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Neural Hamilton: Can A.I. Understand Hamiltonian Mechanics?

Tae-Geun Kim, Seong Chan Park

TL;DR

Compared to traditional numerical methods based on the fourth-order Runge-Kutta (RK4) algorithm, this model demonstrates improved computational efficiency and accuracy.

Abstract

We propose a novel framework based on neural network that reformulates classical mechanics as an operator learning problem. A machine directly maps a potential function to its corresponding trajectory in phase space without solving the Hamilton equations. Most notably, while conventional methods tend to accumulate errors over time through iterative time integration, our approach prevents error propagation. Two newly developed neural network architectures, namely VaRONet and MambONet, are introduced to adapt the Variational LSTM sequence-to-sequence model and leverage the Mamba model for efficient temporal dynamics processing. We tested our approach with various 1D physics problems: harmonic oscillation, double-well potentials, Morse potential, and other potential models outside the training data. Compared to traditional numerical methods based on the fourth-order Runge-Kutta (RK4) algorithm, our model demonstrates improved computational efficiency and accuracy. Code is available at: https://github.com/Axect/Neural_Hamilton

Neural Hamilton: Can A.I. Understand Hamiltonian Mechanics?

TL;DR

Compared to traditional numerical methods based on the fourth-order Runge-Kutta (RK4) algorithm, this model demonstrates improved computational efficiency and accuracy.

Abstract

We propose a novel framework based on neural network that reformulates classical mechanics as an operator learning problem. A machine directly maps a potential function to its corresponding trajectory in phase space without solving the Hamilton equations. Most notably, while conventional methods tend to accumulate errors over time through iterative time integration, our approach prevents error propagation. Two newly developed neural network architectures, namely VaRONet and MambONet, are introduced to adapt the Variational LSTM sequence-to-sequence model and leverage the Mamba model for efficient temporal dynamics processing. We tested our approach with various 1D physics problems: harmonic oscillation, double-well potentials, Morse potential, and other potential models outside the training data. Compared to traditional numerical methods based on the fourth-order Runge-Kutta (RK4) algorithm, our model demonstrates improved computational efficiency and accuracy. Code is available at: https://github.com/Axect/Neural_Hamilton

Paper Structure

This paper contains 55 sections, 5 theorems, 49 equations, 8 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work & Background
  3. Operator Learning
  4. Hamilton's Equations
  5. Hamilton's Equations as an Operator
  6. Methods
  7. Data Generation
  8. Potential Function Generation
  9. Trajectory Generation
  10. Model Architectures
  11. DeepONet
  12. VaRONet
  13. TraONet
  14. MambONet
  15. Experiments
  16. ...and 40 more sections

Key Result

Theorem 1

Suppose that $X$ is a Banach space, $K_1 \subset X, K_2 \subset \mathbb{R}^d$ are two compact sets in $X$ and $\mathbb{R}^d$, respectively, $V$ is a compact set in $C(K_1)$. Assume that $G: V \rightarrow C(K_2)$ is a nonlinear continuous operator. Then, for any $\epsilon > 0$, there exist positive i holds for all $u \in V$ and $y \in K_2$.

Figures (8)

  • Figure 1: (Red dashed box) Architectures of DeepONet, (Green solid box) VaRONet, (Blue dashed box) TraONet, and (Purple solid box) MambONet. Each model processes the input potential function V and time t to predict the position $\hat{q}(\textbf{t})$ and momentum $\hat{p}(\textbf{t})$ trajectories. DeepONet uses simple feed-forward networks, VaRONet incorporates variational sampling and LSTM layers, TraONet utilizes transformer blocks, and MambONet combines Mamba blocks with transformer decoders.
  • Figure 2: Example of a generated potential function and corresponding trajectories. (a) shows the potential function $V(q)$ generated using Algorithm \ref{['alg:potential']}. (b) and (c) display the corresponding position $q(t)$ and momentum $p(t)$ trajectories generated from the potential function using Algorithm \ref{['alg:trajectory']}.
  • Figure 3: Distribution of total test losses ($\mathcal{L}_{\text{tot}}$) for different models. RK4 (Gray), DeepONet (Yellow), TraONet (Green), VaRONet (Red), and MambONet (Blue).
  • Figure 4: Comparison of RK4 (top row) and MambONet (bottom row) solutions for the Double Well potential. Transparent gray lines represent the ground truth, while scatter points transitioning from black to orange over time show the predicted solutions.
  • Figure 5: Distribution of total test losses ($\mathcal{L}_{\text{tot}}$) for TraONet trained on different dataset sizes. RK4 (Gray), TraONet($10^4$) (Green), TraONet($10^5$) (Red), TraONet($10^6$) (Blue).
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 2
  • proof
  • Theorem 3
  • proof