用平衡传播训练拉格朗日动力系统，无需反向传播时间
Equilibrium Propagation for Learning in Lagrangian Dynamical Systems
- 通过作用量极值原理扩展平衡传播至动态轨迹
- 在周期边界条件下实现量子平衡传播的半经典极限
- 适合周期性或固定初末态系统，高效更新参数
我们提出一种基于平衡传播的方法，用于训练由拉格朗日力学支配的动力系统。该方法将原本用于能量模型的平衡传播扩展到动态轨迹，利用作用量极值原理实现训练。通过轻微扰动轨迹至目标状态，并测量与待训练参数共轭变量的响应来完成学习。该方法特别适用于具有周期边界条件或固定初始与终态的系统，可在不依赖显式时间反向传播的情况下实现高效参数更新。在周期边界条件下，该方法可导出量子平衡传播的半经典极限。同时，本文也讨论了含耗散系统的应用。
原文摘要 · Abstract (English)
We propose a method for training dynamical systems governed by Lagrangian mechanics using Equilibrium Propagation. Our approach extends Equilibrium Propagation - initially developed for energy-based models - to dynamical trajectories by leveraging the principle of action extremization. Training is achieved by gently nudging trajectories toward desired targets and measuring how the variables conjugate to the parameters to be trained respond. This method is particularly suited to systems with periodic boundary conditions or fixed initial and final states, enabling efficient parameter updates without requiring explicit backpropagation through time. In the case of periodic boundary conditions, this approach yields the semiclassical limit of Quantum Equilibrium Propagation. Applications to systems with dissipation are also discussed.
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