  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).
