Year: 2026
Type: article
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.
DOI: 10.1137/25m1734543
Institution: University of Cambridge
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Nonlocal Methods for Arbitrary Data Sources, Robust and Efficient Analysis Approaches of Remote Imagery for Assessing Population and Forest Health in India, The Mathematics of Deep Learning, G434337789, Cambridge Mathematics of Information in Healthcare (CMIH), EPSRC Centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging, Nonlocal Methods for Arbitrary Data Sources, Combining Knowledge And Data Driven Approaches to Inverse Imaging Problems, PET++: Improving Localisation, Diagnosis and Quantification in Clinical and Medical PET Imaging with Randomised Optimisation
OpenAlex ID: W4405766526