Symplectic Neural Networks for learning Generalized Hamiltonians

• URL: http://arxiv.org/abs/2606.27029v1
• Date: Thu, 25 Jun 2026 13:36:44 GMT
• Title: Symplectic Neural Networks for learning Generalized Hamiltonians
• Abstract summary: Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency.<n>For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, it is essential to use symplectic iteration, which preserve the system's geometric structure.<n>In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator.
• Score: 5.793251439782513
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

Related papers

• Gradient Descent as a Perceptron Algorithm: Understanding Dynamics and Implicit Acceleration [67.12978375116599]
  We show that the steps of gradient descent (GD) reduce to those of generalized perceptron algorithms.<n>This helps explain the optimization dynamics and the implicit acceleration phenomenon observed in neural networks.
  arXiv  Detail & Related papers  (2025-12-12T14:16:35Z)
• Symplectic Neural Flows for Modeling and Discovery [9.786274281068815]
  SympFlow is a time-dependent symplectic neural network designed using parameterized Hamiltonian flow maps.<n>It allows for backward error analysis and ensures the preservation of the symplectic structure.<n>We demonstrate the effectiveness of SympFlow on diverse problems, including chaotic and dissipative systems.
  arXiv  Detail & Related papers  (2024-12-21T22:02:00Z)
• 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)
• 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)
• Structure-Preserving Learning Using Gaussian Processes and Variational Integrators [62.31425348954686]
  We propose the combination of a variational integrator for the nominal dynamics of a mechanical system and learning residual dynamics with Gaussian process regression. We extend our approach to systems with known kinematic constraints and provide formal bounds on the prediction uncertainty.
  arXiv  Detail & Related papers  (2021-12-10T11:09:29Z)
• Symplectic Learning for Hamiltonian Neural Networks [0.0]
  Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach. We exploit the symplectic structure of Hamiltonian systems with a different loss function. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn.
  arXiv  Detail & Related papers  (2021-06-22T13:33:12Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• Sparse Symplectically Integrated Neural Networks [15.191984347149667]
  We introduce Sparselectically Integrated Neural Networks (SSINNs) SSINNs are a novel model for learning Hamiltonian dynamical systems from data. We evaluate SSINNs on four classical Hamiltonian dynamical problems.
  arXiv  Detail & Related papers  (2020-06-10T03:33:37Z)
• 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)
• Hamiltonian neural networks for solving equations of motion [3.1498833540989413]
  We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. A symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network in order to achieve the same order of the numerical error.
  arXiv  Detail & Related papers  (2020-01-29T21:48:35Z)

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.