  We give a continuous perspective on the Inertial Corrected Primal-Dual
Proximal Splitting (IC-PDPS) proposed by Valkonen ({\it SIAM J. Optim.}, 30(2):
1391--1420, 2020) for solving saddle-point problems. The algorithm possesses
nonergodic convergence rate and admits a tight preconditioned proximal point
formulation which involves both inertia and additional correction. Based on new
understandings on the relation between the discrete step size and rescaling
effect, we rebuild IC-PDPS as a semi-implicit Euler scheme with respect to its
iterative sequences and integrated parameters. This leads to two novel
second-order ordinary differential equation (ODE) models that are equivalent
under proper time transformation, and also provides an alternative
interpretation from the continuous point of view. Besides, we present the
convergence analysis of the Lagrangian gap along the continuous trajectory by
using proper Lyapunov functions.
