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Learning Trajectories of Hamiltonian Systems with Neural Networks

Katsiaryna Haitsiukevich, Alexander Ilin

TL;DR

The paper addresses learning conservative dynamical systems by combining Hamiltonian neural networks with a continuous-time trajectory estimator. They introduce Deep Hidden Hamiltonian (DHH), which learns a trajectory $s(t)=(q(t),p(t))$ via a neural network while learning the Hamiltonian $\mathcal{H}$ through an HNN, enforcing $\frac{d q}{dt}=\frac{\partial \mathcal{H}}{\partial p}$ and $\frac{d p}{dt}=-\frac{\partial \mathcal{H}}{\partial q}$; training combines a supervision loss, a dynamics-consistency loss, and a soft energy-conservation term $\mathcal{L}_{extra}$. Experiments on mass-spring, pendulum, and N-body systems show improved accuracy under low sampling rates and noisy observations, outperforming baselines such as HNN with finite-difference or simulator derivatives, NSSNN, Neural ODE, and DHPMs. The work highlights a robust alternative for Hamiltonian learning with continuous-time trajectory estimation, while acknowledging limitations to canonical-coordinate, conservative settings and pointing to future work on relaxing assumptions and integrating physics priors.

Abstract

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.

Learning Trajectories of Hamiltonian Systems with Neural Networks

TL;DR

The paper addresses learning conservative dynamical systems by combining Hamiltonian neural networks with a continuous-time trajectory estimator. They introduce Deep Hidden Hamiltonian (DHH), which learns a trajectory via a neural network while learning the Hamiltonian through an HNN, enforcing and ; training combines a supervision loss, a dynamics-consistency loss, and a soft energy-conservation term . Experiments on mass-spring, pendulum, and N-body systems show improved accuracy under low sampling rates and noisy observations, outperforming baselines such as HNN with finite-difference or simulator derivatives, NSSNN, Neural ODE, and DHPMs. The work highlights a robust alternative for Hamiltonian learning with continuous-time trajectory estimation, while acknowledging limitations to canonical-coordinate, conservative settings and pointing to future work on relaxing assumptions and integrating physics priors.

Abstract

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.
Paper Structure (7 sections, 8 equations, 4 figures)

This paper contains 7 sections, 8 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Modeling Hamiltonian systems
  3. Hamiltonian neural networks
  4. Physics-informed neural networks as an integrator for HNN
  5. Related work
  6. Experiments
  7. Conclusion and future work

Figures (4)

  • Figure 1: (a)--(c): Estimated state trajectories for the mass-spring system with DHH (green) and HNN (blue). (d): The trajectory obtained by integrating the system equations \ref{['eq:hamilt']} with the Runge-Kutta integrator using the Hamiltonian $\mathcal{H}$ learned by DHH for the data from Fig. \ref{['fig:spring_traj']}c.
  • Figure 2: Estimated trajectories and the corresponding vector field by DHH for the mass-spring system with noisy observations of ${\bm{q}}$ and no observations of ${\bm{p}}$.
  • Figure 3: Log-mean-squared errors between the estimated trajectories and the ground truth as a function of the sampling rate on clean (first row) and noisy (second row) observations.
  • Figure 4: Log-mean-squared errors between the estimated trajectories and the ground truth as a function of the sampling rate on clean (first row) and noisy (second row) observations for different multipliers for $\mathcal{L}_\text{extra}$.