# A Bulirsch-Stoer algorithm using Gaussian processes

## TL;DR

This paper introduces a probabilistic numerics approach using Gaussian processes within the Bulirsch-Stoer algorithm to improve adaptive step-size selection and uncertainty quantification, especially in complex error scenarios.

## Contribution

It embeds Gaussian process regression into a numerical integration scheme, enabling robust error estimation and adaptive control beyond traditional polynomial methods.

## Key findings

- GPR-based approach matches traditional methods on smooth error surfaces.
- The method provides reasonable solutions near poles or chaotic regions.
- It enhances uncertainty quantification in numerical integration.

## Abstract

In this paper, we treat the problem of evaluating the asymptotic error in a numerical integration scheme as one with inherent uncertainty. Adding to the growing field of probabilistic numerics, we show that Gaussian process regression (GPR) can be embedded into a numerical integration scheme to allow for (i) robust selection of the adaptive step-size parameter and; (ii) uncertainty quantification in predictions of putatively converged numerical solutions. We present two examples of our approach using Richardson's extrapolation technique and the Bulirsch-Stoer algorithm. In scenarios where the error-surface is smooth and bounded, our proposed approach can match the results of the traditional polynomial (parametric) extrapolation methods. In scenarios where the error surface is not well approximated by a finite-order polynomial, e.g. in the vicinity of a pole or in the assessment of a chaotic system, traditional methods can fail, however, the non-parametric GPR approach demonstrates the potential to continue to furnish reasonable solutions in these situations.