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REVIEW 6 major objections 5 minor 55 references

A unified framework for data-driven construction of stochastic reduced models with state-dependent memory

T0 review · 6 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that a generalized Langevin equation with state-dependent memory, trained from conditional two-point correlation functions collected under consensus-based sampling, reproduces the transition dynamics of an alanine dipeptid

desk verdict A useful framework for state-dependent memory in GLE models with a real out-of-sample test, but the correlation validation is in-sample and the fluctuation-dissipation parameterization has a factor-of-two issue worth fixing. read the letter →

arxiv 2509.07264 v1 pith:TMVVSHUW submitted 2025-09-08 physics.comp-ph

Zhiyuan She , Liyao Lyu , Bryan Ronain Smith , Huan Lei This is my paper
classification physics.comp-ph
keywords Stochasticreducedmodeldata-drivenmodelinggeneralizedLangevinequationstate-dependentmemoryconsensus-basedsamplingcoarse-grainedmoleculardynamicstransitionrare-eventkinetics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to show that the memory kernel in a generalized Langevin reduced model must depend on the coarse-grained variables themselves to predict non-equilibrium kinetics, and that such a state-dependent memory can be learned from two-point correlation functions alone. The proposed framework combines a consensus-based enhanced sampling scheme, which supplies both equilibrium free-energy data and conditional initial-state samples from the same restrained simulations, with an extended Markovian formulation in which auxiliary 'non-Markovian features' encode the unresolved degrees of freedom. The central numerical claim is that the resulting state-dependent GLE reproduces conditional velocity and position correlations and committor-based transition-time distributions for a two-dimensional alanine dipeptide model, whereas the standard homogeneous-kernel GLE does not. If the claim holds, it removes a major obstacle to building predictive coarse-grained models in systems with rugged free-energy landscapes.

What carries the argument

The central object is an extended Markovian system over (q, p, χ), where χ are non-Markovian features built as convolutions of past momenta. The coupling and dissipation matrix J(q), whose skew-symmetric and lower-triangular parts are represented by neural networks, is trained by matching the differential form of the conditional two-point correlation equations (Eq. 24). The time-scale-separation approximation in Eq. (23) freezes J at the conditioning state q*, reducing memory construction to two-point statistics while keeping the invariant distribution consistent through a multiplicative noise obeying the fluctuation–dissipation relation.

What would settle it

Run the same training protocol on a coarse-grained system with a fast position variable (for example a stiff bond coordinate) where C_qq decays on the same timescale as C_vv; if the conditional correlation matching then fails to reproduce full-MD correlations, the time-scale-separation assumption is the culprit. Alternatively, compare the learned J(q*) against J obtained by directly solving Eq. (23) without freezing J, at several states.

Watch

Extended reading notes

Core claim

The central claim is that the standard simplification of the Mori–Zwanzig memory term to a homogeneous kernel is the reason reduced models miss rare transitions. The paper constructs a state-dependent generalized Langevin equation (SD-GLE) whose memory term varies across coarse-grained space. Rather than parameterizing the kernel directly, the model embeds a set of auxiliary variables, learned as convolution filters over past momenta, into an extended Markovian system; the coupling matrix J(q) is trained by matching conditional two-point correlations, so no three-point statistics are needed. A multiplicative noise satisfying a fluctuation–dissipation relation preserves the Boltzmann invarian

Load-bearing premise

The load-bearing assumption is that, in every region of coarse-grained space used for training, the position correlation decays much more slowly than the velocity correlation, so the coupling matrix J(q(t)) can be replaced by J(q*) under the conditional expectation; the paper checks this separation globally but not per state.

Editorial extensions

If this is right

  • SD-GLE predictions from conditional two-point correlations match full MD conditional correlations at multiple metastable points, indicating that spatial heterogeneity in dissipation is captured without three-point statistics.
  • Committor-based transition-time distributions between metastable states are reproduced, so state-dependent memory appears necessary for rare-event kinetics, not just equilibrium averages.
  • Because the same unbiased conditional samples build the free energy, mass matrix, and memory kernel, the framework extends existing CG workflows by sharing sampling effort across all modeling terms.
  • With constant coupling and dissipation matrices, the formulation reduces to the standard GLE, making SD-GLE a strict generalization rather than a separate model.
  • The constructed reduced model preserves the full Boltzmann invariant distribution via the multiplicative noise, giving long-time thermodynamic consistency.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The time-scale separation underlying Eq. (23) is validated only globally (Fig. 4); a per-state test could reveal CG regions where the learned J(q*) is fitted to the wrong correlation target, and an adaptive residual-aware sampling could remedy it.
  • Because the method needs only two-point statistics, extending it to higher-dimensional CG sets mainly adds sampling cost, making state-dependent memory feasible for collective variables beyond torsion angles.
  • The success on committor transition times suggests the framework could also be tested on path-space observables such as transition path ensembles, not just first-passage durations.
  • The auxiliary features χ are fixed as discrete convolution weights here, but the formulation does not require that specific choice; other feature representations could be explored while keeping the same conditional-correlation training objective.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

