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.

我们利用Bayesian filtering and smoothing的框架，基于iterated extended Kalman smoothers的时间并行公式，提出了一种并行时间概率数值ODE求解器，将动力系统的模拟从顺序处理转变为并行处理的形式，从而将时间步长的计算复杂度从线性降低到对数级别。我们在多种ODE问题上展示了这种方法的有效性，并将其与经典的和概率的数值ODE求解器进行了比较。