The multi-dimensional generalized Langevin equation for conformational motion of proteins

Coarse-Graining Conformational Dynamics with Multidimensional Generalized Langevin Equation: How, When, and Why

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained (CG) conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build CG models in dynamical consistency (DC) with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term DC. Case studies of a toy polymer, with 20 CG sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling CG conformational dynamics in practice.

Data-Driven Learning of the Generalized Langevin Equation with State-Dependent Memory

We present a data-driven method to learn stochastic reduced models of complex systems that retain a state-dependent memory beyond the standard generalized Langevin equation with a homogeneous kernel. The constructed model naturally encodes the heterogeneous energy dissipation by jointly learning a set of state features and the non-Markovian coupling among the features. Numerical results demonstrate the limitation of the standard generalized Langevin equation and the essential role of the broadly overlooked state-dependency nature in predicting molecule kinetics related to conformation relaxation and transition.

Derivation of the nonequilibrium generalized Langevin equation from a time-dependent many-body Hamiltonian

It has become standard practice to describe systems that remain far from equilibrium even in their steady state by Langevin equations with colored noise which is chosen independently from the friction contribution. Since these Langevin equations are typically not derived from first-principle Hamiltonian dynamics, it is not clear whether they correspond to physically realizable scenarios. By exact Mori projection in phase space we derive the nonequilibrium generalized Langevin equation (GLE) for an arbitrary phase-space dependent observable A from a generic many-body Hamiltonian with a time-dependent external force h(t) acting on the same observable A. This is the same Hamiltonian from which the standard fluctuation-dissipation theorem is derived, which reflects the generality of our approach. The observable A could, for example, be the position of an atom, of a molecule or of a macroscopic object, the distance between two such entities or a more complex phase-space function such as the reaction coordinate of a chemical reaction or of the folding of a protein. The Hamiltonian could, for example, describe a fluid, a solid, a viscoelastic medium, or even a turbulent inhomogeneous environment. The GLE, which is a closed-form equation of motion for the observable A, is obtained in explicit form to all orders in h(t) and without restrictions on the type of many-body Hamiltonian or the observable A. If the dynamics of the observable A corresponds to a Gaussian process, the resultant GLE has a similar form as the equilibrium Mori GLE, and in particular the friction memory kernel is given by the two-point autocorrelation function of the sum of the complementary and the external force h(t). This is a nontrivial and useful result, as many observables that characterize nonequilibrium systems display Gaussian statistics. For non-Gaussian nonequilibrium observables correction terms appear in the GLE and in the relation between the force autocorrelation and the friction memory kernel, which are explicitly given in terms of cubic correlation functions of A. Interpreting the external force h(t) as a stochastic process, we derive nonequilibrium corrections to the fluctuation-dissipation theorem and present methods to extract all GLE parameters from experimental or simulation time-series data, thus making our nonequilibrium GLE a practical tool to study and model general nonequilibrium systems.

Ab initio generalized Langevin equation

We introduce a machine learning-based approach called ab initio generalized Langevin equation (AIGLE) to model the dynamics of slow collective variables (CVs) in materials and molecules. In this scheme, the parameters are learned from atomistic simulations based on ab initio quantum mechanical models. Force field, memory kernel, and noise generator are constructed in the context of the Mori-Zwanzig formalism, under the constraint of the fluctuation-dissipation theorem. Combined with deep potential molecular dynamics and electronic density functional theory, this approach opens the way to multiscale modeling in a variety of situations. Here, we demonstrate this capability with a study of two mesoscale processes in crystalline lead titanate, namely the field-driven dynamics of a planar ferroelectric domain wall, and the dynamics of an extensive lattice of coarse-grained electric dipoles. In the first case, AIGLE extends the reach of ab initio simulations to a regime of noise-driven motions not accessible to molecular dynamics. In the second case, AIGLE deals with an extensive set of CVs by adopting a local approximation for the memory kernel and retaining only short-range noise correlations. The scheme is computationally more efficient than molecular dynamics by several orders of magnitude and mimics the microscopic dynamics at low frequencies where it reproduces accurately the dominant far-infrared absorption frequency.

