  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.
