Large-scale computational molecular models provide scientistsameans to investigate the effect of microscopic details on emergentmesoscopic behavior. Elucidating the relationship between variationson the molecular scale and macroscopic observable properties facilitatesan understanding of the molecular interactions driving the propertiesof real world materials and complex systems . As a result, discoveringan explicit, systematic connection between microscopic nature andemergent mesoscopic behavior is a fundamental goal for this type ofinvestigation. The molecular forces critical to driving the behaviorof complex heterogeneous systems are often unclear. More problematically,simulations of representative model systems are often prohibitivelyexpensive from both spatial and temporal perspectives, impeding straightforwardinvestigations over possible hypotheses characterizing molecular behavior.While the reduction in resolution of a study, such as moving froman atomistic simulation to that of the resolution of large coarse-grained(CG) groups of atoms, can partially ameliorate the cost of individualsimulations, the relationship between the proposed microscopic detailsand this intermediate resolution is nontrivial and presents new obstaclesto study. Small portions of these complex systems can be realisticallysimulated. Alone, these smaller simulations likely do not provideinsight into collectively emergent behavior. However, by proposingthat the driving forces in both smaller and larger systems  have an explicit connection,systematic bottom-up CG techniques can be used to transfer CG hypothesesdiscovered using a smaller scale system to a larger system of primaryinterest. The proposed connection between different CG systems isprescribed by (i) the CG representation (mapping) and (ii) the functionalform and parameters used to represent the CG energetics, which approximatepotentials of mean force (PMFs). As a result, the design of CG methodsthat facilitate a variety of physically relevant representations,approximations, and force fields is critical to moving the frontierof systematic CG forward. Crucially, the proposed connection betweenthe system used for parametrization and the system of interest isorthogonal to the optimization used to approximate the potential ofmean force present in all systematic CG methods. The empirical efficacyof machine learning techniques on a variety of tasks provides strongmotivation to consider these approaches for approximating the PMFand analyzing these approximations. To complementthese experimental approaches, theorists leverage classical moleculardynamics (MD) simulations to investigate dynamical phenomena at highspatial resolution, most commonly at the atomistic level.7 However, within the space of MD simulation techniques,coarse-grained (CG) modeling and simulation are particularly attractivefor the study of systems with hierarchical length and time scalessuch as biomolecular systems14 .Understanding how molecularphenomena translate into emergent mesoscopicand macroscopic behavior is a common theme throughout biology, chemistry,physics, materials science, and engineering. Experimental techniqueshave offered microscopic insights into systems from these fields.For example, ensemble-averaged atomic structures can be resolved athigh-resolution using X-ray crystallography or cryo-electron microscopy.mapping. Effective interactions between the CG sites are parametrized to retain the essential aspects of the systemof interest under the chosen equations of motion. However, definingthese essential aspects depends upon the scientific question at handand the reference data available, variations in which have led tothe development of the top-down and bottom-up approaches, which are the two general classes of CG models thatwe will discuss later. CG simulations have three primary benefitscompared to FG simulations. First, these models enable simulationsof larger systems at appropriate length scales by virtue of the reducednumber of particles. Second, a larger integration time step can beused in CG simulations since the removal of highly fluctuating shortwavelength atomistic degrees of freedom results in a smoother CG freeenergy surface that accelerates the sampling under Hamiltonian mechanics.Finally, the construction of useful CG models grants tacit insightinto molecular features  and energetics  that areessential for understanding mesoscopic and macroscopic behavior. Forthese reasons, CG simulations provide perspectives that would otherwisebe inaccessible from more detailed atomistic MD simulations, whichhas driven their continued use and development.By design, CG models are reduced representations of fine-grained(FG) atomistic resolution molecules, where CG sites represent groupsof corresponding FG atoms through a process that can be called 27 Studies of self-assembling systems have also benefited from top-downapproaches. Using simplified geometries and interactions, it has beenpossible to broadly explore how morphologies may be dictated by afew adjustable parameters.45 However, as these approaches neglect the direct validationof microscopic details by design, it is unclear if the resultant CGmodels faithfully reproduce microscopic physics. For example, theoriginal MARTINI model, by construction, lacks a rigorous CG mappingfrom atomistic degrees of freedom by design46 and also may not reflect the underlying nature of atomisticallymapped interactions onto the CG representation, such as the correctenthalpy–entropy decomposition for certain calculated potentialsof mean force (PMF).48The top-downstrategy is perhaps the more typical CG modeling approach.Often, the scientific question posed by top-down CG studies is todetermine if a particular set of interactions is capable of reproducingspecific macroscopic properties. For example, the original MARTINICG model for lipids was parametrized to recapitulate partition coefficients.thermodynamic consistency, is to systematically parametrize CG models such that the sampleddistribution recapitulates the multidimensional configurational distributionof their FG counterparts when mapped to the CG phase space.33 Under this criterion, the ideal effective CG Hamiltoniansare the conditioned (or CG mapped) many-body PMFs expressed in theCG coordinates or configurations. Reproducing the many-body PMFs usingan arbitrarily complex set of functions or “basis set”,34 however, is challenging and sometimes problematic,as even if it is computationally feasible to capture the propertiesof the simulation being analyzed for parametrization, the resultingpotential must also describe the larger system of primary interestto the study at hand—an extrapolative task that becomes increasinglydifficult as the basis set grows in complexity. Instead, bottom-upstudies have explored if CG models that recapitulate reduced setsof microscopic statistics using similarly simplified basis functionsare also capable of collectively recapitulating mesoscopic and macroscopicbehavior.51 Unlike static properties,dynamical processes are correlated with both temporal and spatialvariables and thus can be difficult to represent at CG resolution.Extracting the time evolution of CG systems from the FG referenceprovides a strategy for rigorously integrating the many-body natureof time-dependent processes into CG models.54Bottom-up approaches usethe opposite strategy and attempt to reproducemicroscopic (mapped atomistic) statistics. The underlying principlesof most bottom-up approaches are that properties observed in referencesimulations are to be captured by the correct CG equations of motiondescribing equilibrium and certain nonequilibrium processes. For example,static properties are to be reproduced by the effective CG interactionsas determined by equilibrium statistical mechanical principles. Themajority of bottom-up CG approaches aim to reproduce static correlations.One common strategy, which we refer to as 56 Algorithms for high-dimensionalregression have been applied as the building blocks for force fieldsthat recapitulate high order correlations in CG mapped atomistic data.60 Simultaneously, ML techniques focused on directly generating samplesfrom high-dimensional distributions have provided approaches for quantifyingthe error in existing CG models and more efficient methods for generatingatomistic configurations. These novel generative approaches have additionallyshown promise in producing atomistic configurations from CG simulations(an approach often referred to as “backmapping”).69Bottom-upCG models are created to generate samples that systematicallyapproximate high-dimensional data produced by a reference model. Thisapproach is fundamentally similar to those of contemporary methodsin machine learning (ML).In this Review, we summarize recent advances in bottom-upCG modeling and discuss promisingfuture directions. In particular, we focus on how bottom-up CG modelscan be derived by establishing a systematic connection between themicroscopic (atomistic) and the reduced descriptions. We first reviewfundamental concepts in CG models that have been proposed over thepast two decades. We then discuss limitations and challenges in CGmodeling in terms of consistency, representability, and transferability.We briefly survey recent scientific findings and breakthroughs thatbenefited from methodological advances, e.g., concepts from ML. Weconclude with a brief overarching summary and future outlook towardthe next generation of bottom-up CG modeling.2Two ingredients are necessary for any bottom-up CG modeling recipe.While intertwined, these processes are typically performed separately.First, one needs to define the CG mapping that formally defines thecorrespondence between the FG and reduced resolutions. Then, oncea mapping is selected, the CG mechanics need to be defined in thedesired CG phase space on the basis of the FG statistics mapped ontothat space. Various techniques can be applied for parametrizing theinteractions governing the CG equations of state. These two ingredientsare not only essential to construct CG models but also to providethe theoretical basis to understand the challenges underlying bottom-upCG modeling, i.e., consistency, transferability, and representability.Below, we briefly describe these essential steps required for bottom-upCG modeling and discuss how they are related to the aforementionedchallenges.2-1The processof applying a CG mapping reduces the high-dimensional atomistic phasespace to a low-dimensional CG phase space. Ideally, this should involvemapping over both configurational and momentum variables in phasespace, but most molecular CG models do not involve momentum in theirequilibrium distribution consistency, and under this assumption, themajority of CG mappings can be readily applied to only the configurationalvariables. The field of chemistry has an established history of breakingcomplex molecules into moieties and functional groups in order topredict and understand atomistic behavior. This process provides anintuitive basis for designing CG configurational mappings. Even thoughthere are numerous mapping schemes that one can take in order to preservethe desired behavior in CG models, the resultant CG interaction associatedwith the specific mapping should be designed based on the statisticalmechanical principles given by the CG methodology, which will be reviewedin 2-1.AN linear functions M(r):(M1(r),...,MN(r)), where the FG coordinates can bemapped into CG site I followingRepresentingeach chemical moiety as a group of CG sites is perhaps most easilyrealized by defining each CG site as a weighted average of the configurationsof various atoms. For these real particle-based mappings, the CG mappingoperator on configurational variables is expressed as a set of cIi ∝ mi),enabling one to retain important molecular configurations and momentaat the reduced level. For force-based CG methodologies, the center-of-massmapping provides thermodynamically consistent forces that act on thecenter-of-mass phase space variables in comparison to atomistic forces.32 However, information beyond configurations maybe lost using the center-of-mass mapping. Recent advances have suggestedthat one can alternatively perform center-of-charge mapping, whichis a reweighted mass based on the partial charges of atoms withinCG sites,70 to better encode electrostaticinformation72 for systems in which electrostaticinteractions play a major role, e.g., ionic liquids.73 It is also conceivable that the geometry of the