High-fidelity simulations of unsteady incompressible flows are computationally expensive due to the nested iterations required by implicit solvers: time stepping, Newton iterations, linear solver iterations, and, in many cases, preconditioner sub-iterations. In this work, we propose a cycle-based machine-learning-enhanced time-stepping (MLT) algorithm that reduces computational cost by enabling larger effective time steps while preserving the underlying flow dynamics. While ML-based acceleration has been explored for steady-state problems, nonlinear elliptic problems and stiff parabolic systems, its extension to dynamic incompressible flows remains non-trivial due to error accumulation and stability constraints. The proposed approach addresses this challenge by introducing ML predictions only at selected time steps, allowing the solver to correct accumulated errors and maintain stability. To establish a rigorous theoretical foundation for this strategy in a setting amenable to detailed analysis, we study the Allen–Cahn equation as a prototypical nonlinear dynamical system. We derive sensitivity estimates for implicit time-stepping schemes that quantify how perturbations in the initial state propagate to the converged solution. These theoretical predictions are corroborated by numerical experiments, demonstrating that errors introduced by ML-based initial guesses remain bounded and propagate linearly under implicit time integration, thereby establishing the stability and accuracy of the MLT algorithm. The MLT algorithm is implemented within the OpenIFEM multiphysics code and validated on the flow past a cylinder with vortex shedding at Reynolds numbers unseen during training, achieving approximately a 21% reduction in core computational cost while maintaining accuracy in key aerodynamic quantities such as lift, drag, and pressure differentials. A detailed error analysis further illustrates how the frequency of ML initialization and subsequent correction phases governs error growth. The proposed framework provides a practical pathway for accelerating dynamic solvers and is readily extensible to more complex single- and multi-physics systems.