Hamiltonian Matching for Symplectic Neural Integrators

TL;DR

This paper introduces SympFlow, a neural network-based symplectic integrator that learns to accurately simulate Hamiltonian systems over long timescales, maintaining energy conservation similar to traditional methods.

Contribution

The paper presents a novel neural network architecture for symplectic integration, enabling Hamiltonian matching and improved long-term accuracy in simulating Hamiltonian systems.

Findings

SympFlow demonstrates energy conservation comparable to classical symplectic integrators.

The architecture allows for Hamiltonian matching via a training objective.

Numerical experiments show promising long-term simulation accuracy.

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.