  Recent years have seen a growing interest in understanding acceleration
methods through the lens of ordinary differential equations (ODEs). Despite the
theoretical advancements, translating the rapid convergence observed in
continuous-time models to discrete-time iterative methods poses significant
challenges. In this paper, we present a comprehensive framework integrating the
inertial systems with Hessian-driven damping equation (ISHD) and learning-based
approaches for developing optimization methods through a deep synergy of
theoretical insights. We first establish the convergence condition for ensuring
the convergence of the solution trajectory of ISHD. Then, we show that provided
the stability condition, another relaxed requirement on the coefficients of
ISHD, the sequence generated through the explicit Euler discretization of ISHD
converges, which gives a large family of practical optimization methods. In
order to select the best optimization method in this family for certain
problems, we introduce the stopping time, the time required for an optimization
method derived from ISHD to achieve a predefined level of suboptimality. Then,
we formulate a novel learning to optimize (L2O) problem aimed at minimizing the
stopping time subject to the convergence and stability condition. To navigate
this learning problem, we present an algorithm combining stochastic
optimization and the penalty method (StoPM). The convergence of StoPM using the
conservative gradient is proved. Empirical validation of our framework is
conducted through extensive numerical experiments across a diverse set of
optimization problems. These experiments showcase the superior performance of
the learned optimization methods.