6 major / 5 minor

Summary. The manuscript proposes a data-driven framework for constructing stochastic reduced models with state-dependent memory from full Hamiltonian molecular dynamics. The method combines consensus-based adaptive sampling (CAS) to collect free-energy data and conditional two-point correlation functions, and an extended Markovian model with learned auxiliary variables to represent the state-dependent memory kernel. The framework is applied to a polymer chain and to a two-dimensional alanine dipeptide model. The central numerical claims are that the state-dependent GLE (SD-GLE) reproduces conditional velocity/position correlations and committor-based transition-time distributions, while a standard homogeneous-memory GLE (SI-GLE) does not.

Significance. If the framework is correct, it is a meaningful contribution: it extends GLE construction to state-dependent memory using only two-point conditional statistics, provides a shared sampling strategy for equilibrium and dynamical data, and demonstrates on a realistic biomolecular model that homogeneous memory can fail for transition kinetics. The paper is transparent about its main closure assumption, and the inclusion of transition-time distributions as an out-of-sample observable is a strength. However, several load-bearing technical issues need to be resolved before the central claim can be accepted: an inconsistency in the fluctuation-dissipation construction, an unvalidated per-state closure assumption, and insufficient uncertainty quantification for the key kinetic comparison.

major comments (6)
  1. [Sec. 2.3, Eq. (23)] The derivation replaces J(q(t)) by J(q*) under the conditional expectation based on the assumption that C_qq decays much more slowly than C_vv. The only support is the global comparison in Fig. 4 (right); no per-state verification is given. The four states in Fig. 12 are metastable minima, but transition paths pass through regions where q changes on a timescale comparable to velocity relaxation. If the separation fails in those regions, L_k fits J(q*) to a correlation target with the wrong time dependence, biasing the very state-dependent memory that the kinetic comparison is meant to demonstrate. Please provide state-resolved C_qq/C_vv checks at barrier and basin-edge states, or otherwise control this closure error.
  2. [Sec. 2.3, Eq. (20) and Prop. 2] The stated construction is internally inconsistent. The fluctuation-dissipation condition requires D D^T = -β^{-1}(Γ+Γ^T). With Γ = -L L^T + J_a and D = β^{-1/2} L, one obtains D D^T = β^{-1} L L^T, while -β^{-1}(Γ+Γ^T) = 2 β^{-1} L L^T. Equality holds only for L=0. A missing factor (1/2 in Γ or sqrt(2) in D) appears to be present. As written, the invariant distribution (21) is not the stationary law of (18), contradicting a central thermodynamic-consistency claim. Please correct and re-derive.
  3. [Sec. 2.3, Eqs. (22)-(24)] Eq. (22) defines chi_i(t) as a deterministic functional of past momenta, while in the reduced model (18) chi is an independent stochastic state variable. The loss uses <chi_i(t) v(0)^T> computed from Eq. (22) as if it were the correlation of the model auxiliary variables, but the initial law of chi under the conditional expectation is not specified, and no consistency is shown between the model chi and the filtered-momentum features. If the model chi has a different initial distribution or dynamics, the target C2 in Eq. (24) is not the actual model correlation. Please specify initial conditions and verify that the learned model reproduces the encoder representation, or reformulate the loss using model-generated chi.
  4. [Sec. 3, Figs. 6-11 and Fig. 12] The conditional correlation comparisons in Figs. 6-11 are largely in-sample, because L_k fits J(q), H(q), L(q) and encoder weights to the same conditional two-point correlations shown in those figures. The genuinely out-of-sample evidence is Fig. 12, the transition-time distributions, which are not part of the loss. However, Fig. 12 has no error bars, no transition-event counts, and no discrepancy metric, so it is not possible to assess whether the SD-GLE improvement over SI-GLE is statistically significant. Please provide error bars or confidence intervals and a quantitative error measure for the transition-time predictions.
  5. [Sec. 3, model comparison] The number of auxiliary variables is fixed at n=18 for both SD-GLE and SI-GLE, but no convergence or sensitivity study in n is reported. Since the finite Markovian embedding is an approximation, the failure of SI-GLE could in principle be due to an under-resolved embedding rather than to the homogeneity assumption. A short n-convergence study, at least for the alanine dipeptide conditional correlations or transition times, is needed to support the central claim that state dependence, not embedding expressiveness, is the key missing ingredient.
  6. [Eq. (17) and Eq. (4)] The mass-matrix loss appears dimensionally inconsistent. Eq. (4) defines m(q)^{-1} = β C_vv(0,q), so the loss should compare m(q) with β^{-1} C_vv(0,q)^{-1}, or compare the inverses. As written, L_m minimizes ||m(q) - β^{-1} C_vv(0,q)||^2, which targets the wrong object. Since m(q) enters the free energy (19) and the dynamics (18), this needs correction or clarification.
minor comments (5)
  1. [Algorithm 1] The symbol 'cV' in the update for q_{i,t+1} is undefined; presumably it denotes the inverse covariance or a computed matrix. The moving-average bias correction is also not fully explained.
  2. [Eq. (22)] The notation N_w is used both for the number of CAS particles and for the number of discrete time-lags in the encoder weights. Please rename one of them to avoid confusion.
  3. [Fig. 4] The right panel plots four normalized correlation functions but the curves are not clearly distinguished in the text description. A legend with explicit line styles would improve readability.
  4. [Fig. 12] Please state the number of transition events and the MD trajectory length used to construct the histograms, as well as the level of statistical uncertainty of the full-MD reference.
  5. [Appendix B] The training details omit the neural-network architecture (number of layers, activations) and the loss weights lambda_c and lambda_Lambda. These are needed for reproducibility.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: SD-GLE is fitted to conditional two-point correlations and those same correlations are then presented as successful predictions; the transition-time result is an independent target.