systemcan be better conserved by performing center-of-geometry mapping as cIi = 1/nI, where nI denotes the number of FG particles involved in CG site I.One example is thecenter-of-mass mapping , enhancing the fidelity of structuralcorrelations and transferability. These early successes demonstratethe utility of virtual sites and motivate the need for additionalefforts to determine systematic rules for virtual site mappings andeffective interactions for complex molecular systems, e.g., polymers.88 Another approach that shares a similar physicalprinciple is to introduce virtual sites that help to represent theeffects of explicit solvent in implicit solvent models.89 These so-called “solvent-free”CG models can potentially provide an accurate and transferable CGmodeling for biomolecules, e.g., amphiphilic assemblies of lipids.Virtual sites in CG models havebeen increasingly utilized in recent years, most predominantly intop-down CG models.2-1.C93 However, the same strategy cannotbe applied for unbonded systems since the construction of CG blobsbecomes a nonlinear and time-dependent procedure. As developed byEspañol and co-workers, who introduced the Voronoi cell representation,94 suitable mesoscopic representations for unbondedsystems must be obtained via alternate approaches such as clusteringmethods other than conventional center-of-mass and related mappings.One such example is the application of the k-meansclustering algorithm to unbonded fluids,96 and another option proposed by Praprotnik et al.98 uses spatial tessellation at adaptive resolutions. By adjustingthe center of each Voronoi cell based on its center-of-mass, theseclustering methods allow for mesoscopic CG blobs to faithfully representtheir FG counterparts. Yet, conformation-based clustering suffersfrom the nonanalytical nature of the mapping process, which makesthe derivation of CG interactions impractical, and requires frequentreclustering over the simulation. This limits the development of highlyCG models to study mesoscopic behavior.99 Hence, several alternatives have been reported in the literature,including a spherical CG blob mapping by Ayton et al.100 This latter work was also extended  to treat biomolecular membranes,with and without bound proteins.104On larger mesoscopic scales, CG particles can instead be representedas supramolecular “blobs”, and the CG mapping at thisresolution becomes less clear than at the molecular level. If thetarget system is composed of bonded systems, one can still employa linear center-of-mass mapping.105 To note, this mapping scheme is based on a Lagrangian descriptionto track individual fluid particles, but a complementary Euleriandescription can also be established in a similar vein.106 In turn, this new approach allows stable propagationof the CG blobs over time, indicating its applicability to variousfluids,107 e.g., heterogeneous multiphasesystems.In general, the nonlinear, nonanalytic, and iterative natureof the mesoscopic CG mapping is considered a major bottleneck in mesoscalemodeling of unbonded molecules . Recently, a “dynamicmapping” scheme was developed by mapping velocities insteadof configurations, which only requires the initial configuration fromsmoothed centroidal Voronoi tessellations.2-1.D108 In prior studies, backmapped structures from CG simulationshave been used as starting points for FG simulations in order to improveFG sampling.69 Generally, the procedure is performed in two steps. First, an initialmodel for the backmapped FG structure is predicted using geometricalgorithms. Then, the structure is equilibrated using a short MD simulation.However, due to the degeneracy associated with the CG mapping, thisprocess is highly nontrivial, and some recent advances have introducedML techniques to perform backmapping , this resolution may itselfprove to be too computationally expensive. As a result, it is oftendesirable to recapture the full FG details by “backmapping”the CG configurations onto FG configurations.ping see 6. Early 2-1.E114Along with CGmapping, the resolution of CG models impacts the resultant CG modelphase space and the performance of CG models in reproducing the valuesof key observables. These factors are particularly important for modelingcomplex biomolecules, where grouping different atomistic entitiesbecomes less clear. While there is no definitive answer to this problem,the optimal resolution for the desired CG models can be chosen tomaximally reproduce the key observables from CG models or the lossof information from the coarse-graining process.115 ED-CG can be employed inconjunction with principal component analysis (PCA) of atomistic trajectories115 or CG models of large proteins116 described by an heterogeneous elastic networkmodel (HeteroENM).117 Various determinationprotocols have also been suggested recently to determine the optimalCG representation, e.g., constrained minimization,118 stepwise optimization,119 andfluctuation maximization.120Notably, essential dynamics coarse-graining (ED-CG)has been developed to systematically estimate the optimal partitioningscheme for large biomolecules by recapitulating key dynamics .123 As this information loss is intrinsically relatedto the mapping entropy from the coarse-graining process, ref  in a way to preserve the maximum possible FG information. We emphasizethat the mapping entropy introduced here is one of the most centralquantities not only to determine the CG mapping but also to correctlyunderstand the representability and transferability of CG models aswill be discussed in On theother hand, an alternative approach can be achieved by minimizingthe loss of information accompanied by the coarse-graining process.References  and 122 ess, ref  suggests129 Then, the coupling between each resolutionis designed based on the ensemble of interest and desired resolutionranging from quantum to hydrodynamics levels. For the molecular level,the original AdResS used a force coupling,129 and then Hamiltonian AdResS (H-AdResS) was developed by designingthe global Hamiltonian.132 For different spatial levels,several AdResS-based schemes have been developed, such as for largerproteins,133 continuum hydrodynamics,135 systems with quantized nuclei,137 open systems,139 and the grand canonical ensemble.141 One particularlynontrivial question would be to determine the thermodynamically consistentbottom-up interactions at the multiple resolutions. We will brieflydiscuss this in Beyond a single resolution, CG models withmultiple resolutionscan be a way to capture different levels of detail in inhomogeneoussystems. However, these so-called adaptive resolution models alsorequire a concurrent coupling among multiple levels of resolutionand changing the particles’ resolution on the fly. One of themost commonly practiced frameworks, the adaptive resolution simulation(AdResS) approach, allows for a smooth transition between differentresolutions by introducing a hybrid transition region.2-2NVT) ensemble, the effective CG interactions are only dependent on theCG configurations. The essential building blocks for CG models aresummarized in Once themapping operator is decided, CG mechanics must be determined via (1)the equations of motion for the CG phase space variables and (2) theeffective CG interactions from the chosen equation of motion in phasespace, which will then enable CG simulations to be performed. Similarto the process of CG mapping, in the canonical  requires equations of motion accountingfor nonconservative interactions, e.g., the generalized Langevin equation.In this section, we specifically discuss the performance of CG modelsin terms of static correlations. The related discussion on dynamicalproperties is presented in CG equations of motion and energetics affect the fidelity of CG modelsin different ways. In order to correctly address dynamical properties,CG models should faithfully represent the friction and fluctuationsobserved in the reference FG system.2-2.BIn general, it is impractical to determine exact forms of the renormalizedmany-body CG interactions. Therefore, constructing approximate bottom-upCG models is performed in two sequential steps. The first step isto design the form of the CG Hamiltonian in terms of configurationalvariables. This is often done by adopting molecular mechanics functionalforms similar to that of the atomistic description, such that theapproximate CG interactions are written as a combination of analyticaland tractable forms, including bond, angle, torsion, and pair nonbondedinteractions. Various design principles for the CG Hamiltonian willbe described in detail in 2-32-3.AIn orderto address a myriad of chemical and biological systems, most CG methodologiesprovide a general principle to determine CG interaction parametersregardless of the CG Hamiltonian form. This is often manifested inthe bottom-up manner by enforcing certain statistical mechanical principlesto maintain the fundamental properties of the FG system. Dependingon the microscopic target of interest (thermodynamic properties orstatic correlations), various bottom-up CG methodologies have beenproposed in the field. In this subsection, we briefly survey someof the leading strategies for approximating CG energetics, and wewill review naturally emerging issues in 2-3.B142 and from Izvekov, Parrinello,Burnham, and Voth143 to define molecularmechanics force fields on an ad hoc basis from ab initio calculations. The extension of force-matchingto CG configurational space was established with the Multiscale Coarse-Graining(MS-CG) method developed first by Izvekov and Voth for biomolecularsystems30 and liquids.29 This advance was accomplished through a recognition that force-matchingcould be carried out along with a resolution reduction (coarse-graining)and that this would be a variational route to determine the many-bodyPMF for the CG variables. Later, in a series of papers,151 the MS-CG method was more fully developed and explored. By design,the MS-CG methodology determines the effective force field actingon CG site I, FI(M(rn)), by minimizing the least-squared force residual χ2[F] between a target CG model and the FG counterpart.Here, M denotes a mapping operator that maps the FG configuration rn to the CG configuration RN. These force differences areoften expressed as a quadratic residual, and thus, a systematic determinationis possible by variationally minimizing this force metricFI(M(rn)) is the unknown CG forces at the CG configuration, and fI(rn) is the projected microscopic forces on the CG site I. The unknown CG forces can be linearly expressed throughthe CG force field parameters FI(RN) = –∇IUCG(RN). In general, a two-body (pairwise) approximation is often adoptedto express the CG force field basis sets, and subsequent algorithmicadvances152 introduced spline interpolation to describeforce field parameters as 153 resultingin the following matrix equationAs noticed from the CG equations of motion, forces are central toequilibrium thermodynamics, and several methodologies have been designedbased on the conservative forces. The idea of employing force-matching(without coarse-graining) originated from the early work of Ercolessiand AdamsF is the force matrixcalculated from the CG configurations and the CG force field parametersϕ, and the column vector f represents the projectedFG forces at the CG resolution. From a statistical mechanical perspective,the MS-CG methodology satisfies the thermodynamic consistency betweenthe FG and CG phase spaces.33 It is important tonote that this approach to training a bottom-up CG force field fromFG data can be considered an early example of “machine learning”,which has become very popular in recent times (including for coarse-graining),albeit a deep neural network59 was notutilized in that early MS-CG work of almost 20 years ago.In 41 Relative entropy is defined as the differences between the FG andCG probability distributions, given by the Kullback–Leiblerdivergence154Smap⟩CG denotes the mapping entropy (introducedin Smap = ln ∫δ[M(r) – R]dr. Thus, minimizing Srel enforces minimizing the log difference between the FG and CG probabilitydistributions. Based on this metric, relative entropy minimization(REM) can be expressed within the canonical ensemble as Srel = β⟨UFG – UCG⟩CG – (AFG – ACG) + ⟨Smap⟩, where U and A denote the internal energy and free energy, respectively.Then, a set of CG model parameters {λi} can be variationally determined by minimizing therelative entropy differences between the FG and CG systems, resultingin