ODE. Its most
popular version, which employs an integrated Brownian motion prior, uses Taylor
expansions of the mean to extrapolate forward and has the same convergence
rates as classical numerical methods. As the solution of many important ODEs
are periodic functions (oscillators), we raise the question whether Fourier
expansions can also be brought to bear within the framework of Gaussian ODE
filtering. To this end, we construct a Fourier state space model for ODEs and a
`hybrid' model that combines a Taylor (Brownian motion) and Fourier state space
model. We show by experiments how the hybrid model might become useful in
cheaply predicting until the end of the time domain.
