Parallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our paper proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal's single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.

翻译：摘要：时间并行算法为基于时间依赖微分方程的数值积分模型提供了额外的并发层。诸如Parareal等跨多个时间步并行化的方法，依赖计算成本低且粗化的积分器向前传播信息，而可并行的昂贵细传播子则提供精度。通常，粗方法采用较低分辨率、降阶或简化模型的数值积分器。本文提出改用物理信息神经网络（PINN）。我们以计算金融领域的偏微分方程——布莱克-舒尔斯方程为例，证明采用PINN粗传播子的Parareal算法比数值粗传播子具有更优的加速比。神经网络的训练与评估均为计算模式高度适配GPU的任务，而基于网格的算法因计算强度低难以发挥性能优势。我们进一步表明，将粗传播子PINN部署至GPU，同时在CPU上运行数值细传播子，可显著提升Parareal的单节点性能。这提示：将机器学习技术融入时间并行积分方法，并利用其计算模式差异，有望更高效地利用异构计算架构。