We derive and solve an ``Equation of Motion'' (EoM) for deep neural networks (DNNs), a differential equation that precisely describes the discrete learning dynamics of DNNs. Differential equations are continuous but have played a prominent role even in the study of discrete optimization (gradient descent (GD) algorithms). However, there still exist gaps between differential equations and the actual learning dynamics of DNNs due to discretization error. In this paper, we start from gradient flow (GF) and derive a counter term that cancels the discretization error between GF and GD. As a result, we obtain EoM, a continuous differential equation that precisely describes the discrete learning dynamics of GD. We also derive discretization error to show to what extent EoM is precise. In addition, we apply EoM to two specific cases: scale- and translation-invariant layers. EoM highlights differences between continuous-time and discrete-time GD, indicating the importance of the counter term for a better description of the discrete learning dynamics of GD. Our experimental results support our theoretical findings.

翻译：我们推导并求解了深度神经网络（DNN）的“运动方程”（EoM），该微分方程精确描述了DNN的离散学习动力学。微分方程虽为连续形式，但在离散优化（梯度下降算法）研究中占据重要地位。然而，由于离散化误差的存在，微分方程与DNN实际学习动力学之间仍存在差距。本文从梯度流（GF）出发，推导出可抵消GF与梯度下降（GD）之间离散化误差的补偿项，从而得到EoM——一种精确描述GD离散学习动力学的连续微分方程。我们还推导了离散化误差，以明确EoM的精确程度。此外，我们将EoM应用于两类特例：尺度不变层与平移不变层。EoM揭示了连续时间GD与离散时间GD之间的差异，表明补偿项对更准确描述GD离散学习动力学的重要性。实验结果支持了我们的理论发现。