We propose a probabilistic algorithm for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Our motivation emanates from statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.

研究了一种基于概率的数值方法，用于解决边界和初值问题，并返回解的联合高斯过程后验。提出的方法可用于非解析常微分方程上的流形统计学问题，通过较少精度推算的边际化方法，可以使统计结果不那么敏感。该方法可以用于均值计算和主要测地线分析等新的Riemann算法，也可以通过结果不那么精确的方法比点估计更快地完成。该方法认为，在整个机器学习算法流程中应跟踪数值计算引起的不确定性。

微分方程的概率解及其在黎曼统计学中的应用