Active Uncertainty Calibration in Bayesian ODE Solvers

TL;DR

This paper introduces Bayesian Quadrature filtering (BQF), a new probabilistic ODE solver that actively learns gradient measurement uncertainty to improve calibration while balancing computational cost.

Contribution

It proposes BQF, a filtering-based probabilistic ODE solver that actively calibrates uncertainty by learning gradient imprecision, bridging the gap between cost and calibration.

Findings

BQF improves probabilistic calibration over classical methods.

Active learning of gradient uncertainty enhances solver accuracy.

The method balances computational efficiency with uncertainty quantification.

Abstract

There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.