where Z = exp[-βU (q)]dq is a partition function of the system with respect to q; β = 1/k B T , k B is Boltzmann constant and T denotes temperature. The primary objective of structure-based coarse-graining lies in approximating the (many-body) potential of mean force (PMF) with effective potentials that reproduce a set of reference distribution functions from AA models. Hence, deriving the effective CG potentials can be considered as solving an inverse problem: The forward function from the potential U (q) to the probability distribution functions P (q) is well established (through MD simulation), while the opposite direction, from P (q) to U (q), is to be determined. Furthermore, this inverse problem can be treated as an (iterative) optimization problem: (iteratively) optimize the effective potentials until it reproduces all target observables (e.g., distribution functions) through the CG-MD simulation. It should be noted that the Boltzmann-inverted potential serves as an exact solution only for non-interacting systems, such as ideal gases, but it often fails to reproduce the reference g ref (r) for condensed-phase systems due to strong many-body correlations. Therefore, optimization is necessary to improve the initial Boltzmann-inverted potential.