  1. fitted input called prediction [Sec. 2.3, Eq. (24) and loss L_k; Sec. 3.2, Figs. 6–11]
    "the correlation matrices C0(t,q*), C1(t,q*) and C2(t,q*) can be directly evaluated for each q* in the training set. This enables us to establish a joint learning of both the non-Markovian weights {w_ij} and the function J(q) ... by minimizing the empirical loss function Lk ... the predictions from the present SD-GLE model show better agreement with the full MD results than the standard SI-GLE model."

    The SD-GLE parameters J(q) (i.e., H(q), L(q), J_a(q)) and the non-Markovian weights {w_ij} are optimized against L_k, whose terms C0, C1, C2 are exactly the conditional two-point correlation functions at the training states q*. The figures then compare simulated SD-GLE conditional velocity/position correlations at those same states against the MD data used to build the loss. Agreement is therefore an in-sample consistency check enforced by the fit, not an independent prediction. The paper's kinetic conclusion does not reduce to this fit, because the committor-based transition-time distributions in Fig. 12 are not part of the loss; that comparison provides independent content. Eq. (23)'s C_qq << C_vv separation is an unvalidated-per-state correctness assumption, not itself a circularity.

full rationale

The derivation chain is largely self-contained: the extended Markovian ansatz, the CAS sampling, and the correlation-matching loss are all stated explicitly, and the central kinetic claim is tested against full-MD transition-time distributions that were not used in the loss. The self-citations to Refs. [14], [33], [43] are background/method citations; the current paper's alanine-dipeptide transition-time comparison is a fresh computation against full MD, so there is no load-bearing self-citation chain. The main circular element is limited to the validation of conditional correlation functions: those curves are training targets for J(q) and the non-Markovian weights, so Figs. 6–11 are consistency checks rather than out-of-sample predictions. The time-scale-separation approximation in Eq. (23), verified only globally in Fig. 4, is a legitimate correctness risk—especially near transition barriers—but it is an assumption rather than a reduction of a prediction to its inputs. Overall, the paper has partial circularity in a secondary validation, but its primary transition-dynamics claim retains independent content, so the score is moderate.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The construction rests on the MZ projection ansatz, the finite Markovian embedding ansatz, and the time-scale-separation approximation used to turn correlation functions into training targets. Beyond these modeling assumptions, the memory representation is fitted: encoder weights, neural networks for coupling and dissipation, and multiple hyperparameters are learned or hand-chosen from data. The only introduced entities are the auxiliary chi variables, which are latent internal variables without independent evidence.

