A Fourier State Space Model for Bayesian ODE Filters

TL;DR

This paper introduces a Fourier state space model for Bayesian ODE filtering, enhancing the ability to solve periodic ODEs efficiently by combining Fourier and Taylor models within a probabilistic framework.

Contribution

It develops a novel Fourier state space model and a hybrid model combining Fourier and Taylor approaches for Bayesian ODE filtering.

Findings

Hybrid model improves prediction efficiency for periodic ODEs.

Fourier-based approach extends Gaussian ODE filtering to oscillatory solutions.

Experimental results demonstrate potential computational advantages.

Abstract

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.