the following conditions for local optimality38Incontrast to the force-based metric, Shell and co-workers haveidentified and implemented the information-theoretic relative entropyas a target metric.155 The connection between MS-CG and REM has also been analyzed:156 both will give the exact many-body CG variable PMF if a “perfect”basis set is used to describe the CG interactions, but the two approacheswill differ in complementary ways if more approximate basis sets areused.Due to the systematicnature of REM, several variations were reportedto enhance the predictability of CG models.157 indicatingthat MS-CG with two-body interactions attempts to capture two-bodyand three-body structural correlations.31 In a related fashion, Chaimovich and Shell showed that REM guaranteescapturing any n-body statistical correlations thatare explicitly represented in a corresponding n-bodyCG Hamiltonian.38Even though the aforementioned approaches can variationallydeterminethe effective CG interaction parameters, it is not immediately clearif the resultant CG models will reproduce a particular target atomisticcorrelation correctly given the approximate nature of the CG Hamiltonianas well as the basis set chosen to describe the CG interactions. Recentstudies have, however, shed light on such connections. Noid et al.demonstrated that the MS-CG method determined from 159 The g-YBG approachwas readily applied to CG systems on the basis of the MS-CG framework,where the structural correlations were used instead of forces.162 This observation is based on FT on the left-hand side.This produces the following equationG ≔ FTF and b ≔ FTf. In the g-YBG approach, b is expressed in terms of a set of structural correlationfunctions,160 and G contains the ensembleaverage of cross-correlations between the CG degrees of freedom.By establishinglinks between force-based models and structuralcorrelations, explicit consideration of FG structural correlationscan result in modified strategies for force field parametrization.Notably, Mullinax and Noid provided a practical link by developingthe generalized YBG (g-YBG) framework to determine optimal interactionpotentials for complex classical force fields by utilizing only structuralcorrelation functions.G matrix iteratively by matching theCG forces at a given CG structure to the mapped FG forces. Cho andChu applied this scheme to the MS-CG methodology,163 and Rudzinski and Noid made similar extensions to the g-YBGframework.164 Another iterative treatmentis to recalculate the b column vector by matchingthe CG forces at a given FG structure to the CG forces at a givenCG structure. The latter approach is equivalent to matching expectationsof basis function derivatives, as demonstrated by Lu et al.165 Another recent advance has resulted in a noniterativeparametrization scheme, while still based on 166 Lastly, the parametrization strategies discussed abovecan be improved by leveraging ML techniques. While general principlesand examples of ML-based CG parametrization will be discussed in f-divergence,58 and the force-matchingscheme in 59 Beyond force-matching, the effective flow by combining 167An iterative refinement to 2-3.C169 based on Henderson’suniqueness theorem,170 which asserts thatthere is a unique pair potential that gives rise to a given RDF. Underdilute (low-density) conditions, one can ignore the many-body correlationsin the system, and the effective pair interactions can be approximatedbased on the RDF or gFG(R). This approach is known as the (direct) Boltzmann inversion171Alternatively,several bottom-up CG approaches have been specifically designed tocapture target static correlations (as opposed to the many-body PMFof the CG variables like MS-CG and REM). One of the earliest attemptswas to capture pair correlations, or radial distribution functions(RDFs), from FG systemsgFG(R)). This deviation can be correctedusing an iterative scheme, known as iterative Boltzmann inversion(IBI), where one can iteratively improve the fidelity of the CG modelsby updating the CG interactions according to172 Even though the IBI approach has been appliedto various chemical systems with its relatively simple update scheme, 174 and recent work by Wang et al. systematically demonstrated thatthe RDF can be insensitive to pair interactions,175 as substantiated by previously reported systems using variousCG modeling methods.178 Therefore, several extensions from IBI havebeen designed to surmount these issues, including the multistate generalization(multistate IBI) by McCabe and co-workers180 as well as thermodynamic property-based IBI approaches such as Kirkwood−BuffIBI by van der Vegt and co-workers181 and 182However, most condensedmatter systems are not at the low-density limit, and thus, the modelRDF from the CG trajectory using 184 Yet, theseIMC iterations often suffer from computational overhead due to thesampling of all possible configurations.185 In light of these issues, recent methodologies have been designedbased on an integral equation approach that can approximate the many-bodycorrelations via closure equations. For example, Guenza and co-workershave proposed an Ornstein–Zernike integral equation-based analyticalapproach to determine the effective CG interactions for polymericsystems.188 It is also possible to utilize an invertedintegral equation as an initial guess for IBI to improve parametrizationefficiency.189Similarly, an approach named inverseMonte Carlo (IMC) by Lyubartsevand Laaksonen also targets the reference RDF and determines the CGinteractions in an iterative manner.2-3.D190 By directly extracting theforce information from the atomistic simulations, the CG Hamiltonianis approximated as atomistically averaged forces, and EF-CG can bederived from an averaged description of the MS-CG methodology. Similaraveraging schemes, as well as the EF-CG method, can be employed tocapture structural correlations,194 since these averaged interactionscorrectly account for the existing pair- and many-body correlationsin FG systems. Similarly, in order to correctly capture the underlyingFG energetics, a recent extension of the MS-CG method196  has shown that one couldvariationally determine the pair energy functions by minimizing theenergy differences akin to As opposedto target- or correlation-based approaches, one can also directlyderive CG interactions from reference FG systems. Based on atomisticenergetics, this class of approaches attempts to extract the reducedenergetics, i.e., forces or free energies, directly from the atomisticenergetics. For example, by borrowing from the force-matching philosophy,the effective force CG (EF-CG) approach computes the averaged forcesacting on the groups of atoms by projecting onto corresponding radialvectors.197 This fragment-based CG approach can be advantageousin terms of transferability for different system conditions, yet,by design, it does not guarantee that the static correlations willbe correctly addressed.202A direct assessment of thePMF is also possible by explicitly computing effective pair potentialsfollowing the definition of the PMF in the low-density limit. In thiscase, by adopting a free energy perturbation approach, effective CGinteractions can be computed via conditional reversible work (CRW).2-42-4.Aconsistency, representability, and transferability. The use and importance ofeach of these measures are dependent upon the scientific questionof interest. It is therefore worthwhile to discuss each of these metricswith the understanding that all three contribute to the overall accuracyof a bottom-up CG model. It should be noted that to an extent top-downCG models do not satisfy some or all of the properties discussed below,and it will generally mean that those top-down models are not consistentwith statistical mechanics (meaning they do not provide a direct connectionbetween the FG and CG worlds but are instead primarily models in thelarger sense of the word).The accuracy of bottom-up CG models is undoubtedly important as itreflects the fidelity of CG models relative to the FG reference. Yet,appropriate metrics for accuracy are ambiguous at present. In thissection, we define three separate yet related measures of bottom-upCG model fidelity: 2-4.BConsistency refersto the specific statistical mechanical principle used to generatea bottom-up CG model from a reference FG model. This consistency canserve as the theoretical basis to derive CG force fields or be utilizedas the criterion to evaluate how the constructed CG models captureFG statistics in a consistent manner.Aex = −kBT ln [∫dRNeβUCG(RN)–/VN] that is identical to theFG excess free energy aex = −kBT ln [∫drneβuFG(rn)–/Vn].203 This condition is sufficientto derive the effective CG interaction form in configurational space.However, in order to consider the full phase space, the most commonlyused criterion built upon equilibrium statistical mechanics is thermodynamic consistency,33 indicating that theCG variables pCG shouldhave exactly identical probability distributions as compared to thatof FG variables pFG that are mapped to the specific CG phase variables via the mappingoperator M: → . Both probability distributions follow Boltzmann samplingat equilibrium, and configurational and momentum variables can beseparated as discussed earlier. Mathematically, this can be expressedasSince the emergence ofCG modeling as a “field”, various consistency conditionshave been proposed. For example, the MS-CG methodology was initiallybuilt upon the consistency of excess free energies, asserting thatthe CG model should provide an excess free energy The configurationalconsistency implies that the effective CG interactionpotential should have the following form33 and a rigorous linkbetween n0 FG particles and N0 CG particles), the overall renormalized interaction UFG/CG should be writtenas207 For example, the AdResS treats the overall potential as130λα varies from 0to 1 to couple the FG and CG systems with their bulk (naked) interactions VαFG and VαCG, respectively. While this interpolativeapproach has shown to be able to capture various thermodynamic propertiesat equilibrium, the thermodynamic consistency principle of 204 it can stillbenefit from additional development based on rigorous statisticalmechanical principles. In the remainder of this Review, we will thereforeonly focus on CG models with a single level of resolution.Furthermore, the renormalized natureof the CG interactions shownin 32Whileconfigurational consistency reveals the many-body natureof CG interactions, momentum consistency imposes a rigorous conditionwhen designing CG mapping operators. Unlike configurational interactions,the momentum contribution to the system Hamiltonian is rather simpleif the mapping functions are linear: 2-4.Crepresentability. In the case of dynamical representability,we will refer to how well the CG models reproduce FG dynamical properties,and this is often related to the choice of equations of motion usedthroughout the CG simulations. We specifically discuss dynamical representabilityin The general connectionbetween microscopic statistics and dynamical or thermodynamic observables hasbeen rigorously derived through statistical mechanics. It is theserelationships that allow computer simulations of FG models  to quantitatively predict experimental properties;in this paper, we holistically refer to these relationships as 51 Perhaps themost common example is the trade-off between consistent recapitulationof pairwise correlations  and dramaticoverestimation of pressure as computed from the virial theorem dueto missing degrees of freedom in the CG model.209 From a statistical mechanical perspective, the representabilityproblem is rooted in the nature of the renormalized degrees of freedom.Due to the missing configurational and momentum variables in the CGphase space, the FG observables from the FG ensemble are not alwaysequivalent in value to their CG counterparts if the latter are simplycalculated using the same expressions as in the FG model.195Our particular interest in this sectionis in representability issues that arise from thermodynamic inconsistenciesobserved in various CG systems, which are broadly referred to as the(thermodynamic) representability problem.UCG(RN), in the canonical ensemble under thermodynamicconsistency is equivalent to the many-body PMF.33 As a projectionof the free-energy, UCG has two important attributes. First, UCG(RN) is clearly state point-dependent,i.e., UCG(RN) = f. Second, UCG encodes both energetic (ECG) and entropic (SCG) contributions,i.e., UCG(RN) = ECG – TSCG(RN);148 the entropic contribution represents the entropy“lost” and folded into the CG interactions due to CGmapping, that is, the entropy associated with the FG degrees of freedomthat map to the same CG configuration. Therefore, evaluating ⟨UCG⟩ would not give the FG internal energy.Instead, a reformulated expression ⟨UCG + TSCG⟩ could recoverthe FG internal energy; finding approximations for SCG is an active area of research.124To demonstrate this perspective on representability, considerthefact that the ideal effective CG Hamiltonian in the CG configurationalspace, 49 For example, if the observable of interest, A, only depends upon configurational variables, this inconsistencycan be mathematically formulated by examining the difference between AFG = ⟨A(rn)⟩rn and ACG = ⟨ACG(RN)⟩RN.By introducing thermodynamic consistency into AFG, it can be shown that observables at the CG resolution useequivalent expressions to that of the FG resolution only when thefollowing definition is used:The thermodynamic representabilityproblem becomes more apparentfor observable expressions explicitly mapped from the FG to CG ensembles.AFG ≠ ACG.49 There has been considerable interest in determining the observableincompatibility from a rigorous statistical mechanical perspective.211However, as can beseen from the complex expression in 212 rather than only the reduced CG configurational variables. As such,the evaluation of pressure in a CG model by simply using the virialexpression as if the CG variables were the FG variables has essentiallyno connection to the pressure of the system at the actual FG level.Stated differently, the “pressure” in the CG model israther meaningless unless interpreted as a part of a global modelin which virtually everything is a model, not just the interactionsbetween the CG particles. On the other hand, if the volume dependenceof UCG were known, it would be possibleto construct a modified virial expression to compute the pressuresusing only CG variables that are consistent with the FG pressures.146 One way to resolve this discrepancy is to determine a compatibleobservable expression using correct basis sets according to the thermodynamicproperties of interest. A recent study on determining the correctobservable expression for pressure using particle-wise decompositionssuggested that such an approach can resolve the thermodynamic representabilityissue.213 In related fashion, Lebold andNoid developed a dual-potential approach that combines both structure-and energy-based variational principles, resulting in a more faithfulrecapitulation of FG energetics at CG resolution.196 We believe that this approach is generalizable and that findingappropriate expressions at CG resolution within the parametrized statepoint forms the basis of CG representability. Moreover, these approachesare expected to further reveal the fluctuations underlying fundamentalthermodynamic quantities, e.g., heat capacity and isothermal compressibility,although, similar to CG potentials, they must be able to extrapolateto the system of primary interest.Pressure is an example of a thermodynamic observable thatdoesnot meet the criterion in the equation above. All FG configurationalvariables contribute to pressure according to the virial theorem,2-4.DNVT ensembleis generally chosen in practice. However, this fact does not necessarilymean that bottom-up CG models will have zero transferability. In thisregard, imbuing transferability onto bottom-up CG models needs tobe carried out in a manner consistent with the underlying thermodynamics.For example, based on the entropy-enthalpy decomposition approachintroduced earlier,SCG(RN) at different densities or temperaturesshould be correctly reflected while designing CG models. Most of theseconditions are natural variables of free energy  and system composition (bulk to mixtures with different ratios).Yet, these conditions are inextricably linked. For example, in mixtureconditions, each molecular entity will experience differences in pressureand volume due to the presence of other molecules, resulting in differentinteraction profiles than that of bulk conditions. We will discussrecent advances in dealing with the transferability issue in In CG modeling,the transferability issue naturally emerges from the differences betweenthe FG and CG Hamiltonians and can be defined as a measure of howpredictive or extrapolatory the CG models are to the statistics beyondthe parametrized conditions. Since effective bottom-up CG interactionsare free energy functions, unlike FG Hamiltonians, the CG interactionswill vary at different state points . They are clearly defined as a function of the thermodynamicstate point. For example, the constant 2-5Structural Representability: How canone design CG models to capture higher-order structural correlationscorrectly? Most CG models suffer from this issue due to the use ofrelatively simple pairwise interactions.Thermodynamic Representability: Whilestructural representability can be directly computed from CG simulations,correct representation of CG thermodynamic properties requires a systematictreatment. For example, how can one obtain comparable pressures, internalenergies, or entropies with respect to the FG reference?Dynamical Representability: Since mostbottom-up CG methodologies focus on configurational variables, theresultant CG dynamics is not guaranteed (or even likely) to be consistentwith the FG reference. How can one overcome this inconsistency?Transferability: How canone designbottom-up CG models that can be applied to nonparameterized conditions?This issue is directly related to the applicability of CG models.Machine Learning: How canone benefitfrom emerging ML techniques to reduce the complexity underlying CGmodeling?Currently, major challenges in CG modeling originatefrom the approximate nature of CG models that aim to faithfully describethe complex many-body correlations and properties of atomistic systems.Here, we present some important challenges faced in these areas thathave been and are being actively pursued by a number of researchersin the field.With this in mind, we aim to address each of these issuesin the remainder of this Review.33-1UCG(RN), be exactly equivalentto the many-body PMF in terms of the CG variables.33 However, itis computationally difficult to derive a many-body expression for UCG(RN), and many-body potentials are computationally expensive to use.Instead, low-dimensional basis sets are commonly used by adoptingcommonly used molecular mechanics functional forms, e.g., pairwisenonbonded interactions and bonded interactions .219 In summary, UCG(RN) is often approximated asUnb(2)(RIJ) is the two-body nonbonded potentialthat depends upon the distance RIJ between CG sites I and J, Ub(dIJ) is the two-body bonded potential, Uθ(θIJK) is the three-body angle potential,and Uψ(ψIJKL) is the four-bodydihedral or improper potential.A bottom-up CG model that follows thermodynamic consistency requiresthat the effective CG Hamiltonian, 3-2UCG(RN) ignores explicit contributionsfrom other thermodynamic variables  asintroduced in By approximating the CG force field as 3-2.A89 Thesechoices are often more favorable than introducing nonisotropic descriptionsfor CG particles, e.g., Gay−Berne interactions,220 due to the complexity and computational costof both the CG parametrization and CG simulation. Thus, despite somepreliminary efforts in nonisotropic CG particle representation227 and parametrization strategy,228 we willfocus on efforts to improve CG models using isotropic CG mapping representationsin this section.Dueto their pairwise approximate nature, CG models constructed from 229 an improvement can be achieved by introducinghigher-order interaction terms in the CG HamiltonianUnbn)((·) denotes the n-body nonbonded interactionwith configurational variables (·). For example, studies of bottom-up1-site CG water have shown that pairwise interactions are incapableof recapitulating the local structure due to hydrogen bonds but canbe recovered with the explicit addition of three-body interactions Unb(3).147 The importance of three-body interactionsin molecular systems236 can be seen by applications of Stillinger−Weber interaction-basedmodels,237 including in the top-down CGwater model, e.g., mW.238Alternatively, based on the many-body expansion,150 or IMC239 for three-body Hamiltonians.240 However,introducing many-body interactions inevitably reduces the efficiencygains from CG modeling. In contrast, recent developments have proposedtwo different generalized approaches to include many-body interactionsat a reduced computational cost. The first approach194 implicitly projects the many-body interaction (up to N-body) onto lower-order basis sets based on the conditional probability p(OIn)( | RIJ) of higher-orderconfigurational variables OIn) into the CGHamiltonian. Explicitly evaluating arbitrary higher-order order parameterswill often reduce simulation efficiency, but several order parametersthat can be computed using pairwise statistics can account for many-bodycorrelations with reduced computational cost. Inspired by many-bodyDissipative Particle Dynamics (DPD),256 and material systems,260 several papers have reported that utilizingthe local density enhances the structural representability of CG modelsfor phase separating or interfacial systems.266 Taking a step further, one could incorporate variations in the localdensity in terms of a local density gradient 267Due to notable improvementsin modeling implicit solvents269 as well as order parameters related to liquid crystals270 and glass transitions272) to accurately describe the corresponding correlations. However,the choice of order parameters used is often determined phenomenologically,and thus, a systematic principle should instead be established. Wewill provide a systematic framework in N-body order parameters are generally difficult to regress;ML techniques can help solve such nonlinear problems . Since the effective CG interactions obeying thermodynamicconsistency are free energy variables,274 Recently, more thermodynamically consistent approaches have beenpursued using entropy-enthalpy decomposition (UCG(R) = ECG(R) – TSCG(R). By estimating the pairwise thermodynamic functionals(ECG(R), SCG(R)) and extrapolating to nonparameterizedtemperatures, temperature transferability can be achieved in bothconstant NVT (Helmholtz free energy) and NPT (Gibbs free energy) conditions.275 Even though ECG(R), SCG(R) remains relatively unclear at present.276 A recent report elucidated that these functionals may be deeplyconnected to the thermodynamic representability issue for entropy.276 Alternatively, a more direct approach based on statistical mechanicsthat does not suffer from the ambiguity of pairwise thermodynamicfunctionals was developed by numerically transferring the phase spaceexpectation value at different temperatures.145 It should be noted that reweighting approaches often suffer frominefficient sampling and non-negligible numerical noise.278 Notably, the dual approach by Lebold and Noid circumvented thislimitation in reweighting without sampling other temperatures by employingthe least-squares minimization to the energy quantity to obtain ECG(R), which is analogous toforce-matching.196 Recently, Pretti and Shell showedthat the CG models constructed from microcanonical basis sets in conjunctionwith REM for capturing entropy functions can naturally provide temperaturetransferable CG models by recapitulating atomistic energy fluctuations.41 Altogether, current findings and reports, regardlessof methodological details, emphasize the role of entropy in temperaturetransferable CG models.Early attempts to address temperature transferabilitywere based on heuristics, such as temperature rescaling position 15 under NPT ensemble,as both thermodynamic variables affect the system volume.279 Early improvements were based on rescaling CG interactions withrespect to pressure but lacked theoretical rigor. Notably, one canintroduce a volume-dependent conjugate force into the CG Hamiltonian,a strategy that dates back to the 1970s for liquid metals.280 Das and Andersen showed that introducing volume-dependentinteractions UV(V) to the CG Hamiltonian can correct the virial pressuresin CG models146Temperature transferability is inevitablycoupled with pressuretransferability in the case of the constant This interpretationhas been further explored in refs −283 to i284 Another free