Time-adaptive SympNets for separable Hamiltonian systems

• URL: http://arxiv.org/abs/2509.16026v1
• Date: Fri, 19 Sep 2025 14:33:21 GMT
• Title: Time-adaptive SympNets for separable Hamiltonian systems
• Authors: Konrad Janik, Peter Benner,
• Abstract summary: We provide a universal approximation theorem for separable Hamiltonian systems and show that it is not possible to extend it to non-separable Hamiltonian systems.<n>We fix a mistake in a proof of a substantial theorem for the approximation of symplectic maps in general, but specifically for symplectic machine learning methods.
• Score: 1.0542145271875853
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Measurement data is often sampled irregularly i.e. not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [20] and H\'enonNets [4] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets, which we call TSympNets, was introduced in [20]. We adapt the architecture of TSympNets and extend them to non-autonomous Hamiltonian systems. So far the approximation qualities of TSympNets were unknown. We close this gap by providing a universal approximation theorem for separable Hamiltonian systems and show that it is not possible to extend it to non-separable Hamiltonian systems. To investigate these theoretical approximation capabilities, we perform different numerical experiments. Furthermore we fix a mistake in a proof of a substantial theorem [25, Theorem 2] for the approximation of symplectic maps in general, but specifically for symplectic machine learning methods.

Related papers

• Disordered Dynamics in High Dimensions: Connections to Random Matrices and Machine Learning [52.26396748560348]
  We provide an overview of high dimensional dynamical systems driven by random matrices.<n>We focus on applications to simple models of learning and generalization in machine learning theory.
  arXiv  Detail & Related papers  (2026-01-03T00:12:32Z)
• Time-adaptive HénonNets for separable Hamiltonian systems [1.0542145271875853]
  We propose T-H'enonNets, which is symplectic by design and can handle adaptive time steps.<n>We also extend the T-H'enonNet architecture to non-autonomous Hamiltonian systems.
  arXiv  Detail & Related papers  (2025-09-24T15:04:53Z)
• Hamiltonian Matching for Symplectic Neural Integrators [9.786274281068815]
  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. 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.
  arXiv  Detail & Related papers  (2024-10-23T20:21:56Z)
• Learning Generalized Hamiltonians using fully Symplectic Mappings [0.32985979395737786]
  Hamiltonian systems have the important property of being conservative, that is, energy is conserved throughout the evolution.<n>In particular Hamiltonian Neural Networks have emerged as a mechanism to incorporate structural inductive bias into the NN model.<n>We show that symplectic schemes are robust to noise and provide a good approximation of the system Hamiltonian when the state variables are sampled from a noisy observation.
  arXiv  Detail & Related papers  (2024-09-17T12:45:49Z)
• Symplectic Neural Networks Based on Dynamical Systems [0.0]
  We present and analyze a framework for Symplectic neural networks (SympNets) based on geometric for Hamiltonian differential equations. The SympNets are universal approximators in the space of Hamiltonian diffeomorphisms, interpretable and have a non-vanishing property.
  arXiv  Detail & Related papers  (2024-08-19T09:18:28Z)
• How neural networks learn to classify chaotic time series [77.34726150561087]
  We study the inner workings of neural networks trained to classify regular-versus-chaotic time series. We find that the relation between input periodicity and activation periodicity is key for the performance of LKCNN models.
  arXiv  Detail & Related papers  (2023-06-04T08:53:27Z)
• Port-Hamiltonian Neural Networks with State Dependent Ports [58.720142291102135]
  We stress-test the method on both simple mass-spring systems and more complex and realistic systems with several internal and external forces. Port-Hamiltonian neural networks can be extended to larger dimensions with state-dependent ports. We propose a symmetric high-order integrator for improved training on sparse and noisy data.
  arXiv  Detail & Related papers  (2022-06-06T14:57:25Z)
• Learning Trajectories of Hamiltonian Systems with Neural Networks [81.38804205212425]
  We propose to enhance Hamiltonian neural networks with an estimation of a continuous-time trajectory of the modeled system. We demonstrate that the proposed integration scheme works well for HNNs, especially with low sampling rates, noisy and irregular observations.
  arXiv  Detail & Related papers  (2022-04-11T13:25:45Z)
• The Separation Capacity of Random Neural Networks [78.25060223808936]
  We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity.
  arXiv  Detail & Related papers  (2021-07-31T10:25:26Z)
• Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers [50.85524803885483]
  This work proposes a formal definition of statistically meaningful (SM) approximation which requires the approximating network to exhibit good statistical learnability. We study SM approximation for two function classes: circuits and Turing machines.
  arXiv  Detail & Related papers  (2021-07-28T04:28:55Z)
• On dissipative symplectic integration with applications to gradient-based optimization [77.34726150561087]
  We propose a geometric framework in which discretizations can be realized systematically. We show that a generalization of symplectic to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.
  arXiv  Detail & Related papers  (2020-04-15T00:36:49Z)
• SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems [2.6016814327894466]
  We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules. In particular, we define two classes of SympNets: the LA-SympNets composed of linear and activation modules, and the G-SympNets composed of gradient modules.
  arXiv  Detail & Related papers  (2020-01-11T13:04:34Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.