Accurate estimates of dynamical statistics using memory

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

Investigation of Rare Protein Conformational Transitions via Dissipation-Corrected Targeted Molecular Dynamics

To sample rare events, dissipation-corrected targeted molecular dynamics (dcTMD) applies a constant velocity constraint along a one-dimensional reaction coordinate s, which drives an atomistic system from an initial state into a target state. Employing a cumulant approximation of Jarzynski's identity, the free energy ΔG(s) is calculated from the mean external work and dissipated work of the process. By calculating the friction coefficient Γ(s) from the dissipated work, in a second step, the equilibrium dynamics of the process can be studied by propagating a Langevin equation. While so far dcTMD has been mostly applied to study the unbinding of protein-ligand complexes, here its applicability to rare conformational transitions within a protein and the prediction of their kinetics are investigated. As this typically requires the introduction of multiple collective variables {xj} = x, a theoretical framework is outlined to calculate the associated free energy ΔG(x) and friction Γ(x) from dcTMD simulations along coordinate s. Adopting the α-β transition of alanine dipeptide as well as the open-closed transition of T4 lysozyme as representative examples, the virtues and shortcomings of dcTMD to predict protein conformational transitions and the related kinetics are studied.

Inferring free-energy barriers and kinetic rates from molecular dynamics via underdamped Langevin models

Rare events include many of the most interesting transformation processes in condensed matter, from phase transitions to biomolecular conformational changes to chemical reactions. Access to the corresponding mechanisms, free-energy landscapes and kinetic rates can in principle be obtained by different techniques after projecting the high-dimensional atomic dynamics on one (or a few) collective variable. Even though it is well-known that the projected dynamics approximately follows - in a statistical sense - the generalized, underdamped or overdamped Langevin equations (depending on the time resolution), to date it is nontrivial to parameterize such equations starting from a limited, practically accessible amount of non-ergodic trajectories. In this work we focus on Markovian, underdamped Langevin equations, that arise naturally when considering, e.g., numerous water-solution processes at sub-picosecond resolution. After contrasting the advantages and pitfalls of different numerical approaches, we present an efficient parametrization strategy based on a limited set of molecular dynamics data, including equilibrium trajectories confined to minima and few hundreds transition path sampling-like trajectories. Employing velocity autocorrelation or memory kernel information for learning the friction and likelihood maximization for learning the free-energy landscape, we demonstrate the possibility to reconstruct accurate barriers and rates both for a benchmark system and for the interaction of carbon nanoparticles in water.

Construction of Coarse-Grained Molecular Dynamics with Many-Body Non-Markovian Memory

We introduce a machine-learning-based coarse-grained molecular dynamics model that faithfully retains the many-body nature of the intermolecular dissipative interactions. Unlike the common empirical coarse-grained models, the present model is constructed based on the Mori-Zwanzig formalism and naturally inherits the heterogeneous state-dependent memory term rather than matching the mean-field metrics such as the velocity autocorrelation function. Numerical results show that preserving the many-body nature of the memory term is crucial for predicting the collective transport and diffusion processes, where empirical forms generally show limitations.

Data-driven construction of stochastic reduced dynamics encoded with non-Markovian features

One important problem in constructing the reduced dynamics of molecular systems is the accurate modeling of the non-Markovian behavior arising from the dynamics of unresolved variables. The main complication emerges from the lack of scale separations, where the reduced dynamics generally exhibits pronounced memory and non-white noise terms. We propose a data-driven approach to learn the reduced model of multi-dimensional resolved variables that faithfully retains the non-Markovian dynamics. Different from the common approaches based on the direct construction of the memory function, the present approach seeks a set of non-Markovian features that encode the history of the resolved variables and establishes a joint learning of the extended Markovian dynamics in terms of both the resolved variables and these features. The training is based on matching the evolution of the correlation functions of the extended variables that can be directly obtained from the ones of the resolved variables. The constructed model essentially approximates the multi-dimensional generalized Langevin equation and ensures numerical stability without empirical treatment. We demonstrate the effectiveness of the method by constructing the reduced models of molecular systems in terms of both one-dimensional and four-dimensional resolved variables.

