Hamiltonian Matching for Symplectic Neural Integrators

Priscilla Canizares, Davide Murari, Carola-Bibiane Schönlieb, Ferdia Sherry, Zakhar Shumaylov

Keywords: Hamiltonian systems, backward error analysis, physics-informed machine learning, scientific machine learning

TL;DR: We introduce a novel symplectic neural network architecture and corresponding Hamiltonian matching loss function to train it to approximate flow maps.

Abstract: Hamilton’s equations of motion form a fundamental framework in various branches of physics, including astronomy, quantum mechanics, particle physics, and climate science. Classical numerical solvers are typically employed to compute the time evolution of these systems. However, when the system spans multiple spatial and temporal scales numerical errors can accumulate, leading to reduced accuracy. To address the challenges of evolving such systems over long timescales, we propose SympFlow, a novel neural network-based symplectic integrator, which is the composition of a sequence of exact flow maps of parametrised time-dependent Hamiltonian functions. This architecture allows for a backward error analysis: we can identify an underlying Hamiltonian function of the architecture and use it to define a Hamiltonian matching objective function, which we use for training. In numerical experiments, we show that SympFlow exhibits promising results, with qualitative energy conservation behaviour similar to that of time-stepping symplectic integrators.

Submission Number: 77