## Introduction

The recent years have witnessed a growing number of studies focused on increasing the range of space and time scales attainable in computer simulations. One of the most crucial points in pursing this goal is that the acceleration would not be provided only by improving the computer architecture, but rather by utilizing smart computational schemes to bridge the gap of space and time scales. More importantly, this target is meant to be achieved without losing accuracy in describing (at least part of) the physicochemical properties of the systems under study.

In this perspective, we discuss a technique that allows to parametrize coarse-grained force fields, using as reference some properties of the system simulated at a finer grain of description. Initially known as force matching (FM) method,[1] slightly different flavors of its applications have been recently called potential matching,[2] combined function derivative approximation,[3] adaptive FM,[4] matching algorithm,[5] force balance,[6] or multiscale coarse graining.[7] In all the above listed approaches, the sought parametrization is obtained through a minimization procedure of an objective function, that, in the seminal paper of Ercolessi and Adams on FM, was a sum of least-square residuals of atomic forces (*vide infra*).

In principle, FM could be applied to a hierarchy of scales starting from high level *ab initio* molecular dynamics (MD), passing through atomistic force field (polarizable if needed), to higher levels of coarse graining. Potentially, if the computational and methodological issues still open will be solved, given its intrinsic degree of automatization, FM would represent an efficient and systematic approach to tackle the space and time scales problem. Current research is focused on solving these weaknesses and on improving the performances of the method. Once this goal will be achieved, FM would become a strong candidate to face the study of a wide range of systems, in particular in the realms of material science and biophysics. As far as multiscale methods are concerned, FM is direct competitor of other methods based on the distribution matching approaches such as the iterative inverse Boltzmann[8] or the inverse Monte Carlo,[9] where the distribution functions of a fine-grained simulation are fitted to the coarse-grained force field.

Since the first study, dating back to 20 years ago,[1] FM has grown to include molecular torques,[4, 5] stress tensors,[10, 11] and total energy[6, 11, 12]; for more complex systems, as proteins, it has been coupled to charge fitting methods based on the electrostatic potential to improve its efficiency.[13] The implementation of FM with well-established methods to compute the value of some parameters, such as charges, polarizabilities and, in some cases, dispersion terms, might reduce the risk of cancellation errors, hence improving the robustness of the procedure. It is also important to mention that free implementations of FM are available, either as standalone packages[14] or as part of MD codes.[15]

We hereby discuss merits and shortcomings of FM, by identifying three main routes along which the method would become a sound, automated, and possibly reliable multiscale approach. More in detail, in the following sections, we are going to discuss about the minimization procedure, the empirical potential, and the reference data.