Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the span cost from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.

翻译：概率数值求解器将常微分方程的数值模拟视为贝叶斯状态估计问题。除了生成关于常微分方程解的后验分布以量化方法本身的数值逼近误差外，该理论框架的一个较少被提及的优势在于，通过将数值模拟构建为贝叶斯滤波和平滑框架，从而获得了算法灵活性。本文利用这一灵活性，基于迭代扩展卡尔曼平滑器的时间并行公式，提出了一种面向时间维度的并行概率数值常微分方程求解器。与现有概率求解器按时间顺序依次模拟动力系统不同，所提方法并行处理所有时间步，从而将时间跨度成本从线性降低至时间步数的对数级别。我们在多种常微分方程上验证了该方法的有效性，并与经典及概率数值常微分方程求解器进行了对比。