free parameters (6)
  • Non-Markovian encoder weights w_ij = not reported
    Define chi_i in Eq. (22) and are optimized in the loss L_k; they encode the memory representation.
  • Neural-network parameters for H(q), L(q), J_a(q) = not reported
    Parameterize the state-dependent coupling and dissipation matrix J(q) in Eq. (20) and are fit to conditional correlation data.
  • Number of auxiliary variables n = 4 (polymer), 18 (alanine dipeptide)
    Hand-chosen truncation of the memory approximation; no convergence study over n is reported.
  • CAS sampling parameters kappa_l, kappa_h, gamma, beta_1, beta_2, delta_t = kappa_l=10, kappa_h=1; others not reported
    Control the particle sampling dynamics and therefore which states contribute training data, though they are not part of the reduced model itself.
  • Loss weights lambda_c, lambda_Lambda = not reported
    Balance correlation-matching against the chi covariance constraint; no sensitivity analysis is given.
  • Restrained spring constant k in Eq. (9) = not reported
    Controls how sharply the biased distribution approximates the conditional distribution in Eq. (10); finite-k bias is not quantified.
assumptions (6)
  • domain assumption The full Hamiltonian system samples the canonical distribution rho_0 proportional to exp(-beta H), and the Mori-Zwanzig projection formalism applies.
    Sec. 2.1; the reduced equations (2)-(5) inherit this structure and the fluctuation-dissipation relation.
  • domain assumption The projected reduced dynamics has memory depending only on the CG coordinates q and admits the multiplicative-noise GLE form of Eq. (2).
    Sec. 2.1; this is a modeling simplification of the exact MZ memory term.
  • ad hoc to paper A finite Markovian embedding with linear auxiliary variables chi and white noise, Eqs. (18)-(20), can represent the state-dependent memory kernel.
    No approximation theorem is given for this ansatz; expressive capacity is assumed and validated only numerically.
  • domain assumption Time-scale separation: position correlations decay much more slowly than velocity correlations, justifying J(q(t)) approximately J(q*) in Eq. (23).
    Sec. 2.3; verified globally for alanine dipeptide in Fig. 4 but not per state.
  • domain assumption Restrained dynamics at finite spring constant k yields samples close enough to the conditional distribution in Eq. (10).
    Eqs. (9)-(11); the equivalence is exact only in the k to infinity limit.
  • standard math Convergence of the consensus-based sampling distribution follows from the conditions in Ref. [33].
    Sec. 2.2 relies on Proposition 2.5 of Ref. [33] for the invariant distribution of the particle system.
invented entities (1)
  • Non-Markovian memory features chi_i (auxiliary variables)
    purpose: Latent variables in the extended Markovian system that encode unresolved orthogonal dynamics and approximate the state-dependent memory K(q,t).
    No independent physical observable is provided; their covariance is constrained to the identity by the loss term L_Lambda, an internal modeling choice.

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Cite this review

Pith. "Pith review of A unified framework for data-driven construction of stochastic reduced models with state-dependent memory." pith.science (2026). https://pith.science/paper/TMVVSHUW

@misc{pith2026250907264,
  author       = {Pith},
  title        = {Pith review of: A unified framework for data-driven construction of stochastic reduced models with state-dependent memory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMVVSHUW}},
  note         = {Machine review of arXiv:2509.07264}
}
read the original abstract

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: (\rmnum{1}) accurately modeling heterogeneous non-Markovian effects where the memory function depends on the coarse-grained (CG) variables beyond the standard homogeneous kernel, and (\rmnum{2}) 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.

Figures

Figures reproduced from arXiv: 2509.07264 by the authors.

Figure 1
Figure 1. Velocity autocorrelation function Cvvptq obtained from the full MD, the present model (SD-GLE) and the standard GLE (SI-GLE), where the CG coordinate q is defined as the end-to-end distance of a polymer molecule. 10−1 100 101 102 t −0.2 0.1 0.4 0.7 1.0 Cvv (t, q ) / Cvv(0, q ) MD 10−1 100 10 1 10 2 t SD-GLE 10−1 100 101 102 t SI-GLE [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Conditional velocity autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The rare-event distribution of the time duration for the polymer molecule under a large extension with the [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Left: The 2D free energy function Upqq for the reduced model of an alanine dipeptide molecule in solvent, where the CG coordinates q “ rϕ, ψs denote the two backbone torsion angles of the molecule. The four red points denote the individual metastable states separated b…
Figure 5
Figure 5. Figure 5: The 2D state-dependent mass matrix mpqq for the reduced model of an alanine dipeptide molecule in solvent. 10−3 10−2 10−1 t[ps] −0.5 0.0 0.5 1.0 Cφ˙φ˙(t, φ, ψ)/Cφ˙φ˙(0, φ, ψ) 10−3 10−2 10−1 t[ps] −4 −2 0 2 Cφ˙ψ˙(t, φ, ψ)/Cψ˙φ˙(0, φ, ψ) MD SD-GLE SI-GLE 10−3 10−2 10−1 t…
Figure 6
Figure 6. Figure 6: The conditional velocity correlation function [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The conditional velocity correlation function [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: The conditional velocity correlation function [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: The conditional position correlation function [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: The conditional position correlation function [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: The conditional position correlation function [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: The distribution of the committor-based transition time (defined by Eq. (25)) that characterizes the duration [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

