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

TL;DR

This paper introduces a neural network framework that models Hamiltonian mechanics as an operator learning problem, enabling direct trajectory prediction without iterative solving, reducing error accumulation, and improving efficiency over traditional methods.

Contribution

It presents a novel neural network approach with two architectures, VaRONet and MambONet, for direct phase space trajectory prediction in classical mechanics.

Findings

Outperforms RK4 in accuracy and efficiency

Effectively models various potential energy functions

Reduces error propagation in trajectory prediction

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