A Fourier State Space Model for Bayesian ODE Filters
This is an incremental improvement for probabilistic numerical methods in ODE solving, specifically targeting oscillatory systems.
The paper tackles the problem of solving ordinary differential equations (ODEs) with periodic solutions by introducing a Fourier state space model within Gaussian ODE filtering, and experiments show the hybrid model can cheaply predict until the end of the time domain.
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.