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Reference graph

Works this paper leans on

55 extracted references · 54 canonical work pages

  1. [1]

    Jerry B Abrams and Mark E Tuckerman. Efficient and direct generation of multidimen- sional free energy surfaces via adiabatic dynamics without coordinate transformations.The Journal of Physical Chemistry B, 112(49):15742–15757, 2008. 19

    work page 2008

  2. [2]

    Dalton, and Roland R

    Cihan Ayaz, Laura Scalfi, Benjamin A. Dalton, and Roland R. Netz. Generalized langevin equation with a non-linear potential of mean force and non-linear memory friction from a hybrid projection scheme.Physical Review E, 105:054138, 2022

    work page 2022

  3. [3]

    Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel.The Journal of chemical physics, 139(4):044107, 2013

    Andrew D Baczewski and Stephen D Bond. Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel.The Journal of chemical physics, 139(4):044107, 2013

    work page 2013

  4. [4]

    Canonical sampling through velocity rescaling.The Journal of Chemical Physics, 126(1), 2007

    Giovanni Bussi, Davide Donadio, and Michele Parrinello. Canonical sampling through velocity rescaling.The Journal of Chemical Physics, 126(1), 2007

    work page 2007

  5. [5]

    Langevin equation with colored noise for constant-temperature molecular dynamics simulations.Physical review letters, 102(2):020601, 2009

    Michele Ceriotti, Giovanni Bussi, and Michele Parrinello. Langevin equation with colored noise for constant-temperature molecular dynamics simulations.Physical review letters, 102(2):020601, 2009

    work page 2009

  6. [6]

    Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules.J

    Minxin Chen, Xiantao Li, and Chun Liu. Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules.J. Chem. Phys., 141:064112, 2014

    work page 2014

  7. [7]

    Calculating free energies using average force.The Journal of Chemical Physics, 115(20):9169–9183, 2001

    Eric Darve and Andrew Pohorille. Calculating free energies using average force.The Journal of Chemical Physics, 115(20):9169–9183, 2001

    work page 2001

  8. [8]

    Computing generalized Langevin equations and generalized fokker-planck equations.Proc

    Eric Darve, Jose Solomon, and Amirali Kia. Computing generalized Langevin equations and generalized fokker-planck equations.Proc. Natl. Acad. Sci., 106(27):10884–10889, 2009

    work page 2009

Show all 55 references
  1. [9]

    The heterognous multiscale methods.Commun

    Weinan E and Bjorn Engquist. The heterognous multiscale methods.Commun. Math. Sci., 1(1):87–132, 03 2003

    2003

  2. [10]

    Het- erogeneous multiscale methods: A review.Communications in Computational Physics, 2(3):367–450, 2007

    Weinan E, Bjorn Engquist, Xiantao Li, Weiqing Ren, and Eric Vanden-Eijnden. Het- erogeneous multiscale methods: A review.Communications in Computational Physics, 2(3):367–450, 2007

    2007

  3. [11]

    Transition-path theory and path-finding algorithms for the study of rare events.Annual Review of Physical Chemistry, 61(1):391–420, 2010

    Weinan E and Eric Vanden-Eijnden. Transition-path theory and path-finding algorithms for the study of rare events.Annual Review of Physical Chemistry, 61(1):391–420, 2010

    2010

  4. [12]

    Active learning based sampling for high-dimensional nonlinear partial differential equations.Journal of Computational Physics, 475:111848, 2023

    Wenhan Gao and Chunmei Wang. Active learning based sampling for high-dimensional nonlinear partial differential equations.Journal of Computational Physics, 475:111848, 2023

    2023

  5. [13]

    Failure-informed adaptive sampling for pinns

    Zhiwei Gao, Liang Yan, and Tao Zhou. Failure-informed adaptive sampling for pinns. SIAM Journal on Scientific Computing, 45(4):A1971–A1994, 2023

    2023

  6. [14]