energy-based direction would be to introduce a chemicalpotential-like term Δμ as an analog to the chemical potential term in the Helmholtz freeenergy. Eventually, addressing the aforementioned transferabilityissues will elucidate how to achieve chemical transferability wherethe CG Hamiltonian can be determined a priori bygrouping atoms into molecular building blocks and sampling these groups at various configurations and state variables.287 For example, an extended ensemble approachby Mullinax and Noid suggested the use of topology in compositionand chemical transferability.285 Combinedwith ML techniques, such a systematic treatment to predict CG interactionsfor new molecules is deemed possible by sufficiently sampling numeroussmall molecules.287Transferability across compositionis a more complicated issue,where both temperature and pressure transferability come into play.The ultimate goal would be to correctly address the reduced pressurein mixture conditions, which is a nonlinear process due to nonidealinteractions, and to design the cross-interactions between differentmolecular moieties. While this direction has not been actively pursueddue to its complexity, several preliminary reports have paved theway for developing so-called mixing rules for cross-interactions,indicating that having a correct description of the CG thermodynamicquantities is essential for achieving such transferability.3-2.CAn accurate CG Hamiltonian should be able toaccount for both many-body correlations and transferability issues.On the basis of the efforts described above, we argue that the idealeffective CG Hamiltonian may be expressed asUnbn)((RN) accounts for the nonbondedinteractions up to n-body, which can be explicitlycast based on n-bodyorder parameter OI underthe pairwise approximation, this term would be 251 The last three termsin Unbn)((RN) is independent of state point, while the latterthree terms Utemp, Upress,and Uchem are examples of state-dependent potentials.Both classes are examples of efforts to improve the expressivity of UCG*.In ACG = ⟨UCG*⟩. This interpretation is inline with the state-dependent potential derived from the free energyperspective.289 Recently, some CG models havebeen developed based on the above principles and shown favorable results,such as the combination of density-dependent interactions with volume-dependentterms290 or the UCG models in the mean-fieldansatz.275Alternatively, 284 By deliberatelyextending the simulation ensemble, matching a response of the systemto perturbations has recently shown to sample a broad range of parameters.This wide parameter space with more abundant information allows fordetermining the most transferable CG interactions by maximizing aninformation metric, e.g., Fisher information.291 In the future, it would be informative to rigorously elucidatethe physical nature of these pairwise thermodynamic functionals andtheir relationship to CG systems at different thermodynamic statepoints and ensemble conditions.While 3-3UCG, it is important to remain mindful of thetrade-offs between model complexity and computational cost, the latterof which includes the cost of parametrization, implementation, andruntime. The development of effective CG interactions bears resemblanceto “Jacob’s Ladder” as seen in density functionalapproximations across quantum chemistry.292 Each degree of complexity can be thought of as a new “rung”on the “ladder” that represents the field of CG modeling.We note, however, that climbing successive rungs does not necessarilyguarantee improvement in the overall accuracy of a given CG model.For instance, it is possible that the increased complexity enhancesthe adherence to CG model consistency at the expense of CG thermodynamicrepresentability and transferability.156 To avoid these pitfalls, a standardized method to derive, explore,and validate the benefit of increasingly complex basis sets is stillneeded. Our focus in the next two sections will be to discuss frameworksthat can systematically increase the expressivity of CG force fieldsusing theoretical and algorithmic procedures.As researchers continueto explore different avenues to increase the expressivity of 44-1294 The idea underpinning the UCGapproaches is to introduce internal quantum-like “states”into the CG sites, and thus, the CG Hamiltonian can effectively accountfor the driving forces associated with these discrete changes thatare missing in conventional CG force field treatments by modulatingthe internal state interactions . The basic idea of a UCG model is to utilize a kindof isomorphism with quantum mechanics, i.e., in the latter case thesystem nuclei can evolve on multiple potential energy functions, whichin turn depend on the quantum mechanical state space of the systemand its underlying dynamics, and this in turn is tied back to theevolution of the dynamics of the nuclei. “Simple” nonadiabaticdynamics with surface hopping between electronic states296 is one example of this behavior, albeit there can be other examples.The basic notion of the UCG approach is to utilize this quantum isomorphismto increase the expressivity of the CG model so that the influencesof the processes that become implicit at the CG level are still includedto a certain degree.In 300The importance of the underlying molecularnature beneath the CG resolution is pronounced in many systems, especiallyfor highly coarsened representations. As indicated by refs  and 298 4-2293 Since the chemicalor physical changes in the molecular system have clearly separatedrelaxation times, a separation of time scales can be introduced. Forexample, physical changes such as bond breaking or formation happenmuch slower than the CG particle mass translations. On the other hand,chemical changes in electronic states or solvation will directly affectthe CG system without any relaxation in state dynamics. Thus, insteadof seeking a generalized Hamiltonian involving both CG configurationand state variable dynamics, two specific approximations at each timescale limit can be designed,294 as described below. A UCG idea can be mathematically formulated byintroducing internal state variables into CG interactions but in principlerequires rigorous formulation and parametrization of the correct equationof motion for the state variables, which can significantly increasecomputational cost.4-2.A294 can thus be utilized to approximate the instantaneous rateof switching from states α to β that are defined by theorder parameter m .Underthe slow state transition (SST) limit, the internal states rarelychange with the characteristic relaxation time due to conformationaltransitions that take place at time scales of nanoseconds or longer.A kinetic Metropolis–Hastings-like approachIn 4-2.BsN configurations, where s is thenumber of internal states per CG particle, and N representsthe number of CG particles. However, under the RLE limit, one canchoose a local order parameter to decouple state correlations betweendifferent particles, resulting in the expressionp(sI|RN) denotes the probability of CG site I being sI, and UsI, sJ(2)(RIJ) is thestate-wise interaction between sI and sJ.301 Unlike the SST limit, the RLE limit explicitly accounts for thedriving forces (the term −∇p(sI|RN) in 263 Notably, the UCG force fieldfrom 302 and conformational surface hopping methods.304Theopposite limit of the SST limit is when internal states undergo rapidstate transitions or the so-called rapid local equilibrium (RLE).In this case, in an analogy to Ehrenfest dynamics, the effective UCGHamiltonian can be expressed as a mixed interaction with respect toits state probabilities that follow quasi-equilibrium distributions.Ideally, one should consider the overall 305It must be noted that the UCG approach has already achievednotable success in treating realistic and highly complex biomolecularsystems, an example of which is the assembly of the HIV-1 virus capsidfrom the more than 1,000 copies of its capsid (CA) protein component.4-34-3.A294 and the characterization of ATP hydrolysisand phosphate-release reactions in actin filaments.300 In a coarser description (lower resolution), UCG models based largelyon the SST limit have been applied to capture the dynamic self-assemblybehavior originating from many-protein HIV viral capsids.305 As such, when conformationaltransitions in complex biomolecules rarely occur, the SST limit caneffectively embed the finely detailed chemical nature into a reducedlevel of representation.Even though mostcomputationally efficient CG models are not able to describe changesin the chemical nature, a key attribute of UCG models in the SST limit,by design, is the ability to explicitly model chemical transformationswhile avoiding the use of many-body force fields by introducing distinctstates that represent different chemical phenomena. This advance hasenabled UCG models to capture previously missing conformational transitionsbetween gauche- and anti- conformations in 1,2-dichloroethane4-3.B306Introducingflexibility into CG interaction forms enables the UCG models to improvethe description of the many-body correlations necessary for more accurateCG models. For the SST limit, chemical reactions due to complex many-bodycorrelations that involve many-particle nonbonded and electrostaticinteractions can be faithfully folded into the UCG models. Namely,the SST limit implicitly encodes the many-body correlations into distinctstate-dependent pairwise UCG potentials, which have a strong similarityto a “polarizable” CG model.31 but three-body correlationsand even higher-order correlations are often not well-captured incomplex CG systems, e.g., water.194 Even though the UCGmodels at the RLE limit are built upon the pairwise basis sets, thesestate-wise interactions are linked via local order parameters. Bychoosing these local order parameters properly, the resultant UCGmodel can faithfully capture complex many-body correlations. Basedon the observation that the local density is an N-body property, several studies have reported that the local density-basedUCG models can readily reproduce the many-body phenomena of interest.For example, the solvophobic association of hydrophobic solutes dueto many-body correlations between solute molecules was captured withthe UCG formalism as well as pair correlations and clustering behavior.263 The local density can also be an importantorder parameter for distinguishing different phases in heterogeneoussystems. This inhomogeneity is pronounced in interfacial systems,where the conventional CG models failed to properly describe two ormore phases at their interfaces. Notably, UCG interface models candifferentiate distinct characteristics emergent in the system, e.g.,liquid and vapor states in liquid/vapor interfaces with well-reproducedliquid “slab” density profiles and structural correlations.265 Extending beyond interfacial UCG models, itwas recently demonstrated that the single UCG model is able to encompassthe structural correlations emergent from distinct bulk phases, resultingin multiphase CG models.275Other types of many-body correlations in terms of nonbondedstructures can be practically impossible to address using only pairwisebasis sets. We note, as before, that pairwise MS-CG models attemptto capture two-body and three-body correlations concurrently by satisfyingthe YBG equation,307 with thehydrogen-bonding not being explicitly resolved at the CG level.Sincethe local density of CG moieties reflects the various chemicaland physical natures , it is conceivable that the local densitycan be utilized as a generalized order parameter to indirectly representthe nonuniform nature of a given system. For example, it has beenshown that effects of FG hydrogen-bonding interactions can be faithfullycaptured in a local density-based UCG model by differentiating thedonor and acceptor states,4-4308 or the empirical valence bond theory,309 where a systematic connection between these different theorieshas been recently demonstrated.310The main advantageof UCG modeling not only lies in being able to reproduce importantcorrelations faithfully described by the order parameters but alsoin its flexible interaction form. This is especially true in the RLElimit, where the UCG Hamiltonian inherits transferability by design;several distinct state-wise interactions are folded into a singleHamiltonian form, allowing for transferring to different state conditions.This flexibility is akin to Ehrenfest dynamics265First, the flexibility of 275 Sincenumber density directly responds