提出计算惯性概念，揭示优化过程中的能量守恒规律。
Computational Inertia as a Conserved Quantity in Frictionless and Damped Learning Dynamics
- 定义计算惯性为参数速度与损失之和的标量
- 在无摩擦训练中该量保持不变，有阻尼时呈解析衰减
- 适用于分析收敛性、稳定性及训练轨迹几何
我们识别出连续时间优化动力学中的一个守恒量，称为计算惯性。它定义为动能（参数速度）与势能（损失）之和，该标量在理想无摩擦训练中保持不变。我们形式化了这一守恒律，推导其在阻尼和随机扰动下的解析衰减行为，并在合成系统中验证其表现。该不变量提供了一种简洁视角来解释学习轨迹，可能有助于构建分析收敛性、稳定性和训练几何的理论工具。
原文摘要 · Abstract (English)
We identify a conserved quantity in continuous-time optimization dynamics, termed computational inertia. Defined as the sum of kinetic energy (parameter velocity) and potential energy (loss), this scalar remains invariant under idealized, frictionless training. We formalize this conservation law, derive its analytic decay under damping and stochastic perturbations, and demonstrate its behavior in a synthetic system. The invariant offers a compact lens for interpreting learning trajectories, and may inform theoretical tools for analyzing convergence, stability, and training geometry.
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