Pair-Reaction Dynamics in Water: Competition of Memory, Potential Shape, and Inertial Effects

When described by a one-dimensional reaction coordinate, pair-reaction rates in a solvent depend, in addition to the potential barrier height and the friction coefficient, on the potential shape, the effective mass, and the friction relaxation spectrum, but a rate theory that accurately accounts for all of these effects does not exist. After a review of classical reaction-rate theories, we show how to extract all parameters of the generalized Langevin equation (GLE) and, in particular, the friction memory function from molecular dynamics (MD) simulations of two prototypical pair reactions in water, the dissociation of NaCl and of two methane molecules. The memory exhibits multiple time scales and, for NaCl, pronounced oscillatory components. Simulations of the GLE by Markovian embedding techniques accurately reproduce the pair-reaction kinetics from MD simulations without any fitting parameters, which confirms the accuracy of the approximative form of the GLE and of the parameter extraction techniques. By modification of the GLE parameters, we investigate the relative importance of memory, mass, and potential shape effects. Neglect of memory slows down NaCl and methane dissociation by roughly a factor of 2; neglect of mass accelerates reactions by a similar factor, and the harmonic approximation of the potential shape gives rise to slight acceleration. This partial error cancellation explains why Kramers' theory, which neglects memory effects and treats the potential shape in harmonic approximation, describes reaction rates better than more sophisticated theories. In essence, all three effects, friction memory, inertia, and the potential shape nonharmonicity, are important to quantitatively describe pair-reaction kinetics in water.

Assessing transition rates as functions of environmental variables

We present a method to estimate the transition rates of molecular systems under different environmental conditions that cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios," governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then use the Square Root Approximation method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis method to coarse-grain the kinetic model. This permits efficiently estimating the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work, we formalize the theoretical framework of the procedure, and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies that take into account how the cellular environment influences biochemical processes.

Position-dependent memory kernel in generalized Langevin equations: Theory and numerical estimation

Generalized Langevin equations with non-linear forces and position-dependent linear friction memory kernels, such as commonly used to describe the effective dynamics of coarse-grained variables in molecular dynamics, are rigorously derived within the Mori–Zwanzig formalism. A fluctuation–dissipation theorem relating the properties of the noise to the memory kernel is shown. The derivation also yields Volterra-type equations for the kernel, which can be used for a numerical parametrization of the model from all-atom simulations.

Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a generalized Langevin equation (GLE) for a general interacting many-body system. The resulting GLE includes (i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, (ii) a random force term that explicitly depends on the initial state of the system, and (iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general nonlinear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear memory friction contribution is shown to be related to correlations between the reaction-coordinate velocity and the random force. We present a numerical method to compute all parameters of our GLE, in particular the nonlinear memory friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example, we demonstrate that nonlinear memory friction is present and that the random force exhibits significant non-Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase-space of the underlying many-body system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system.

Likelihood-based non-Markovian models from molecular dynamics

SignificanceThe analysis of complex systems with many degrees of freedom generally involves the definition of low-dimensional collective variables more amenable to physical understanding. Their dynamics can be modeled by generalized Langevin equations, whose coefficients have to be estimated from simulations of the initial high-dimensional system. These equations feature a memory kernel describing the mutual influence of the low-dimensional variables and their environment. We introduce and implement an approach where the generalized Langevin equation is designed to maximize the statistical likelihood of the observed data. This provides an efficient way to generate reduced models to study dynamical properties of complex processes such as chemical reactions in solution, conformational changes in biomolecules, or phase transitions in condensed matter systems.

