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Discovery of Symbolic Hamiltonian Expressions with Buckingham-Symplectic Networks

Joe Germany, Joseph Bakarji, Sara Najem

Abstract

Hamiltonian systems lie at the heart of modeling the physical world. Their defining scalar, the Hamiltonian, encodes both energy conservation and symplectic geometry in its phase-space trajectories. Recent deep learning approaches model Hamiltonian systems by embedding their properties either in the architecture or in the loss function. However, they typically ignore that: i) a Hamiltonian carries units of energy and/or ii) that every integrable Hamiltonian admits a canonical transformation to action-angle coordinates in which the dynamics reduce to a simple rotation on an invariant torus. We propose BuSyNet, a deep learning architecture that combines these two constraints via a dimensionally-consistent, symplectic transformation. A symplectic layer maps input trajectories to lower-dimensional latent action-angle variables, which are then combined with system parameters to discover a symbolic Hamiltonian expression in units of energy. Evaluated on the harmonic oscillator and the Kepler two-body problem (in 2D and 3D), BuSyNet recovers concise, closed-form Hamiltonians that outperform state-of-the-art neural architectures in long-term prediction accuracy and stability, while maintaining interpretability.

Discovery of Symbolic Hamiltonian Expressions with Buckingham-Symplectic Networks

Abstract

Hamiltonian systems lie at the heart of modeling the physical world. Their defining scalar, the Hamiltonian, encodes both energy conservation and symplectic geometry in its phase-space trajectories. Recent deep learning approaches model Hamiltonian systems by embedding their properties either in the architecture or in the loss function. However, they typically ignore that: i) a Hamiltonian carries units of energy and/or ii) that every integrable Hamiltonian admits a canonical transformation to action-angle coordinates in which the dynamics reduce to a simple rotation on an invariant torus. We propose BuSyNet, a deep learning architecture that combines these two constraints via a dimensionally-consistent, symplectic transformation. A symplectic layer maps input trajectories to lower-dimensional latent action-angle variables, which are then combined with system parameters to discover a symbolic Hamiltonian expression in units of energy. Evaluated on the harmonic oscillator and the Kepler two-body problem (in 2D and 3D), BuSyNet recovers concise, closed-form Hamiltonians that outperform state-of-the-art neural architectures in long-term prediction accuracy and stability, while maintaining interpretability.

Paper Structure

This paper contains 19 sections, 20 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Buckingham-Symplectic Networks (BuSyNet)
  4. Background
  5. Symplectic maps via (G-)SympNet.
  6. Dimensional consistency via BuckiNet.
  7. Architecture
  8. Grounding Action Calculations.
  9. Unit-Aware BuckiNet Head.
  10. SympNet Losses.
  11. Training loss.
  12. Latent-space rollout (forecasting).
  13. Architecture Parameters.
  14. Experimental Results
  15. Harmonic Oscillator
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Illustration of the BuSyNet architecture. Measurements of position ($q_k$) and momentum ($p_k$) at time index $k$ are fed into a symplectic network, transformed into actions $I_k$ and angles $\theta_k$. Actions are then fed into a BuckiNet head to reconstruct an analytical expression for the Hamitonian $H$ with known equations of motion. At inference, an Euler step is applied to $\theta_k$, while $I_k$ stays the same, and $p_{k+1}$ and $q_{k+1}$ are obtained via the inverse of SympNet.
  • Figure 2: Discovered Hamiltonian $\hat{H}$ versus time on the training and test sets for the BuSyNet method.
  • Figure 3: Graphs comparing the performance of BuSyNet with the other methods for the temporal evolution of the position $q(t)$ of the harmonic oscillator and the phase space (in Cartesian coordinates) for the 2D Kepler problem.
  • Figure 4: The learned transformation to action-angle coordinates on the training and test sets for the BuSyNet method.