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We present a unified framework for the data-driven construction of stochastic reduced models with state-dependent memory for high-dimensional Hamiltonian systems. The method addresses two key challenges: (i) accurately modeling heterogeneous non-Markovian effects where the memory function depends on the coarse-grained (CG) variables beyond the standard homogeneous kernel, and (ii) efficiently exploring the phase space to sample both equilibrium and dynamical observables for reduced model construction. Specifically, we employ a consensus-based sampling method to establish a shared sampling strategy that enables simultaneous construction of the free energy function and collection of conditional two-point correlation functions used to learn the state-dependent memory. The reduced dynamics is formulated as an extended Markovian system, where a set of auxiliary variables, interpreted as non-Markovian features, is jointly learned to systematically approximate the memory function using only two-point statistics. The constructed model yields a generalized Langevin-type formulation with an invariant distribution consistent with the full dynamics. We demonstrate the effectiveness of the proposed framework on a two-dimensional CG model of an alanine dipeptide molecule. Numerical results on the transition dynamics between metastable states show that accurately capturing state-dependent memory is essential for predicting non-equilibrium kinetic properties, whereas the standard generalized Langevin model with a homogeneous kernel exhibits significant discrepancies. 

In this paper, we present a generalized, data-driven collisional operator for one-component plasmas, learned from molecular dynamics simulations, to extend the collisional kinetic model beyond the weakly coupled regime. The proposed operator features an anisotropic, non-stationary collision kernel that accounts for particle correlations typically neglected in classical Landau formulations. To enable efficient numerical evaluation, we develop a fast spectral separation method that represents the kernel as a low-rank tensor product of univariate basis functions. This formulation admits an O(N log N) algorithm via fast Fourier transforms and preserves key physical properties, including discrete conservation laws and the H-theorem, through a structure-preserving central difference discretization. Numerical experiments demonstrate that the proposed model accurately captures plasma dynamics in the moderately coupled regime beyond the standard Landau model while maintaining high computational efficiency and structure-preserving properties. 

One essential goal of constructing coarse-grained molecular dynamics (CGMD) models is to accurately predict nonequilibrium processes beyond the atomistic scale. While a CG model can be constructed by projecting the full dynamics onto a set of resolved variables, the dynamics of the CG variables can recover the full dynamics only when the conditional distribution of the unresolved variables is close to the one associated with the particular projection operator. In particular, the model's applicability to various nonequilibrium processes is generally unwarranted due to the inconsistency in the conditional distribution. Here, we present a data-driven approach for constructing CGMD models that retain certain generalization ability for nonequilibrium processes. Unlike the conventional CG models based on preselected CG variables (e.g., the center of mass), the present CG model seeks a set of auxiliary CG variables similar to the time-lagged independent component analysis to maximize the velocity correlation. This effectively minimizes the entropy contribution of unresolved variables and ensures the distribution under a broad range of nonequilibrium conditions approaches the one under equilibrium. Numerical results of a polymer melt system demonstrate the significance of this broadly overlooked metric for the model's generalization ability, and the effectiveness of the present CG model for predicting the complex viscoelastic responses under various nonequilibrium flows. 

We propose a generative model-based framework for learning collective variables (CVs) that faithfully capture the individual metastable states of the fulldimensional molecular dynamics (MD) systems. Unlike most existing approaches based on various feature extraction strategies, the new framework transfers the exhausting efforts of feature selection into a generative task of reconstructing the full-dimensional probability density function (PDF) from a set of CVs under a prior distribution with pre-assigned local maxima. By pairing the CVs with a set of auxiliary Gaussian random variables, we seek an invertible mapping that recovers the full-dimensional PDF and meanwhile, preserves the correspondence between the metastable states of the MD space and individual local maxima of the prior distribution. Through identifying the metastable states within MD space that are generally unknown and imposing the correspondence between the two spaces, the constructed CVs retain clear physical interpretations and provide kinetic insight for the molecular systems on the collective scale. We demonstrate the effectiveness of the proposed method with the alanine dipeptide in the aqueous environment. The constructed CVs faithfully capture the essential metastable states of the full MD systems, which show good agreement with kinetic properties such as the transition from the ballistic to the plateau regime for the mean square displacement. 

We introduce a data-driven approach to learn a generalized kinetic collision operator directly from molecular dynamics. Unlike the conventional (e.g., Landau) models, the present operator takes an anisotropic form that accounts for a second energy transfer arising from the collective interactions between the pair of collision particles and the environment. Numerical results show that preserving the broadly overlooked anisotropic nature of the collision energy transfer is crucial for predicting the plasma kinetics with non-negligible correlations, where the Landau model shows limitations. 

In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state and further show that if the system is highly degenerate, then the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and hence generally applies to nonequilibrium systems driven by stochastic forces. These theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that our heat transfer model yields better predictions than the Green-Kubo formula for systems far from the equilibrium. 

We introduce a machine-learning-based coarse-grained molecular dynamics (CGMD) model that faithfully retains the many-body nature of the inter-molecular dissipative interactions. Unlike the common empirical CG 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 auto-correlation 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. 

A hallmark of meso-scale interfacial fluids is the multi-faceted, scale-dependent interfacial energy, which often manifests different characteristics across the molecular and continuum scale. The multi-scale nature imposes a challenge to construct reliable coarse-grained (CG) models, where the CG potential function needs to faithfully encode the many-body interactions arising from the unresolved atomistic interactions and account for the heterogeneous density distributions across the interface. We construct the CG models of both single- and two-component polymeric fluid systems based on the recently developed deep coarse-grained potential [Zhang et al., J. Chem. Phys. 149, 034101 (2018)] scheme, where each polymer molecule is modeled as a CG particle. By only using the training samples of the instantaneous force under the thermal equilibrium state, the constructed CG models can accurately reproduce both the probability density function of the void formation in bulk and the spectrum of the capillary wave across the fluid interface. More importantly, the CG models accurately predict the volume-to-area scaling transition for the apolar solvation energy, illustrating the effectiveness to probe the meso-scale collective behaviors encoded with molecular-level fidelity.