    Data-driven learning of the generalized langevin equation with state-dependent memory.Physical Review Letters, 133(7):077301, 2024

    Pei Ge, Zhongqiang Zhang, and Huan Lei. Data-driven learning of the generalized langevin equation with state-dependent memory.Physical Review Letters, 133(7):077301, 2024

    2024

  7. [15]

    Francesca Grogan, Huan Lei, Xiantao Li, and Nathan A. Baker. Data-driven molecular modeling with the generalized langevin equation.J. Comput. Phys., 418:109633–109641, 2020. 20

    2020

  8. [16]

    Lincs: A linear constraint solver for molecular simulations.Journal of Computational Chemistry, 18(12):1463–1472, 1997

    Berk Hess, Henk Bekker, Herman JC Berendsen, and Johannes GEM Fraaije. Lincs: A linear constraint solver for molecular simulations.Journal of Computational Chemistry, 18(12):1463–1472, 1997

    1997

  9. [17]

    Mori– Zwanzig formalism as a practical computational tool.Faraday discuss., 144:301–322, 2010

    Carmen Hijón, Pep Español, Eric Vanden-Eijnden, and Rafael Delgado-Buscalioni. Mori– Zwanzig formalism as a practical computational tool.Faraday discuss., 144:301–322, 2010

    2010

  10. [18]

    Comparison of multiple amber force fields and development of improved pro- tein backbone parameters.Proteins: Structure, Function, and Bioinformatics, 65(3):712– 725, 2006

    Viktor Hornak, Robert Abel, Asim Okur, Bentley Strockbine, Adrian Roitberg, and Carlos Simmerling. Comparison of multiple amber force fields and development of improved pro- tein backbone parameters.Proteins: Structure, Function, and Bioinformatics, 65(3):712– 725, 2006

    2006

  11. [19]

    Thomas Hudson and Xingjie H. Li. Coarse-Graining of Overdamped Langevin Dynamics via the Mori–Zwanzig Formalism.Multiscale Modeling&Simulation, 18(2):1113–1135, 2020

    2020

  12. [20]

    Iterative reconstruction of memory kernels.Journal of Chemical Theory and Computation, 13(6):2481–2488, 2017

    Gerhard Jung, Martin Hanke, and Friederike Schmid. Iterative reconstruction of memory kernels.Journal of Chemical Theory and Computation, 13(6):2481–2488, 2017

    2017

  13. [21]

    Kevrekidis, C

    Ioannis G. Kevrekidis, C. William Gear, James M. Hyman, Panagiotis G Kevrekidid, Olof Runborg, and Constantinos Theodoropoulos. Equation-free, coarse-grained multiscale computation: Enabling mocroscopic simulators to perform system-level analysis.Com- mun. Math. Sci., 1(4):715...

    2003

  14. [22]

    Kevrekidis and Giovanni Samaey

    Ioannis G. Kevrekidis and Giovanni Samaey. Equation-free multiscale computation: Algo- rithms and applications.Annual Review of Physical Chemistry, 60(1):321–344, 2009

    2009

  15. [23]

    Adam: A method for stochastic optimization.Interna- tional Conference on Learning Representations (ICLR), 12 2015

    Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization.Interna- tional Conference on Learning Representations (ICLR), 12 2015

    2015

  16. [24]

    Viktor Klippenstein and Nico F. A. van der Vegt. Cross-correlation corrected friction in (generalized) langevin models.The Journal of Chemical Physics, 154(19):191102, 2021

    2021

  17. [25]

    Rosenberg, Djamal Bouzida, Robert H

    Shankar Kumar, John M. Rosenberg, Djamal Bouzida, Robert H. Swendsen, and Pe- ter A. Kollman. The weighted histogram analysis method for free-energy calculations on biomolecules. i. the method.Journal of Computational Chemistry, 13(8):1011–1021, 1992

    1992

  18. [26]

    Escaping free-energy minima.Proceedings of the National Academy of Sciences, 99(20):12562–12566, 2002

    Alessandro Laio and Michele Parrinello. Escaping free-energy minima.Proceedings of the National Academy of Sciences, 99(20):12562–12566, 2002

    2002

  19. [27]

    Collective Langevin dynamics of conformational motions in proteins.J

    Oliver F Lange and Helmut Grubmüller. Collective Langevin dynamics of conformational motions in proteins.J. Chem. Phys., 124:214903, 2006

    2006

  20. [28]

    Hee Sun Lee, Surl-Hee Ahn, and Eric F. Darve. The multi-dimensional generalized Langevin equation for conformational motion of proteins.The Journal of Chemical Physics, 150(17):174113, 2019

