{"title":"Learning Generalized Hamiltonians using fully Symplectic Mappings","authors":"Harsh Choudhary, Chandan Gupta, Vyacheslav kungrutsev, Melvin Leok, Georgios Korpas","doi":"arxiv-2409.11138","DOIUrl":null,"url":null,"abstract":"Many important physical systems can be described as the evolution of a\nHamiltonian system, which has the important property of being conservative,\nthat is, energy is conserved throughout the evolution. Physics Informed Neural\nNetworks and in particular Hamiltonian Neural Networks have emerged as a\nmechanism to incorporate structural inductive bias into the NN model. By\nensuring physical invariances are conserved, the models exhibit significantly\nbetter sample complexity and out-of-distribution accuracy than standard NNs.\nLearning the Hamiltonian as a function of its canonical variables, typically\nposition and velocity, from sample observations of the system thus becomes a\ncritical task in system identification and long-term prediction of system\nbehavior. However, to truly preserve the long-run physical conservation\nproperties of Hamiltonian systems, one must use symplectic integrators for a\nforward pass of the system's simulation. While symplectic schemes have been\nused in the literature, they are thus far limited to situations when they\nreduce to explicit algorithms, which include the case of separable Hamiltonians\nor augmented non-separable Hamiltonians. We extend it to generalized\nnon-separable Hamiltonians, and noting the self-adjoint property of symplectic\nintegrators, we bypass computationally intensive backpropagation through an ODE\nsolver. We show that the method is robust to noise and provides a good\napproximation of the system Hamiltonian when the state variables are sampled\nfrom a noisy observation. In the numerical results, we show the performance of\nthe method concerning Hamiltonian reconstruction and conservation, indicating\nits particular advantage for non-separable systems.","PeriodicalId":501301,"journal":{"name":"arXiv - CS - Machine Learning","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Machine Learning","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11138","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Many important physical systems can be described as the evolution of a
Hamiltonian system, which has the important property of being conservative,
that is, energy is conserved throughout the evolution. Physics Informed Neural
Networks and in particular Hamiltonian Neural Networks have emerged as a
mechanism to incorporate structural inductive bias into the NN model. By
ensuring physical invariances are conserved, the models exhibit significantly
better sample complexity and out-of-distribution accuracy than standard NNs.
Learning the Hamiltonian as a function of its canonical variables, typically
position and velocity, from sample observations of the system thus becomes a
critical task in system identification and long-term prediction of system
behavior. However, to truly preserve the long-run physical conservation
properties of Hamiltonian systems, one must use symplectic integrators for a
forward pass of the system's simulation. While symplectic schemes have been
used in the literature, they are thus far limited to situations when they
reduce to explicit algorithms, which include the case of separable Hamiltonians
or augmented non-separable Hamiltonians. We extend it to generalized
non-separable Hamiltonians, and noting the self-adjoint property of symplectic
integrators, we bypass computationally intensive backpropagation through an ODE
solver. We show that the method is robust to noise and provides a good
approximation of the system Hamiltonian when the state variables are sampled
from a noisy observation. In the numerical results, we show the performance of
the method concerning Hamiltonian reconstruction and conservation, indicating
its particular advantage for non-separable systems.