  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.
