Parallel-in-time methods with machine learning based coarse propagators

This thesis investigates the integration of machine learning into parallel-in-time algorithms to accelerate the numerical solution of partial differential equations (PDEs) in computational finance. We focus on the Parareal algorithm and propose using neural network models as coarse propagators to improve its convergence and parallel efficiency. In particular, a Physics-Informed Neural Network (PINN) is implemented as the coarse solver for the single-asset Black–Scholes option pricing PDE, and a Physics-Informed Neural Operator (PINO) based on the Fourier Neural Operator is developed for the two-asset Black–Scholes PDE. These learned coarse models are trained to approximate the PDE solution while enforcing physical constraints, enabling fast evaluations on GPUs. We demonstrate that the PINN coarse propagator significantly accelerates Parareal’s convergence, achieving speedups beyond the limits of spatial-only parallelization. Extending to a two-dimensional Black–Scholes equation, the PINO coarse model allows space-time parallelism that outperforms traditional fine solvers, even when spatial parallelization alone saturates. Through theoretical analysis on a representative hyperbolic PDE (the advection equation), we identify key properties for effective coarse integrators– notably, preserving wave phase speed and stability. Neural coarse models trained with these insights yield markedly improved convergence on challenging problems where standard Parareal would diverge. Overall, the results indicate that machine-learning-based coarse propagators can substantially enhance parallel-in-time methods for PDEs, offering a promising avenue to tackle computationally intensive problems in finance and beyond.