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变微分方程模型的数值积分提供了额外的并发层次。像Parareal这类跨多个时间步并行化的方法，依赖于计算成本较低且粗糙的积分器来向前传播信息，而可并行化的高精度精细传播子则提供准确性。通常，粗方法采用低分辨率、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作为替代。我们以计算金融领域的偏微分方程——Black-Scholes方程为例，证明使用PINN粗传播子的Parareal算法相比数值粗传播子能获得更好的加速效果。神经网络训练与评估的计算模式均高度适配GPU，而基于网格的算法因计算强度低而性能欠佳。研究表明，在CPU运行数值精细传播子的同时，将粗传播子PINN迁移至GPU可进一步优化Parareal的单节点性能。这表明，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，有望更好地利用异构架构。