    2019

  21. [29]

    H. Lei, B. Caswell, and G. E. Karniadakis. Direct construction of mesoscopic models from microscopic simulations.Phys. Rev. E, 81:026704, 2010. 21

    2010

  22. [30]

    Huan Lei, N. A. Baker, and X. Li. Data-driven parameterization of the generalized Langevin equation.Proc. Natl. Acad. Sci., 113(50):14183–14188, 2016

    2016

  23. [31]

    Incorporation of memory ef- fects in coarse-grained modeling via the Mori-Zwanzig formalism.The Journal of chemical physics, 143(24):243128, 2015

    Zhen Li, Xin Bian, Xiantao Li, and George Em Karniadakis. Incorporation of memory ef- fects in coarse-grained modeling via the Mori-Zwanzig formalism.The Journal of chemical physics, 143(24):243128, 2015

    2015

  24. [32]

    Gromacs 2019.2 source code, April 2019

    Lindahl, Abraham, Hess, and van der Spoel. Gromacs 2019.2 source code, April 2019

    2019

  25. [33]

    Consensus-based adaptive sampling and approximation for high- dimensional energy landscapes.arXiv preprint arXiv:2311.05009, 2023

    Liyao Lyu and Huan Lei. Consensus-based adaptive sampling and approximation for high- dimensional energy landscapes.arXiv preprint arXiv:2311.05009, 2023

    2023 arXiv

  26. [34]

    Construction of coarse-grained molecular dynamics with many- body non-markovian memory.Physical Review Letters, 131(17):177301, 2023

    Liyao Lyu and Huan Lei. Construction of coarse-grained molecular dynamics with many- body non-markovian memory.Physical Review Letters, 131(17):177301, 2023

    2023

  27. [35]

    On the generalization ability of coarse-grained molecular dynam- ics models for nonequilibrium processes.Multiscale Modeling&Simulation, 23(2):816– 837, 2025

    Liyao Lyu and Huan Lei. On the generalization ability of coarse-grained molecular dynam- ics models for nonequilibrium processes.Multiscale Modeling&Simulation, 23(2):816– 837, 2025

    2025

  28. [36]

    The derivation and approximation of coarse-grained dynamics from Langevin dynamics.The Journal of Chemical Physics, 145(20):204117, 2016

    Lina Ma, Xiantao Li, and Chun Liu. The derivation and approximation of coarse-grained dynamics from Langevin dynamics.The Journal of Chemical Physics, 145(20):204117, 2016

    2016

  29. [37]

    Coarse-graining langevin dynamics using reduced- order techniques.Journal of Computational Physics, 380:170–190, 2019

    Lina Ma, Xiantao Li, and Chun Liu. Coarse-graining langevin dynamics using reduced- order techniques.Journal of Computational Physics, 380:170–190, 2019

    2019

  30. [38]

    A temperature accelerated method for sam- pling free energy and determining reaction pathways in rare events simulations.Chemical Physics Letters, 426(1):168 – 175, 2006

    Luca Maragliano and Eric Vanden-Eijnden. A temperature accelerated method for sam- pling free energy and determining reaction pathways in rare events simulations.Chemical Physics Letters, 426(1):168 – 175, 2006

    2006

  31. [39]

    Settle: An analytical version of the shake and rattle algorithm for rigid water models.Journal of Computational Chemistry, 13(8):952– 962, 1992

    Shuichi Miyamoto and Peter A Kollman. Settle: An analytical version of the shake and rattle algorithm for rigid water models.Journal of Computational Chemistry, 13(8):952– 962, 1992

    1992

  32. [40]

    Transport, collective motion, and Brownian motion.Progress of Theoretical Physics, 33(3):423–455, 1965

    Hazime Mori. Transport, collective motion, and Brownian motion.Progress of Theoretical Physics, 33(3):423–455, 1965

    1965

  33. [41]

    Polymorphic transitions in single crystals: A new molecular dynamics method.Journal of Applied Physics, 52(12):7182–7190, 1981

    Michele Parrinello and Aneesur Rahman. Polymorphic transitions in single crystals: A new molecular dynamics method.Journal of Applied Physics, 52(12):7182–7190, 1981

    1981

  34. [42]

    Deep learning as closure for irreversible processes: A data-driven generalized Langevin equation.arXiv preprint arXiv:1903.09562, 2019

    Antonio Russo, Miguel A Durán-Olivencia, Ioannis G Kevrekidis, and Serafim Kalliadasis. Deep learning as closure for irreversible processes: A data-driven generalized Langevin equation.arXiv preprint arXiv:1903.09562, 2019

