  Deep Equilibrium Models (DEQs) and Neural Ordinary Differential Equations
(Neural ODEs) are two branches of implicit models that have achieved remarkable
success owing to their superior performance and low memory consumption. While
both are implicit models, DEQs and Neural ODEs are derived from different
mathematical formulations. Inspired by homotopy continuation, we establish a
connection between these two models and illustrate that they are actually two
sides of the same coin. Homotopy continuation is a classical method of solving
nonlinear equations based on a corresponding ODE. Given this connection, we
proposed a new implicit model called HomoODE that inherits the property of high
accuracy from DEQs and the property of stability from Neural ODEs. Unlike DEQs,
which explicitly solve an equilibrium-point-finding problem via Newton's
methods in the forward pass, HomoODE solves the equilibrium-point-finding
problem implicitly using a modified Neural ODE via homotopy continuation.
Further, we developed an acceleration method for HomoODE with a shared
learnable initial point. It is worth noting that our model also provides a
better understanding of why Augmented Neural ODEs work as long as the augmented
part is regarded as the equilibrium point to find. Comprehensive experiments
with several image classification tasks demonstrate that HomoODE surpasses
existing implicit models in terms of both accuracy and memory consumption.
