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We derive and solve an ``Equation of Motion'' (EoM) for deep neural networks(DNNs), a differential equation that precisely describes the discrete learningdynamics of DNNs. Differential equations are continuous but have played aprominent role even in the study of discrete optimization (gradient descent(GD) algorithms). However, there still exist gaps between differentialequations 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 thatcancels the discretization error between GF and GD. As a result, we obtain EoM,a continuous differential equation that precisely describes the discretelearning dynamics of GD. We also derive discretization error to show to whatextent EoM is precise. In addition, we apply EoM to two specific cases: scale-and translation-invariant layers. EoM highlights differences betweencontinuous-time and discrete-time GD, indicating the importance of the counterterm for a better description of the discrete learning dynamics of GD. Ourexperimental results support our theoretical findings.

Toward Equation of Motion for Deep Neural Networks: Continuous-time Gradient Descent and Discretization Error Analysis