Learning Deterministic and Stochastic Forced Hamiltonian Systems

Benedikt Brantner; Tomasz Tyranowski

math.NAAug 20, 2026

Learning Deterministic and Stochastic Forced Hamiltonian Systems

Authors: Benedikt Brantner, Tomasz Tyranowski

Organizations: Max-Planck-Institut für Plasmaphysik Boltzmannstraße 2, 85748 Garching, Germany · Technische Universität München, Zentrum Mathematik Boltzmannstraße 3, 85748 Garching, Germany · University of Twente, Department of Applied Mathematics PO Box 217, 7500AE Enschede, The Netherlands

Abstract

We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.

Figures & tables

Figure 6.1: Convergence of the training loss for a representative training run using data generated from the linearly damped harmonic oscillator with damping parameter

\nu=0.1

Figure 6.2: Comparison of trajectories generated by the GFHNN and ResNet flows for the linearly damped harmonic oscillator. The left column corresponds to the case $\nu=0.5$ , while the right column shows the case $\nu=0.001$ . In both cases, trajectories are generated from the same initial condition $(\bar{q}_{0},\bar{p}_{0})=(-0.75,-0.75)$ . The first row displays the position $q(t)$ , the second row the momentum $p(t)$ , and the third row the Hamiltonian $H(t)$ evaluated along the corresponding trajectories.

Figure 6.3: Comparison of trajectories generated by the GFHNN and ResNet flows for the linearly damped harmonic oscillator with

\nu=0.005

. The left column corresponds to the initial condition

(\bar{q}_{0},\bar{p}_{0})=(-0.75,-0.75)

, while the right column corresponds to

(\bar{q}_{0},\bar{p}_{0})=(0.25,0.25)

. In both cases, the first row displays the position

q(t)

, the second row the momentum

p(t)

, and the third row the Hamiltonian

H(t)

evaluated along the corresponding trajectories.

Figure 6.4: Averaged trajectory error $\epsilon_{\mathrm{traj}}(t)$ for trajectories generated by the GFHNN and ResNet flows for the linearly damped harmonic oscillator. For each value of the damping coefficient $\nu$ , the error is computed by averaging over 49 test initial conditions distributed in the square $-1\leq q,p\leq 1$ in phase space, none of which were used during training. Each subplot corresponds to a distinct value of $\nu$ , as indicated in the legend.

Figure 6.5: Averaged Hamiltonian error $\epsilon_{H}(t)$ for trajectories generated by the GFHNN and ResNet flows for the linearly damped harmonic oscillator. The error is computed by averaging over the same set of 49 test initial conditions used in Figure 6.4 . Each subplot corresponds to a distinct value of the damping coefficient $\nu$ , as indicated in the legend.

Figure 6.6: Comparison of trajectories generated by the GFHNN and ResNet flows for the quadratically damped pendulum. The left column corresponds to the case $\nu=0.05$ , while the right column shows the case $\nu=0.01$ . In both cases, trajectories are generated from the same initial condition $(\bar{q}_{0},\bar{p}_{0})=(-0.75,-0.75)$ . The first row displays the position $q(t)$ , the second row the momentum $p(t)$ , and the third row the Hamiltonian $H(t)$ evaluated along the corresponding trajectories.

Figure 6.7: Averaged trajectory error $\epsilon_{\mathrm{traj}}(t)$ for trajectories generated by the GFHNN and ResNet flows for the quadratically damped pendulum. For each value of the damping coefficient $\nu$ , the error is computed by averaging over 49 test initial conditions distributed in the square $-0.75\leq q,p\leq 0.75$ in phase space, none of which were used during training. Each subplot corresponds to a distinct value of $\nu$ , as indicated in the legend.

Figure 6.8: Averaged Hamiltonian error $\epsilon_{H}(t)$ for trajectories generated by the GFHNN and ResNet flows for the quadratically damped pendulum. The error is computed by averaging over the same set of 49 test initial conditions used in Figure 6.7 . Each subplot corresponds to a distinct value of the damping coefficient $\nu$ , as indicated in the legend. Also compare this to the case of the linearly damped harmonic oscillator in Figure 6.5 .

Figure 6.9: Comparison of trajectories generated by the PGFHNN and parametric ResNet flows for the time-dependent damped harmonic oscillator with

V_{0}=1

\Omega=3.5

, and

\nu=0.25

. The left column corresponds to the initial condition

(\bar{q}_{0},\bar{p}_{0})=(-0.75,-0.75)

, while the right column corresponds to

(\bar{q}_{0},\bar{p}_{0})=(0.25,0.25)

. In both cases, the first row displays the position

q(t)

, the second row the momentum

p(t)

, and the third row the Hamiltonian

H(t)

evaluated along the corresponding trajectories. Note that the plots corresponding to the ResNet with

N_{\mathrm{train}}=4761

and the PGFHNN overlap very closely and are therefore nearly indistinguishable in the figure.

Figure 6.10: Averaged trajectory

\epsilon_{\mathrm{traj}}(t)

( Top ) and Hamiltonian

\epsilon_{H}(t)

( Bottom ) errors for trajectories generated by the PGFHNN and parametric ResNet flows for the time-dependent damped harmonic oscillator with

V_{0}=1

\Omega=3.5

, and

\nu=0.25

. The errors are computed by averaging over 49 test initial conditions distributed in the square

-1\leq q,p\leq 1

in phase space, none of which were used during training. In order to reach a comparable accuracy, the non-structure-preserving ResNet needed a 7.6-times larger training data set than the geometric PGFHNN.

Figure 6.11: Comparison of trajectories generated by the PGFHNN and parametric ResNet flows for the forced Kubo oscillator with

\beta=0.5

and

\nu=0.25

. The left column corresponds to the initial condition

(\bar{q}_{0},\bar{p}_{0})=(-0.75,-0.75)

, while the right column corresponds to

(\bar{q}_{0},\bar{p}_{0})=(0,0.25)

. In both cases, the first row displays the position

q(t)

, the second row the momentum

p(t)

, and the third row the Hamiltonian

H(t)

evaluated along the corresponding trajectories.

Figure 6.12: Comparison of trajectories generated by the PGFHNN and parametric ResNet flows for the forced Kubo oscillator with

\beta=0.5

and

\nu=0.01

. The left column corresponds to the initial condition

(\bar{q}_{0},\bar{p}_{0})=(-0.75,-0.75)

, while the right column corresponds to

(\bar{q}_{0},\bar{p}_{0})=(-0.25,-0.25)

. In both cases, the first row displays the position

q(t)

, the second row the momentum

p(t)

, and the third row the Hamiltonian

H(t)

evaluated along the corresponding trajectories.

Figure 6.13: Averaged trajectory

\epsilon_{\mathrm{traj}}(t)

( Top ) and Hamiltonian

\epsilon_{H}(t)

( Bottom ) errors for trajectories generated by the PGFHNN and parametric ResNet flows for the forced Kubo oscillator with

\beta=0.5

. The left column corresponds to the friction coefficient

\nu=0.25

, while the right column corresponds to

\nu=0.01

. In both cases, the errors are computed by averaging over 49 test initial conditions distributed in the square

-1\leq q,p\leq 1

in phase space, and 49 independent sample paths of the Wiener process, none of which were used during training.

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Symplectic Neural Networks for learning Generalized Hamiltonians

Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta, Melvin Leok+1

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Dynamical Systems Symplectic RNNs Hamiltonian Neural Networks System Identification Neural ODEs Hamiltonian Dynamics

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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.

Abstract

Structure-Preserving Neural Networks

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