We introduce a novel framework for optimization based on energy-conserving Hamiltonian dynamics in a strongly mixing (chaotic) regime and establish its key properties analytically and numerically. The prototype is a discretization of Born-Infeld dynamics, with a squared relativistic speed limit depending on the objective function. This class of frictionless, energy-conserving optimizers proceeds unobstructed until slowing naturally near the minimal loss, which dominates the phase space volume of the system. Building from studies of chaotic systems such as dynamical billiards, we formulate a specific algorithm with good performance on machine learning and PDE-solving tasks, including generalization. It cannot stop at a high local minimum and cannot overshoot the global minimum, yielding an advantage in non-convex loss functions, and proceeds faster than GD+momentum in shallow valleys.

本文提出一种基于混沌状态下的能量守恒哈密顿动力学的优化框架，并通过理论和数值分析证明其关键性质。该优化器依靠无摩擦、节能的算法流程来穿越高纬度相空间，并能够在非凸损失函数中快速到达局部最小值点。通过应用于机器学习和偏微分方程求解等任务中，该优化算法比传统的GD+momentum更具优势。

Born-Infeld (BI) for AI: 能量守恒下降法 (ECD) 优化