to the system condition, local density-basedUCG models can be good candidates for properly describing the conjugateforces exerted by the thermodynamic variables of the system. For morecomplex systems exhibiting various conformations, determining thecorrect order parameter would be the most important step toward transferability;this will potentially benefit from advances in ML techniques thatwill be described in Alternatively, one can introduce the already determined MS-CGinteractionsinto the UCG framework and utilize adequate order parameters to distinguisheach interaction. For example, one can utilize the local density inorder to mix bulk interactions at high temperatures and low temperatures,resulting in temperature transferable UCG models. A recent study showedthat such an approach has a direct link to the energy-entropy decompositionunder the mean-field ansatz and further demonstrated that one canpossibly design phase transferable UCG models by combining liquidand gas phase MS-CG interactions.4-5Intermediate State Dynamics. Variouschemical and physical processes may still exhibit intermediate statedynamics that does not fall into either the SSL or RLE limit. Oneillustrative example would be proteins that slowly fold but have rapidlychanging behaviors due to their interactions with the environmentor solvents. Even though one direct direction for modeling intermediatestate dynamics would be to incorporate equations of motion for statevariables and to develop the corresponding UCG Hamiltonian, we proposea hybrid UCG model as a more efficient alternative by simultaneouslyaccounting for both the SST and RLE limits. For the hybrid UCG model,defining two distinct UCG types that undergo different dynamical limitscan benefit from existing dimensionality reduction methods. For example,time-lagged independent component analysis (TICA) can be utilizedto extract the most rapidly varying and the slowest varying orderparameters.311 With the recent successof such dimensionality reduction techniques in Markov State Models(MSM) for CG dynamics modeling,317 the combination of the UCG methodology with TICA is expected tofurther extend the range of dynamical transitions explicitly includedin CG modeling.Toward LargeBiomolecules. Despiterecent developments in UCG models having been mostly focused on relativelysimple liquids, the UCG methodology in principle can be extended tolarger spatial scales, and MS-CG-based UCG models can be also extendedto much larger systems by correctly differentiating the chemical environmentsusing local density parameters. For example, the numerous conformationsin protein folding result in complex energy landscapes, e.g., dodecaalanine,318 and UCG models that have internal states designedbased on the local Cα density could be expected to accuratelycapture the folded and unfolded states. Similarly, the effect of solventsin a solvated system can be faithfully modulated by solvent density-basedUCG models, e.g., lipid bilayers.In contrast to conventionalCG methodologies, UCG theory provides a generalized framework to greatlyenhance the fidelity of CG models, and the practical realization ofUCG state dynamics imparts a physical presence of the missing degreesof freedom in terms of internal states at the reduced CG resolution.The UCG method has recently been shown to be successful for describingvarious systems ranging from liquids to biomolecules, demonstratingits applicability and versatility. We conclude this section by providingopen problems for future developments.319 Relatedly, a network-based UCG model was reportedto effectively assess the mechanical properties of microtubules.320On the other hand, for more complex biomolecules, effectivelarge-scale UCG models can be built upon coarser descriptions of state-wiseCG interactions. Since the UCG methodology does not specify any ofthe interaction forms but rather provides a systematic framework forembedding internal state information into CG models, combining severalCG interactions from different methodologies with the UCG theory cancorrectly account for the state dynamics. Recently, the UCG modelsbased on the fluctuation maximization with harmonic interactions weredeveloped for the glutamine-binding protein and lactoferrin and wereable to correctly describe the protein conformational transitions.55-154 In this section,we elaborate on such difficulties arising from the dynamics of CGmodels and showcase recent advances in this area.Whilethe thermodynamic properties of CG models differ from their FG counterparts,thermodynamic representability can be systematically improved basedon the FG and CG observable expressions, e.g., pressure and energy.However, dynamical properties pose completely different problems incomparison to those from thermodynamic properties.324 The idea behind the Mori−Zwanzig formalism is to projectthe relevant dynamic variables that are left in the CG systems, resultingin a generalized Langevin equation (GLE) form of the equation of motion.325 The following integro-differential equationthus faithfully describes the dynamics of the CG model329w(RN) is a normalized partition function of the microscopic configurations at RN. Seeref 349 and the smoothed DPD351 based on the discretized Navier–Stokes equation can resolvethe momentum conservation issue by introducing pairwise frictionalkernels to A more accuratedescription of frictional and stochastic forces present in the CGequation of motion can be established in two steps. First, one needsto choose an appropriate stochastic differential equation as the equationof motion. Then, based on the chosen equation of motion, one needsto parametrize the friction kernels and associated stochastic forcesin a “bottom-up” manner. Typically, the Mori−Zwanzigequation of motion is approximated using a single-particle GLE orDPD-like equation of motion. A single-particle GLE assumes that thereare no spatial correlations for the random forces acting on differentCG particles. Such an approximation reduces 92 This was further extended to the non-MarkovianDPD regime by Yoshimoto,338 and in recentyears, Karniadakis and co-workers have established a systematic parametrizationof GLE and DPD equations of motion for both Markovian and non-Markovianlimits.354 In this case, the friction kernels are readily obtained from thestochastic forces, allowing direct utilization of the Mori−Zwanzigformalism. However, pairwise decomposition of instantaneous forcesis only feasible for bonded systems, and thus, this approach cannotbe applied to unbonded systems.Oncethe form of the CG equation of motion is chosen, the remainingstep is to determine the friction kernels to be consistent with theFG reference. Here, we provide a brief discussion on bottom-up approachesfor parametrizing these nonconservative interactions. For bonded systems,e.g., star polymers, a pairwise decomposition of instantaneous forcesinto parallel and perpendicular directions at the FG level is possibleat the CG resolution. Practically, Hijon and Español introducedthe so-called “constraint dynamics” technique to extractthe pair decomposed forces under the Markovian DPD equation of motion.353 In practice, this inverse approach matches the pairwise velocityautocorrelation function and force–velocity cross-correlationfunction, resulting in the Volterra integral equation.344 Still, most of these approaches are limitedto bonded systems. Notably, a recent breakthrough for constructingfriction kernels of unbonded fluids was reported using the dynamicmapping approach.107 By estimating theinstantaneous forces on dynamic blobs based on the velocity Verletalgorithm,355 conservative forces in theform of the many-body DPD interactions251 were determinedusing the MS-CG principle. The deconvolution of correlation functionswas then applied to derive a DPD-like equation of motion in Markovianand non-Markovian limits. This approach further establishes the bottom-uplink between the microscopic origins of fluids and macroscopic physics.Such a bottom-up inference of frictional and stochastic interactionshas not been widely investigated due to its complexity but remainsa promising direction for future research. For example, using theREM framework, Español and Zuñiga designed a variationalapproach to infer drift and diffusion terms in the Fokker–Planckequation.356 Another interesting extensionof the MS-CG method was developed by Davtyan, Andersen, and Voth,where they introduced fictitious particles and coupled them with CGsites to effectively introduce a memory kernel under the GLE,340 which shares a similar physical idea with the auxiliary model laterdeveloped by Karniadakis and co-workers.353Alternatively, the frictionkernel can be constructed indirectlyby inverting the time correlation functions from the FG reference.5-3Alternatively, a computationally less expensive yet challengingdirection would be to perform CG simulations under Hamiltonian mechanicsand then elucidate how the accelerated CG diffusion is related tothe FG (reference) dynamics through various rescaling approaches.5-3.A327 Thisapproach has been reported for polymer systems at different resolutionsand chain lengths but with limited applicability due to the strongassumptions made.364 We note that even though one could assume a uniform scaling ratioand estimate such value naïvely based on the FG and CGdiffusion, there is no theoretical guarantee that such a factor exists365 and may necessitate an explicit considerationof the Mori−Zwanzig formalism.369 Also, the uniform scalar frictionterm obtained from the scaling factor itself is a many-body quantity and differs by system conditions, e.g.,thermodynamic state point, and hinders its applicability to otherchemical systems.A naïveyet straightforward approach is to think of the CG time scales asuniformly accelerated time with respect to the physical time of theFG reference. This uniform time rescaling approach assumes that thefrictional forces are not dependent on time, and thus, the Mori−Zwanzigprojection operator can remove the configuration, momentum, and timedependence in the friction kernel.5-3.B371 e.g., proteinfolding,372 the free energy landscape approachaims to address the dynamical properties underlying barrier-crossingdynamics by correctly representing the CG energy landscape. Whilethe barrier-crossing dynamics is quite different from the microscopicdynamics of the system, recent advances have elucidated the structural-kinetic-thermodynamicrelationships for helix–coil transitions of helix-forming peptides.373 More importantly, such approaches have beenshown to have a direct link to the MSMs, where the system dynamicsis represented by transitions between microstates.377 Notably, based on variational approaches to understanding the conformationaldynamics and then follow-up work,379 Nüskeet al. have developed a spectral matching method that targets thedynamical propagator of CG systems, resulting in correct long-timedynamics.380 Outside of the MSM-based framework,Rudzsinki and Bereau have developed the Bayesian dynamical reweightingscheme381 to correctly recapitulate thekinetics of CG peptides.382 These pioneeringefforts in barrier-cross dynamics have highlighted how existing observeddeficiencies in CG dynamics and kinetics may be due to not only incorrectCG equations of motion but also inaccuracies in the approximationof the conservative forces .Inspiredby the energy-landscape theory,5-3.C385 the excess entropy scaling relationshipis an empirical, semiquantitative relationship that links the dynamicproperty of the system D* to its molar excess entropy sex, which is the entropy differencebetween the system and ideal gas, such thatAnotherimportant recent advance in CG dynamics has been an attempt to understandaccelerated CG dynamics using the excess entropy scaling relationship.First proposed by Rosenfeld,39 and until recently, employing 391 The empirical nature of this scaling relationship further exacerbatesthis limitation when applied to CG dynamics. First, it is not guaranteedthat the FG and CG systems will obey the same scaling relationship,i.e., αFGCG. Also, the correspondencebetween DFG* and DCG* is still unclear because thereis no physical explanation for the D0 term,which is the “entropy-free” coefficient from To date, only a handfulof studies paid attention to the potential usefulness of 394 starting from the excess entropydifference between the FG and CG systems, known as the mapping entropy.51 By computing the excess entropy based on earlierarguments from