Hamiltonian Matching for Symplectic Neural Integrators

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.