Time-Dependent Friction Effects on Vibrational Infrared Frequencies and Line Shapes of Liquid Water

From ab initio simulations of liquid water, the time-dependent friction functions and time-averaged nonlinear effective bond potentials for the OH stretch and HOH bend vibrations are extracted. The obtained friction exhibits not only adiabatic contributions at and below the vibrational time scales but also much slower nonadiabatic contributions, reflecting homogeneous and inhomogeneous line broadening mechanisms, respectively. Intermolecular interactions in liquid water soften both stretch and bend potentials compared to the gas phase, which by itself would lead to a red-shift of the corresponding vibrational bands. In contrast, nonadiabatic friction contributions cause a spectral blue shift. For the stretch mode, the potential effect dominates, and thus, a significant red shift when going from gas to the liquid phase results. For the bend mode, potential and nonadiabatic friction effects are of comparable magnitude, so that a slight blue shift results, in agreement with well-known but puzzling experimental findings. The observed line broadening is shown to be roughly equally caused by adiabatic and nonadiabatic friction contributions for both the stretch and bend modes in liquid water. Thus, the quantitative analysis of the time-dependent friction that acts on vibrational modes in liquids advances the understanding of infrared vibrational frequencies and line shapes.

Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium

We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.

Non-Markovian modeling of protein folding

Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics.

We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

Data-driven coarse-grained modeling of non-equilibrium systems

Modeling a high-dimensional Hamiltonian system in reduced dimensions with respect to coarse-grained (CG) variables can greatly reduce computational cost and enable efficient bottom-up prediction of main features of the system for many applications. However, it usually experiences significantly altered dynamics due to loss of degrees of freedom upon coarse-graining. To establish CG models that can faithfully preserve dynamics, previous efforts mainly focused on equilibrium systems. In contrast, various soft matter systems are known to be out of equilibrium. Therefore, the present work concerns non-equilibrium systems and enables accurate and efficient CG modeling that preserves non-equilibrium dynamics and is generally applicable to any non-equilibrium process and any observable of interest. To this end, the dynamic equation of a CG variable is built in the form of the non-stationary generalized Langevin equation (nsGLE), where the two-time memory kernel is determined from the data of the auto-correlation function of the observable of interest. By embedding the nsGLE in an extended dynamics framework, the nsGLE can be solved efficiently to predict the non-equilibrium dynamics of the CG variable. To prove and exploit the equivalence of the nsGLE and extended dynamics, the memory kernel is parameterized in a two-time exponential expansion. A data-driven hybrid optimization process is proposed for the parameterization, which integrates the differential-evolution method with the Levenberg-Marquardt algorithm to efficiently tackle a non-convex and high-dimensional optimization problem.

Transfer learning of memory kernels for transferable coarse-graining of polymer dynamics

The present work concerns the transferability of coarse-grained (CG) modeling in reproducing the dynamic properties of the reference atomistic systems across a range of parameters. In particular, we focus on implicit-solvent CG modeling of polymer solutions. The CG model is based on the generalized Langevin equation, where the memory kernel plays the critical role in determining the dynamics in all time scales. Thus, we propose methods for transfer learning of memory kernels. The key ingredient of our methods is Gaussian process regression. By integration with the model order reduction via proper orthogonal decomposition and the active learning technique, the transfer learning can be practically efficient and requires minimum training data. Through two example polymer solution systems, we demonstrate the accuracy and efficiency of the proposed transfer learning methods in the construction of transferable memory kernels. The transferability allows for out-of-sample predictions, even in the extrapolated domain of parameters. Built on the transferable memory kernels, the CG models can reproduce the dynamic properties of polymers in all time scales at different thermodynamic conditions (such as temperature and solvent viscosity) and for different systems with varying concentrations and lengths of polymers.

