Iterative parallel-in-time algorithms like Parareal can extend scaling beyond the saturation of purely spatial parallelization when solving initial value problems. However, they require the user to build coarse models to handle the inevitably serial transport of information in time.This is a time consuming and difficult process since there is still only limited theoretical insight into what constitutes a good and efficient coarse model. Novel approaches from machine learning to solve differential equations could provide a more generic way to find coarse level models for parallel-in-time algorithms. This paper demonstrates that a physics-informed Fourier Neural Operator (PINO) is an effective coarse model for the parallelization in time of the two-asset Black-Scholes equation using Parareal. We demonstrate that PINO-Parareal converges as fast as a bespoke numerical coarse model and that, in combination with spatial parallelization by domain decomposition, it provides better overall speedup than both purely spatial parallelization and space-time parallelizaton with a numerical coarse propagator.

翻译：基于迭代的时间并行算法（如Parareal）在求解初值问题时，可突破纯空间并行化的饱和限制实现规模扩展。然而，这类算法需要用户构建粗模型来处理时间维度上不可避免的串行信息传递，而由于目前对优质高效粗模型的理论认知仍十分有限，这一构建过程耗时且困难。机器学习求解微分方程的新方法可能为时间并行算法提供更通用的粗尺度模型构建途径。本文证明，物理信息傅里叶神经算子（PINO）能够作为基于Parareal的双资产Black-Scholes方程时间并行化的有效粗模型。我们验证了PINO-Parareal的收敛速度可与定制化数值粗模型相媲美，并且当结合基于区域分解的空间并行化时，相较于纯空间并行化及采用数值粗传播子的时空并行化，该方法能实现更优的整体加速比。