Parareal with a physics-informed neural network as coarse propagator

Abstract

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.

arXiv comment13 pages, 7 figures

References in corpus (5)

• DGM: A deep learning algorithm for solving partial differential equations
• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
• PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs
• Shared Memory Pipelined Parareal
• Parareal Neural Networks Emulating a Parallel-in-time Algorithm

Cited by in corpus (4)

• Space-time parallel scaling of Parareal with a physics-informed Fourier Neural Operator coarse propagator applied to the Black-Scholes equation
• Regime-Aware Time Weighting for Physics-Informed Neural Networks
• Efficient Numerical Wave Propagation Enhanced By An End-to-End Deep Learning Model
• Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge-Kutta Methods