Upcoming Event: PhD Dissertation Defense

Accelerating Multiscale Hamiltonian Simulations via Deep Learning and Parallel-in-time Methods

Rui Fang, CSEM Ph.D. Candidate

Abstract

Hamiltonian systems with multiple timescales commonly arise in molecular dynamics, classical mechanics, and theoretical physics. Stable and accurate long-time integration of these systems demands very small time steps to resolve the fastest dynamics, resulting in substantial computational costs. Moreover, many practical applications involve simulating ensembles of trajectories for uncertainty quantification, sensitivity analyses, or varied initial conditions, which further intensifies the computational demands. This dissertation introduces two complementary strategies to accelerate both long-time and ensemble simulations of multiscale Hamiltonian systems: (1) deep learning-based flow maps, and (2) data-driven stabilization of the parallel-in-time parareal method.

First, we develop neural network models that directly approximate the flow map of Hamiltonian systems, thereby relaxing strict time step constraints. We consider both fixed-timestep and variable-timestep architectures, trained using supervised data loss and unsupervised ODE residual loss defined by numerical schemes. To improve generalization and long-time accuracy, we introduce energy-based error metrics,  regularization informed by numerical residuals, and phase space sampling strategies for sampling microcanonical ensembles. These learned surrogates enable efficient batch evaluations, substantially accelerating ensemble computations. Moreover, the approach flexibly accommodates parametric dependence, enabling sensitivity studies across varying physical regimes. 

Second, we introduce stabilized variants of the parareal method to improve its performance on highly oscillatory problems. In particular, we develop the Procrustes parareal algorithm for Hamiltonian systems, which enhances convergence and stability by correcting phase mismatches between coarse and fine solvers using data from previous iterations. The neural network-based flow maps can be seamlessly integrated into this framework to further boost computational efficiency.

We validate our approach on several benchmark problems, including a nearly-periodic coupled oscillators problem, the Fermi-Pasta-Ulam-Tsingou problem from chaos theory, the gravitational three-body problem, and the $\alpha$-particle problem relevant to magnetic confinement fusion. These results highlight the potential of the proposed methods for large-scale, long-time simulations of complex Hamiltonian systems.

 

Biography

Rui Fang is a CSEM PhD candidate supervised by Prof. Richard Tsai. Her research interests include machine learning, deep learning and scientific computing. Before joining CSEM, she received her BS degree in physics and astronomy from Haverford College and her ME degree in Computational Science and Engineering from Harvard University.