[论文解读] Symplectic Recurrent Neural Networks

Zhengdao Chen, Jianyu Zhang+2 more|arXiv (Cornell University)|Sep 29, 2019

Model Reduction and Neural Networks参考文献 24被引用 64

一句话总结

SRNN 从轨迹中学习哈密顿动力学，使用带辛积分的神经哈密顿量、多步训练和初始状态优化，提高对噪声的鲁棒性并处理像反弹这样的刚性动力学。

ABSTRACT

We propose Symplectic Recurrent Neural Networks (SRNNs) as learning algorithms that capture the dynamics of physical systems from observed trajectories. An SRNN models the Hamiltonian function of the system by a neural network and furthermore leverages symplectic integration, multiple-step training and initial state optimization to address the challenging numerical issues associated with Hamiltonian systems. We show that SRNNs succeed reliably on complex and noisy Hamiltonian systems. We also show how to augment the SRNN integration scheme in order to handle stiff dynamical systems such as bouncing billiards.

研究动机与目标

• 直接从观测到的位置和动量轨迹学习哈密顿动力学。
• 通过辛积分和循环训练提高对观测噪声的鲁棒性。
• 引入初始状态优化以减小噪声引起的偏差。
• 通过扩增演示对刚性动力学的处理，包括完美反弹。
• 在复杂系统（弹簧链、三体）上展示性能，并与基线 HNN/O-NET 方法进行比较。

提出的方法

• 将哈密顿量建模为神经网络 Hθ(p,q)，其导数给出动力学（Hθ = Kθ1(p) + Vθ2(q)）。
• 使用对称辛跃步积分器来传播(p,q)并通过多个时间步进行反向传播（多步训练）。
• 使用 ODE NET 或哈密顿 NET 形式进行训练，将训练与积分耦合以预测轨迹。
• 引入初始状态优化（ISO）：将初始(p0,q0)视为每条轨迹可训练的变量，通过损失进行优化。
• 通过一个反弹模块对跃步积分进行扩增，以处理刚性动力学，利用视觉线索来建模反弹方向和时机（n, α, γ）。
• 证明相同的积分器/时间步在训练与测试时可得到补偿离散化误差的学习到的修正方程。

实验结果

研究问题

• RQ1 Can SRNNs reliably learn complex Hamiltonian dynamics from noisy trajectory data?
• RQ2 Does using a symplectic (leapfrog) integrator improve stability and accuracy over Euler-based training?
• RQ3 Does recurrent (multi-step) training outperform single-step training under noise?
• RQ4 Can initial state optimization improve predictive accuracy under observation noise?
• RQ5 Can SRNN be extended to handle stiff dynamics such as perfect rebounds (billiards) via augmentation?

主要发现

• SRNN with recurrent (multi-step) training and leapfrog integration achieves the lowest predictive errors on the spring-chain and three-body systems compared with H-NET and O-NET baselines.
• Using leapfrog consistently in training and testing improves stability and accuracy over Euler-based schemes.
• Initial state optimization (ISO) yields the best predictions under noise, outperforming fixed-initial-state variants.
• SRNN-derived dynamics can compensate for discretization errors, sometimes outperforming numerically solving the true ODE with the same step size.
• Augmented SRNNs successfully learn perfect rebound behavior in a heavy billiard, outperforming baselines even when rebound timing is uncertain.