    1903 arXiv

  35. [43]

    Data-driven construction of stochastic reduced dy- namics encoded with non-markovian features.The Journal of Chemical Physics, 158(3), 2023

    Zhiyuan She, Pei Ge, and Huan Lei. Data-driven construction of stochastic reduced dy- namics encoded with non-markovian features.The Journal of Chemical Physics, 158(3), 2023

    2023

  36. [44]

    Adaptive deep density approximation for fokker-planck equations.Journal of Computational Physics, 457:111080, 2022

    Kejun Tang, Xiaoliang Wan, and Qifeng Liao. Adaptive deep density approximation for fokker-planck equations.Journal of Computational Physics, 457:111080, 2022. 22

    2022

  37. [45]

    Adversarial adaptive sam- pling: Unify pinn and optimal transport for the approximation of pdes.arXiv preprint arXiv:2305.18702, 2023

    Kejun Tang, Jiayu Zhai, Xiaoliang Wan, and Chao Yang. Adversarial adaptive sam- pling: Unify pinn and optimal transport for the approximation of pdes.arXiv preprint arXiv:2305.18702, 2023

    2023 arXiv

  38. [46]

    Torrie and J.P

    G.M. Torrie and J.P. Valleau. Nonphysical sampling distributions in monte carlo free- energy estimation: Umbrella sampling.Journal of Computational Physics, 23(2):187–199, 1977

    1977

  39. [47]

    Plumed 2: New feathers for an old bird.Computer physics communications, 185(2):604–613, 2014

    Gareth A Tribello, Massimiliano Bonomi, Davide Branduardi, Carlo Camilloni, and Gio- vanni Bussi. Plumed 2: New feathers for an old bird.Computer physics communications, 185(2):604–613, 2014

    2014

  40. [48]

    Likelihood-based non-markovian models from molecular dynamics.Pro- ceedings of the National Academy of Sciences, 119(13):e2117586119, 2022

    Hadrien Vroylandt, Ludovic Goudenège, Pierre Monmarché, Fabio Pietrucci, and Ben- jamin Rotenberg. Likelihood-based non-markovian models from molecular dynamics.Pro- ceedings of the National Academy of Sciences, 119(13):e2117586119, 2022

    2022

  41. [49]

    Position-dependent memory kernel in gener- alized Langevin equations: Theory and numerical estimation.The Journal of Chemical Physics, 156(24):244105, 06 2022

    Hadrien Vroylandt and Pierre Monmarché. Position-dependent memory kernel in gener- alized Langevin equations: Theory and numerical estimation.The Journal of Chemical Physics, 156(24):244105, 06 2022

    2022

  42. [50]

    Ab initio generalized langevin equation.Pro- ceedings of the National Academy of Sciences, 121(14):e2308668121, 2024

    Pinchen Xie, Roberto Car, and Weinan E. Ab initio generalized langevin equation.Pro- ceedings of the National Academy of Sciences, 121(14):e2308668121, 2024

    2024

  43. [51]

    Coarse-graining conformational dynamics with multidimen- sional generalized langevin equation: How, when, and why.Journal of Chemical Theory and Computation, 20, 2024

    Pinchen Xie and Weinan E. Coarse-graining conformational dynamics with multidimen- sional generalized langevin equation: How, when, and why.Journal of Chemical Theory and Computation, 20, 2024

    2024

  44. [52]

    Learning stochastic dynamics with statistics-informed neural network.Journal of Computational Physics, 474:111819, 2023

    Yuanran Zhu, Yu-Hang Tang, and Changho Kim. Learning stochastic dynamics with statistics-informed neural network.Journal of Computational Physics, 474:111819, 2023

    2023

  45. [53]

    Faber approximation of the Mori-Zwanzig equation

    Yuanran Zhu and Daniele Venturi. Faber approximation of the Mori-Zwanzig equation. Journal of Computational Physics, 372:694–718, 2018

    2018

  46. [54]

    Generalized langevin equations for systems with local interactions.Journal of Statistical Physics, 178:1217, 2020

    Yuanran Zhu and Daniele Venturi. Generalized langevin equations for systems with local interactions.Journal of Statistical Physics, 178:1217, 2020

    2020

  47. [55]

    R. Zwanzig. Statistical mechanics of irreversiblity.Lectures in Theoretical Physics, 3:106– 141, 1961. 23

    1961

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Reviewed August 4, 2026 · model on record in the stance chip above.