With the advent of supercomputers, multi-processor environments and parallel-in-time (PinT) algorithms offer ways to solve initial value problems for ordinary and partial differential equations (ODEs and PDEs) over long time intervals, a task often unfeasible with sequential solvers within realistic time frames. A recent approach, GParareal, combines Gaussian Processes with traditional PinT methodology (Parareal) to achieve faster parallel speed-ups. The method is known to outperform Parareal for low-dimensional ODEs and a limited number of computer cores. Here, we present Nearest Neighbors GParareal (nnGParareal), a novel data-enriched PinT integration algorithm. nnGParareal builds upon GParareal by improving its scalability properties for higher-dimensional systems and increased processor count. Through data reduction, the model complexity is reduced from cubic to log-linear in the sample size, yielding a fast and automated procedure to integrate initial value problems over long time intervals. First, we provide both an upper bound for the error and theoretical details on the speed-up benefits. Then, we empirically illustrate the superior performance of nnGParareal, compared to GParareal and Parareal, on nine different systems with unique features (e.g., stiff, chaotic, high-dimensional, or challenging-to-learn systems).

翻译：随着超级计算机的出现，多处理器环境和并行时间（PinT）算法为解决常微分方程和偏微分方程（ODE和PDE）在长时间间隔上的初值问题提供了途径，而这在现实时间范围内通过顺序求解器往往难以实现。最近提出的一种方法GParareal，将高斯过程与传统的PinT方法（Parareal）相结合，以实现更快的并行加速。已知该方法在低维ODE和有限数量的计算机核心上优于Parareal。本文提出了一种新颖的数据丰富型PinT集成算法——最近邻GParareal（nnGParareal）。nnGParareal基于GParareal，通过改进其面向高维系统和更多处理器的可扩展性而构建。通过数据缩减，模型复杂度从样本大小的立方级降至对数线性级，从而产生一种快速且自动化的过程，用于在长时间间隔上求解初值问题。首先，我们提供了误差上界以及加速优势的理论细节。然后，我们通过九个具有独特特征（例如，刚性、混沌、高维或难以学习的系统）的系统，实证说明了nnGParareal相比GParareal和Parareal的优越性能。