Symplectic Neural Flows for Modeling and Discovery

Abstract. Hamilton’s equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose, but neural network-based methods that incorporate these principles remain underexplored. This work introduces SympFlow, a time-dependent symplectic neural network designed using parameterized Hamiltonian flow maps. This design allows for backward error analysis and ensures the preservation of the symplectic structure. SympFlow allows for two key applications: (i) providing a time-continuous symplectic approximation of the exact flow of a Hamiltonian system—purely based on the differential equations it satisfies, and (ii) approximating the flow map of an unknown Hamiltonian system relying on trajectory data. We demonstrate the effectiveness of SympFlow on diverse problems, including chaotic and dissipative systems, showing improved energy conservation compared to general-purpose numerical methods and accurate approximations from sparse irregular data. We also provide a thorough theoretical analysis of SympFlow, showing it can approximate the flow of any time-dependent Hamiltonian system, and providing an a posteriori error estimate in terms of energy conservation.