Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective

10/08/2018
by   Filip Tronarp, et al.
10

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consists of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a Bayesian state estimation problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers, which were formulated in terms of generating synthetic measurements of the vector field, come out as specific approximations. We derive novel solvers, both Gaussian and non-Gaussian, from the Bayesian state estimation problem posed in this paper and compare them with other probabilistic solvers in illustrative experiments.

READ FULL TEXT

page 1

page 2

page 3

page 4

08/11/2020

Deep State-Space Gaussian Processes

This paper is concerned with a state-space approach to deep Gaussian pro...
02/09/2022

Adjoint-aided inference of Gaussian process driven differential equations

Linear systems occur throughout engineering and the sciences, most notab...
07/17/2020

A Fourier State Space Model for Bayesian ODE Filters

Gaussian ODE filtering is a probabilistic numerical method to solve ordi...
05/11/2016

Active Uncertainty Calibration in Bayesian ODE Solvers

There is resurging interest, in statistics and machine learning, in solv...
10/26/2016

Probabilistic Linear Multistep Methods

We present a derivation and theoretical investigation of the Adams-Bashf...
09/25/2017

Bayesian Filtering for ODEs with Bounded Derivatives

Recently there has been increasing interest in probabilistic solvers for...
06/09/2020

Bayesian Probabilistic Numerical Integration with Tree-Based Models

Bayesian quadrature (BQ) is a method for solving numerical integration p...