# Parareal with a physics-informed neural network as coarse propagator

arXiv: https://arxiv.org/abs/2303.03848 · HTML: https://mlsanity.com/paper/2303.03848/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2023.

## Built on by (most cited first, 4 of 4)

- [Efficient Numerical Wave Propagation Enhanced By An End-to-End Deep Learning Model](https://mlsanity.com/paper/2402.02304/) (2024)
- [Regime-Aware Time Weighting for Physics-Informed Neural Networks](https://mlsanity.com/paper/2407.21642/) (2024)
- [Parallel-in-Time Solutions with Random Projection Neural Networks](https://mlsanity.com/paper/2408.09756/) (2024)
- [MCMC Informed Neural Emulators for Uncertainty Quantification in Dynamical Systems](https://mlsanity.com/paper/2603.10987/) (2026)

## Abstract (arXiv)

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.