Petrov-Galerkin methods for the construction of non-Markovian dynamics preserving nonlocal statistics

A common observation in coarse-graining a molecular system is the non-Markovian behavior, primarily due to the lack of scale separations. This is reflected in the strong memory effect and the non-white noise spectrum, which must be incorporated into a coarse-grained description to correctly predict dynamic properties. To construct a stochastic model that gives rise to the correct non-Markovian dynamics, we propose a Galerkin projection approach, which transforms the exhausting effort of finding an appropriate model to choosing appropriate subspaces in terms of the derivatives of the coarse-grained variables and, at the same time, provides an accurate approximation to the generalized Langevin equation. We introduce the notion of fractional statistics that embodies nonlocal properties. More importantly, we show how to pick subspaces in the Galerkin projection so that those statistics are automatically matched.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Dynamic coarse-graining of polymer systems using mobility functions

We propose a dynamic coarse-graining (CG) scheme for mapping heterogeneous polymer fluids onto extremely CG models in a dynamically consistent manner. The idea is to use as target function for the mapping a wave-vector dependent mobility function derived from the single-chain dynamic structure factor, which is calculated in the microscopic reference system. In previous work, we have shown that dynamic density functional calculations based on this mobility function can accurately reproduce the order/disorder kinetics in polymer melts, thus it is a suitable starting point for dynamic mapping. To enable the mapping over a range of relevant wave vectors, we propose to modify the CG dynamics by introducing internal friction parameters that slow down the CG monomer dynamics on local scales, without affecting the static equilibrium structure of the system. We illustrate and discuss the method using the example of infinitely long linear Rouse polymers mapped onto ultrashort CG chains. We show that our method can be used to construct dynamically consistent CG models for homopolymers with CG chain lengthN= 4, whereas for copolymers, longer CG chain lengths are necessary.

Data-driven molecular modeling with the generalized Langevin equation

The complexity of molecular dynamics simulations necessitates dimension reduction and coarse-graining techniques to enable tractable computation. The generalized Langevin equation (GLE) describes coarse-grained dynamics in reduced dimensions. In spite of playing a crucial role in non-equilibrium dynamics, the memory kernel of the GLE is often ignored because it is difficult to characterize and expensive to solve. To address these issues, we construct a data-driven rational approximation to the GLE. Building upon previous work leveraging the GLE to simulate simple systems, we extend these results to more complex molecules, whose many degrees of freedom and complicated dynamics require approximation methods. We demonstrate the effectiveness of our approximation by testing it against exact methods and comparing observables such as autocorrelation and transition rates.

Data-driven coarse-grained modeling of polymers in solution with structural and dynamic properties conserved

We present data-driven coarse-grained (CG) modeling for polymers in solution, which conserves the dynamic as well as structural properties of the underlying atomistic system. The CG modeling is built upon the framework of the generalized Langevin equation (GLE). The key is to determine each term in the GLE by directly linking it to atomistic data. In particular, we propose a two-stage Gaussian process-based Bayesian optimization method to infer the non-Markovian memory kernel from the data of the velocity autocorrelation function (VACF). Considering that the long-time behaviors of the VACF and memory kernel for polymer solutions can exhibit hydrodynamic scaling (algebraic decay with time), we further develop an active learning method to determine the emergence of hydrodynamic scaling, which can accelerate the inference process of the memory kernel. The proposed methods do not rely on how the mean force or CG potential in the GLE is constructed. Thus, we also compare two methods for constructing the CG potential: a deep learning method and the iterative Boltzmann inversion method. With the memory kernel and CG potential determined, the GLE is mapped onto an extended Markovian process to circumvent the expensive cost of directly solving the GLE. The accuracy and computational efficiency of the proposed CG modeling are assessed in a model star-polymer solution system at three representative concentrations. By comparing with the reference atomistic simulation results, we demonstrate that the proposed CG modeling can robustly and accurately reproduce the dynamic and structural properties of polymers in solution.