用神经网络直接从势能预测物理轨迹，避免误差累积。
Neural Hamilton: Can A.I. Understand Hamiltonian Mechanics?
- 将哈密顿力学转为算子学习问题，跳过求解微分方程。
- 在谐振子等1D系统上，精度高于四阶龙格-库塔法。
- 提出两种新架构，适合高效处理时序动力学，适合物理模拟研究者。
我们提出一种基于神经网络的新框架，将经典力学重构为算子学习问题。模型直接从势能函数映射到相空间中的轨迹，无需求解哈密顿方程。与传统方法依赖迭代时间积分导致误差累积不同，本方法有效抑制了误差传播。为此，我们设计了两种新神经网络架构：VaRONet 和 MambONet，分别基于变分LSTM序列到序列模型和Mamba模型，以实现高效的时间动态建模。我们在多种一维物理问题上进行了测试，包括谐振子、双阱势、Morse势以及训练数据外的其他势函数模型。相比基于四阶龙格-库塔（RK4）算法的传统数值方法，该模型在计算效率和精度方面均有提升。代码已公开于：https://github.com/Axect/Neural_Hamilton。
原文摘要 · Abstract (English)
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
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