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.

本文使用高斯过程回归嵌入数值积分方案，以实现对数值解的收敛性进行鲁棒性选择和不确定性量化的两种方法，并在 Richardson's extrapolation technique 和 Bulirsch-Stoer 算法上进行了实验。在误差面光滑有界的情况下，该非参数方法表现出与传统多项式（参数）外推方法相似的结果，但在误差面不适于有限阶多项式的情况下，例如在极点附近或混沌系统的评估中，传统方法可能会失败，但非参数 GPR 方法展示了在这些情况下继续提供合理解决方案的潜力。

利用高斯过程的Bulirsch-Stoer算法