Karplus, Lazaridis, and Zielkiewicz,396 ref  or long-wavelength fluctuations . While thehard sphere treatment of CG systems determines the D0CG, the entropy-freediffusion coefficient for FG systems is usually larger than D0CG. This discrepancy can be understood from the degrees of freedomthat are missing at the CG resolution.394 By incorporating the missing rotations and vibrations back intothe single CG-site resolution, the complete dynamic correspondencebetween the FG and CG systems can be recapitulated for liquids. Inpractice, then, one may (1) predict the accelerated CG diffusion underthe Hamiltonian mechanics by estimating the D0CG or (2) recoverthe reference FG diffusion from the CG level by incorporating themissing diffusion into the translational CG diffusion. However, theserecent findings have primarily focused on relatively simple CG systemsof liquids, so considerable effort to extend such dynamic correspondenceto nontrivial, complex CG systems, e.g., biomolecules, will be a challengeand should be pursued. A detailed description of the dynamical correspondencebetween the FG and CG systems using the excess entropy scaling isgiven in Recent progresson the excess entropy scaling has addressed theseaforementioned problems,,396 ref  has conf5-4367 energy renormalization,406 and transfer learning using ML techniques.407In order to correctly address dynamical representability in bottom-upCG models, recent studies have examined the dynamical properties ofCG models using both GLE-like equations of motion and rescaled Hamiltonianmechanics. While the GLE description provides a rigorous statisticalmechanical description of CG dynamics, complexity and numerical stabilityare currently a bottleneck for application to complex molecular CGsystems. In addition, similar to the conservative interactions, dissipativeand stochastic forces present problems with transferability. The majorityof attempts has only been tested on uniform single-component fluids,as multicomponent systems tend to face transferability issues. Also,in the same context as the transferability of the conservative forces, thermodynamic transferability of dynamicproperties should be addressed to impart a high fidelity CG model.This particular area has not been extensively explored at the currentstage, yet several preliminary directions have provided potentialdirections: dynamical rescaling,ad hoc physical scalingprinciples, e.g., excess entropy scaling. In this regard, rigorousphysical scaling principles beyond the hard sphere description forexplaining the excess entropy scaling are a promising area for futureresearch. A few possible directions are based on the mode couplingtheory409 and transition state theory.410 A grand goal in CG dynamics would be to correctlyunderstand and faithfully reproduce transport phenomena under nonequilibriumconditions. Continuous development of a rigorous and practical bottom-upCG theory based on nonequilibrium statistical mechanics would thereforepave the way to a new era of CG modeling.On the other hand, understanding accelerated CG dynamics producedusing Hamiltonian mechanics can lessen complications from frictionalinteractions by introducing 66-1411 For example,in order to design an automated method that can determine the contentof a picture by inspecting its pixels, e.g., whether the image containsa cat or a dog, two strategies could be considered. First, an algorithmcould be explicitly programmed to analyze each pixel in the imageto determine whether the picture is more similar to a cat or a dog—anadmittedly difficult task. On the other hand, one could use ML proceduresto extract patterns from a large set of images that are already labeledas containing a cat or dog. This ML-based approach would then usethis labeled data to learn the connection between individual pixelsand the overall content of the image, resulting in a new algorithmthat is able to discern overall picture content. In other words, MLapproaches can leverage high volumes of data to perform tasks thatseem intractable when using other strategies. These techniques, however,naturally require a large amount of data to succeed and produce solutionswhose quality and accuracy fundamentally depend on the quality ofthe data used for parametrization. Furthermore, the produced solutionsmay be opaque and extrapolate poorly to cases outside the data usedfor parametrization, creating a natural barrier to transferability.Nevertheless, the ability to create complex algorithms from data hasrevolutionized a number of areas such as computer vision and advertisementtargeting.412ML is a subfield of artificial intelligence that uses algorithmsto study and analyze data.418Since bottom-up CGapproaches focus on learning patterns from data sets generated fromFG simulations, it should come as no surprise that algorithms developedin ML have found use in molecular CG modeling. Given the similaritiesbetween CG and FG models, a natural extension is to employ atomisticML techniques for CG systems. Our particular interest in this subsection,however, is to describe applications of ML to CG modeling with a focuson how these methods differ from ML used in the atomistic setting.For a thorough survey of atomistic (as well as some CG) ML methods,we refer readers to recent reviews.6-26-2.A420 These EBMs are algorithmsthat describe a distribution by specifying its probability up to anunknown normalizing constant . However, thedata available for training and the need for scientific extrapolationcreate a specialized domain with its own approaches and difficulties:similar to the atomistic setting, the availability of a noisy estimateof the forces of the many-body PMFs provides an important avenue forparametrization not available in most EBM applications, and the needfor physical transferability impedes the use of many EBM architectures.These two key details underpin the training and design of force fields.Despite these differences, various function representation approacheshave been successfully adapted from ML to increase the flexibilityand accuracy of atomistic force fields for over two decades,411 and many of these same methods have now beentransferred to the CG resolution. Recent reviews have detailed severalML-based force field approaches, including several kernel-based methodssuch as the Gaussian Approximation Potential by John and Csányi,421 the method introduced by Scherer et al. thatuses Gaussian process regression projected onto tabulated potentials,422 and Gradient Domain ML-based methods as describedby Chmiela et al.423 and Wang et al.424In the language typical to ML, the MD-based CG and atomistic modelsdiscussed in this article fall under a class of methods referred toas energy-based models (EBMs).426 We also note that publications such as Schneideret al.427 similarly proposed using neuralnetworks to capture free energy surfaces but did not do so for a high-dimensionalparticle representation. The Deep CG Potential (DeePCG), introducedby Zhang et al.,428 was the first to adaptmore traditional ideas from the atomistic force field community,429 followed by CGnet.59 We note that while typically used with highly general feature sets,these same approaches could also be applied to custom high-dimensionalorder parameters as discussed in 424 Traditionalautoencoders are commonly used in data compression as they temporarilyreduce the feature space within the network.432 This bottleneck feature allows the neural network to find a CG mappingoperator before force-matching.433 In addition,CGSchNet developed by Husic et al.434 andthe recent work by Ruza et al.435 haveintroduced graph neural network-based methods to CG force field development.These approaches can be viewed as combining multilayer perceptron-basedapproaches , which produce a CG force field using user-suppliedfeaturization, with SchNet,436 a graphneural network architecture originally designed to reproduce ab initio forces and energies, and a compatible simulationengine, TorchMD.437 This use of graph neuralnetworks requires no molecular featurization to be supplied by theuser, as the network learns its own features via its graph subnetwork.This also has the added benefit of ideally making CGSchNet transferableacross system composition and accurately modeling the solvation environmentfor biomolecules in a novel manner with ISSNet.438 We note that graph neural network architectures439 and graph-based approaches440 have additionally found widespread use in other moleculartasks such as the selection of CG mapping operators and automaticsampling of atomistic configurations.442 The CGnetarchitecture has also been adapted to only consider many-body interactionsup to a specified order. For example, it is shown that even 5-bodyinteractions notably improve the quality of the resulting CG modelwhen studying a small protein.443A growing number of CG neural networkapproaches have also been introduced for estimating the many-bodyCG variable PMFs, likely beginning with Lemke and Peter who developeda convolutional neural network-based approach to learn correctionsto an existing CG force field through ideas connected with noise contrastiveestimation and adversarial learning.6-2.Bab initio approaches,the force field is often trained via regression to reproduce the connectionbetween molecular configurations and the energies and forces presentin the reference data set. While the initial data set may often begenerated via MD using a reference energy function, this is not required;in fact, configurations outside the stable basins of the referencesystem are often critical for reproducing the barriers fundamentalto chemical behavior.448 These additional structures are either included through human interventionor actively added to the reference set through a variety of strategies,one of which is of particular importance to the current discussionwhen considering neural networks: query by committee (QBC).417 In QBC, multiple neural networks are trainedwith varying initial optimization conditions on a given referencedata set.449 While each of these neuralnetworks can typically reproduce energies on the reference data set,their predictions outside this domain may differ. This disagreementis then used as an engine for adding new structures to the referencedata set: If a candidate structure results in disagreement among thevarious neural networks in the committee, a reference ab initio calculation is performed, and the configuration is used for parametrization.While QBC is an important technique, its construction emphasizes howneural networks trained on finite samples  often exhibit inaccuracyoutside the stable basins of the reference system—thereby highlightingthe importance of complex sampling for high-dimensional force fieldparametrization.When parametrizingan atomistic model to match 450 Doing so creates an avenue to use active sampling strategies suchas those described in the previous paragraph.While this ability to add particular configurations to a data set has proven to be criticalto the creation of high-dimensional atomistic force fields, this routehas not typically been pursued in the creation of CG force fields.In order to understand the barriers to doing so, it is helpful tocompare the settings in which CG force fields are parametrized incomparison to their atomistic counterparts. The atomistic configurationalenergy corresponds to the many-body PMF in terms of the CG variables,which is extremely difficult to evaluate in practice. However, forthe force-based parametrization strategy given in 452 but does not seem to be commonplace). This lack of non-Boltzmannsampling, when combined with the similarity between the model architecturesused for atomistic and CG force fields and the nature of active samplingstrategies, seems to imply that current highly flexible CG potentialstrained using forces may persistently face difficulties outside thestable basins of the system under study. The situation for CG modelsmay indeed be sometimes worse than the atomistic case as implied bythe previously mentioned results of Wang et al.,443 where high order CG potential terms were required to reproduceatomistic results, implying that low capacity model representationsmay often be insufficient. Despite this, however, the preliminarysuccess of the results discussed in earlier subsections provides hopefor quantitatively accurate CG force fields using ML-based algorithms.However, oftendue to computational reasons, most studies insteadresort to canonically distributed nonconstrained atomistic