We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.

翻译：我们展示了如何利用概率数值方法将初值问题转化为由该初值问题动力学参数化的高斯-马尔可夫过程。由此，常微分方程中通常棘手的参数估计问题被简化为高斯-马尔可夫回归中的超参数估计，而后者往往容易得多。本文阐明了该方法与经典数值积分及梯度匹配方法的关系及其优势。特别地，与梯度匹配方法不同，该方法能处理部分观测数据，并具备若干摆脱局部最优的路径而经典数值积分方法则不具备。实验结果表明，该方法在性能上与其他竞争方法持平或略有优势。