Symplectic Learning for Hamiltonian Neural Networks

David, Marco; Méhats, Florian

doi:10.48550/arxiv.2106.11753

Cited by 4 publications

(6 citation statements)

References 14 publications

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“…In this work, models are trained on an approximation of the ODE (11) as made by some discretization method corresponding to a numerical integration scheme, as done in [24,21,8]. That is, given the integrator…”

Section: Choice Of Discretization Methods For Training Datamentioning

confidence: 99%

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Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces

Eidnes¹,

Stasik²,

Sterud³

et al. 2022

Preprint

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Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems. In this work, we stress-test the method on both simple mass-spring systems and more complex and realistic systems with several internal and external forces, including a system with multiple connected tanks. We quantify performance under various conditions and show that imposing different assumptions greatly affect the performance during training presenting advantages and limitations of the method. We demonstrate that port-Hamiltonian neural networks can be extended to larger dimensions with state-dependent ports. We consider learning on systems with known and unknown external forces and show how it can be used to detect deviations in a system and still provide a valid model when the deviations are removed. Finally, we propose a symmetric high-order integrator for improved training on sparse and noisy data.

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Section: Choice Of Discretization Methods For Training Datamentioning

confidence: 99%

“…with the Neural Ordinary Differential Equation method [5]. In this paper we follow an alternative approach and train on an integration scheme directly as done in [24,21,8], see Section 3.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces

Eidnes¹,

Stasik²,

Sterud³

et al. 2022

Preprint

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“…In fact, following the PyTorch implementation of the mean squared error, E 1 is actually divided by 2n. Alternatively, as introduced in David and Méhats (2021), one can compare pointwise values of the approximated and the true Hamiltonian, when known. This gives…”

Section: Description Of the Problemmentioning

confidence: 99%

Learning Hamiltonians of constrained mechanical systems

Celledoni¹,

Leone²,

Murari³

et al. 2022

Preprint

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Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in classical mechanics, where the dynamics is fully determined by one scalar function, the Hamiltonian. The solution trajectories are often constrained to evolve on a submanifold of a linear vector space. In this work, we propose new approaches for the accurate approximation of the Hamiltonian function of constrained mechanical systems given sample data information of their solutions. We focus on the importance of the preservation of the constraints in the learning strategy by using both explicit Lie group integrators and other classical schemes.

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“…The derivatives ∂H k ∂p , ∂H k ∂q are calculated by differentiating the neural network which models the Hamiltonian, while the derivatives dp k dt , dq k dt are either assumed to be known (from the simulator) or approximated with finite differences. Using finite differences to approximate the derivatives dp dt and dq dt is essentially equivalent to Euler integration with a time step being equal to the sampling interval, which limits the accuracy of the trained model [8].…”

Section: Hamiltonian Neural Networkmentioning

confidence: 99%

“…The original HNN model [11] had the limitation of assuming the knowledge of the state derivatives with respect to time or approximating those using finite differences. Many recent works have used numerical integrators for modeling the evolution of the system state and several improvements of the integration procedure have been proposed [5,8,9,16,25,28].…”

Section: Introductionmentioning

confidence: 99%

Learning Trajectories of Hamiltonian Systems with Neural Networks

Haitsiukevich¹,

Ilin²

2022

Preprint

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Modeling of conservative systems with neural networks is an area of active research. A popular approach is to use Hamiltonian neural networks (HNNs) which rely on the assumptions that a conservative system is described with Hamilton's equations of motion. Many recent works focus on improving the integration schemes used when training HNNs. In this work, we propose to enhance HNNs with an estimation of a continuous-time trajectory of the modeled system using an additional neural network, called a deep hidden physics model in the literature. We demonstrate that the proposed integration scheme works well for HNNs, especially with low sampling rates, noisy and irregular observations.

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Symplectic Learning for Hamiltonian Neural Networks

Cited by 4 publications

References 14 publications

Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces

Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces

Learning Hamiltonians of constrained mechanical systems

Learning Trajectories of Hamiltonian Systems with Neural Networks

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