HyperVerlet: a Deep Learning Method for Numerically Solving Initial Value Problems of Hamiltonian Systems
Authors
Madsen, Anders ; Mathiesen, Frederik Baymler
Term
4. term
Education
Publication year
2021
Submitted on
2021-06-10
Pages
70
Abstract
This thesis introduces HyperVerlet, a self-supervised deep learning method (learning without labeled solutions) for numerically solving initial-value problems in Hamiltonian systems—physical systems that conserve quantities like energy and total momentum. We provide a theoretical proof that HyperVerlet has lower local and global truncation error than the widely used velocity Verlet method, meaning it makes smaller errors at each time step and accumulates less error over time. We test HyperVerlet on an undamped spring–mass system, an idealized pendulum, and a chaotic three-body spring–mass system, where conservation laws are crucial. In these experiments, HyperVerlet outperforms velocity Verlet and other hypersolvers, and in some cases matches the accuracy of 4th-order solvers. Depending on the choice of the neural network-based corrector, HyperVerlet can be symplectic or non-symplectic; the symplectic option preserves geometric volume and is well suited to long-duration simulations.
Dette speciale præsenterer HyperVerlet, en selv-superviseret deep learning-metode (uden mærkede facitdata) til numerisk løsning af initialværdiproblemer i Hamiltonske systemer—fysiske systemer, hvor størrelser som energi og total impuls bevares. Vi giver et teoretisk bevis for, at HyperVerlet har lavere lokal og global trunkeringsfejl end den udbredte velocity Verlet-metode; det vil sige mindre fejl pr. tidssteg og mindre akkumuleret fejl over tid. Vi afprøver HyperVerlet på et udæmpet fjeder-masse-system, et idealiseret pendul og et kaotisk tre-legeme fjeder-masse-system, hvor bevaringslove er vigtige. I disse forsøg overgår HyperVerlet velocity Verlet og andre hypersolvere, og i nogle tilfælde præsterer den på niveau med 4.-ordens metoder. Afhængigt af valget af den neurale netværksbaserede korrektor kan HyperVerlet være symplektisk eller ikke-symplektisk; den symplektiske variant bevarer geometrisk volumen og egner sig til langvarige simuleringer.
[This apstract has been rewritten with the help of AI based on the project's original abstract]
Keywords