referencesamples for parametrization .426 Despite these initialconnections, however, explicit cross-pollination between these fieldsremains sparse and is an area for future development.Additional strategies beyond force-matching have been developedsee 2 topara6-36-3.A417 We refer interested readers to the aforementioned reviews. In thissection, we instead focus on two novel applications of ML that focuson problems common to CG simulation: the lack of atomistic detailand the inability to identify high-dimensional structural representabilityissues.CG simulations, similar to their atomistic counterparts, create largeamounts of data, and transforming this data into knowledge and understandingis of utmost importance and yet a difficult task. As mentioned in 6-3.B456 or GANs in orderto create samples at the atomistic resolution that are consistentwith a given CG sample. GANs work by simultaneously training two neuralnetworks which “compete” against each other.426 One network (the “generator”)is trained in the task of creating the backmapped configurations,while the other (the “adversary”) is trained to classifystructures as being generated either via atomistic molecular dynamicsor the generator network. Over the course of training, the generatorideally improves to the point that a fully trained adversary can nolonger distinguish between the two sources of samples. In this idealcase, the generator produces atomistic samples that are Boltzmanndistributed conditioned on a given CG sample. In application, similarto ML-based CG force fields, the network could first be trained ona smaller system and then applied in a larger context, providing away to atomistically interpret emergent behavior discovered in CGsimulations.CG models are more efficient than atomistic simulations due to theirreduced resolution, yet as posed earlier, this simplification inhibitsunderstanding the molecular driving forces underpinning emergent behavior.Approaches that reintroduce atomistic details into CG configurations(aptly referred to as “backmapping” methods) providestrategies to take advantage of the efficiency of CG models whilemaintaining a clear atomistic picture. Several ML-based techniqueshave recently been developed to backmap CG data. Most of these approachesuse generative models457 By converting CGpositions into a two-dimensional image, the method is able to performbackmapping with minimal additions to an already existing framework.458 Another method by Stieffenhoffer, Wand, andBereau utilizes a new network architecture designed specifically forthe task of backmapping.459 In this so-calleddeepBackmap approach, the network not only sees CG configurationsbut also has access to force field information, allowing higher energystructures to be directly penalized. The network also distinguishesbetween different chemical groups such as aromatic rings and backboneatoms and additionally builds the high-resolution structure one particleat a time, which allows information from previous atoms to informthe placement of later atoms. Recently, this method showed promisingresults for complicated systems such as polymer melts and can translatewell to crystalline structures.286 We notethat such stochastic backmapping methods would also allow the force-basedactive learning strategies previously described to be performed ata minimal cost.There are currently multiple examples of GAN-basedbackmapping methods. One method introduced by Li et al. uses the Pix2Pixnetwork architecture which was originally designed as a style transfernetwork for images.460 or advanced ML methodologies, including Bayesian inference,461 Gaussian process regression,462 and autoencoders.433 In particular,the approach of directly generating molecular configurations withoutextensive MD has similarly been exploited by Boltzmann Generators,which are trained on small amounts of molecular dynamics data andthe atomistic force field in an unconditional manner.463 This technique, while not currently implementedfor CG data, could similarly be adapted to the backmapping domainand would have the advantage of taking into account atomistic forcefield information in a rigorous manner.We also note that a number of ML methods otherthan GANs can bealso employed to perform backmapping. To note, various backmappingmethods have been developed by utilizing traditional supervised/unsupervisedmethods6-3.CEven thoughbottom-up CG models are rigorously built upon FG statistics, manyCG models have structural representability issues. In addition, thehigh-dimensional parametrizations typical to bottom-up models canmake it difficult to intuit how two approximate CG models differ beyondtheir qualitative or projected behavior. When parametrizing atomisticforce fields, the per-configuration error in the energy or forcesis known for each entry in the reference data set, and these knowndiscrepancies can provide intuition on the regions of phase spacewell-described by the model at hand. Unfortunately, this type of analysisis not possible for bottom-up CG models, as neither the true many-bodyCG variable PMF nor its gradients are typically known . Instead, the validation of CG models is generally limitedto comparing their performance using low-dimensional free energy surfaces, leaving the differences in their fullhigh-dimensional behavior relatively undescribed.464 has proposed a differentstrategy for capturing and describing the errors present in parametrizedCG models. The output of a calibrated classifier trained to differentiatebetween configurations in the reference atomistic data set and thoseproduced by a candidate CG model can be transformed into an estimateof the difference in the CG force field and the true many-body PMFat each configuration , variationally recapturingthe information present when dealing with atomistic force fields.Furthermore, the applicability of classifiers in this domain opensthe door to exploiting techniques from explainable ML in order tointerpret the inferred configurational errors. These same ideas arealso applicable to resolutions coarser than that of the CG force fieldand provide an avenue for understanding classification-based trainingtechniques, such as those in Lemke and Peter’s work as wellas in adversarial-residual-CG (ARCG).58In this regard,recent workWe summarizethe current advances in ML-based CG approaches in 6-4466 In situations where extrapolation and physical intuition are critical,this trade-off can create problems. For example, while bottom-up MLCG models may recapitulate the ensemble it is parametrized to match,are the derived force field parameters transferable to the primarysystem of interest where emergent behavior is expected? Do backmappingapproaches parametrized in the stable basins of a system extrapolatecorrectly to the transition regions? If a model is inaccurate in portionsof phase space, can this be predicted and scientifically understood?As the parametrization of bottom-up methods is often based on referencedata sets that inherently do not guarantee the performance of CG simulations,these questions are critical to the many ML-based approaches currentlyin development. These issues, of course, are not new to the bottom-upCG community , further work can focuson computational efficiency and ease of use, ideally leading to aclass of CG models that underpin a new wave of research devoted topreviously insurmountable scientific problems. In general, one mayalso expect an explosion of new literature in the future in termsof ML methods applied to coarse-graining.Both bottom-up CG andML use algorithms to discern patterns in data. This similarity hasmotivated the application of a variety of ML algorithms in CG frameworks,ranging from novel force field approaches to systematic methods forreintroducing atomistic detail. Collectively, this increased expressivitycan produce atomistic explanations for previously uninvestigated phenomena.However, this increase in accuracy has come with a loss in model transparency(whether this trade-off is inevitable is a topic of debate).nity see 2 or the 7ad hoc design principles. To move toward true“multiscale” models, such bottom-up principles shouldmaintain accuracy across different physical scales. Notably, somerecent advances are pushing this limit forward to much smaller (quantumregime) and larger (meso- to macro-scopic) regimes. An extension ofthe MS-CG framework to the quantum regime,474 described by quantum Boltzmann statistics, corroborates that sucha multiscale treatment is possible in one extreme, while the otherextreme toward macroscale CG modeling has been actively pursued viamesoscopic fluids.107 Similarly, the Electronic Coarse-Graining(ECG) method476 has been developed to targetconfigurationally dependent electronic structure and applied to semiconductors477 and optoelectronic materials.478 These advances promise to significantly extend the spatiotemporalscale of systematically parametrized simulations, which currentlyencompass biological entities comprising millions to billions of atoms,e.g., the SARS-CoV-2 virion.479 When bridgingacross distinctly different scales, transferability between thesescales is critical to reconcile different emergent physics withina single unified model. While at an early stage, recent success inquantum mechanics/CG molecular mechanics (QM/CG-MM),481 which encompasses both quantum and molecular regimes, implies thatit is possible to design CG mappings, equations of motion, and energeticsbased on the scale of interest. In the other limit, a promising directionwould be to incorporate classical field theory into the bottom-upCG framework.482For the past two decades, bottom-up CG models have been developedby studying the microscopic origins underlying many macroscopic processesand have emerged as efficient, powerful, and multiscale computationaltools in several fields of science. However, due to the enormous complexityof atomistic systems, statistical mechanics-driven CG modeling hasprimarily only been possible for relatively simple systems, e.g.,liquids and small peptides, using various practical approximationsfrom While future effortsshould focus on extending the multiscale regime,an equal amount of attention might be spent developing new CG theoriesat the molecular level. Possible directions for new CG methodologiesinclude the following: (1) stable and extensible nonlinear CG mappingsfor complex mesoscale systems, (2) fully expressive CG energetics,(3) a more complete description of UCG state dynamics, (4) a morecomplete dynamic representability between the FG and CG systems, and(5) advanced ML-based CG methodologies.483 computer software hasbeen inseparable from molecular models. Given the ad hoc nature of many existing CG models, CG modeling software should beamenable to new feature implementation and easy dissemination to facilitateboth access and utility to the general scientific community. Currently,there are several options that satisfy these criteria with differentfeatures and capabilities according to their objectives: VOTCA185 , MAGIC485 , BOCS486 , openMSCG487 , and CGNet59 . Continued development in both CG methodology and software necessitatesstandardized data sets that are used to validate existing or new CGmethods. For example, in the free energy sampling community, alaninedipeptide serves as a standard example to validate new methodologies,488 and ultralong atomistic MD trajectories forfast-folding proteins491 using the ANTON supercomputer by the D.E. Shaw lab492 have been extensively employedin kinetics studies such as MSMs. Even though CG modeling tacklesa variety of chemical and biological systems across many differentscales, there are relatively few data sets that are publicly availablein the CG community. Therefore, by taking inspiration from the MLcommunity, future efforts should also aim to achieve community agreementon which data sets are reliable to be shared and standardized in orderto benchmark various CG methodologies. Altogether, we expect thata continuous exploration along these directions will push the frontiersof bottom-up CG modeling into the exploration and characterizationof increasingly complex molecular systems.Lastly, we concludethis Review by emphasizing the necessity andimportance of infrastructure for computer software and data set handlingfor the next generation of CG modeling. As models and derivation strategiesincrease in complexity, infrastructure for software and data becomesmuch more crucial. Since the naissance of modern computer simulationsfor molecules,