Pith. sign in

REVIEW 4 major objections 6 minor 53 references

Uncertainty Quantification for Free Energy Calculations by Generalized Hierarchical Bayesian Inference

T0 review · 4 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Generalized hierarchical Bayesian inference makes free-energy uncertainty estimates adapt to the amount and quality of simulation data.

desk verdict Hierarchical GP free-energy UQ that propagates hyperparameter uncertainty is a real practical step forward, but the signature claim—uncertainty tracks error—rests on visual heatmaps and needs calibration-style evidence before it should be cited as established. read the letter →

arxiv 2607.22338 v1 pith:UMFSPCNP submitted 2026-07-24 physics.chem-ph

Martin Skorna , Adam Gottfried , Zuzana Janackova , Katarina Baxova , Pavel Jungwirth , Brennon L. Shanks This is my paper
classification physics.chem-ph
keywords freeenergycalculationsuncertaintyquantificationGaussianprocessregressionhierarchicalBayesianinferenceumbrellasamplingmetadynamicshyperparameterpeptide-lipidmembraneinteractions
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

The paper sets out to show that standard ways of putting error bars on computed free energy profiles — umbrella integration with block averaging, and Gaussian processes with fixed hyperparameters — produce uncertainty estimates that do not respond to how much simulation data was collected. The proposed fix is a generalized hierarchical Gaussian process — a probability distribution over smooth free energy surfaces — in which the kernel hyperparameters and observation noise are treated as unknowns, given priors, and sampled by Hamiltonian Monte Carlo rather than fixed or optimized to a point. On a 100-point data-ablation grid for a peptide–membrane system and on a metadynamics trajectory scan, the hierarchical model's predictive standard deviation tracks the actual reconstruction error, while the baselines either stay nearly constant or collapse abruptly. This matters because a trustworthy convergence signal is what would let automated free energy workflows stop sampling once the answer is known well enough.

What carries the argument

The central object is a generalized hierarchical Gaussian process for the free energy profile: a zero-mean GP with a squared-exponential kernel (a covariance function controlling smoothness and correlation length), fed by both histogram free-energy observations and derivative (mean-force) observations in a joint function–derivative observation model. The kernel length scale, kernel amplitude, and the function and derivative noise variances are given log-normal hyperpriors and sampled from a generalized hyperposterior using Hamiltonian Monte Carlo; the likelihood is a leave-one-out pseudo-likelihood. Hyperparameter samples are propagated into the predictive distribution through the law of tot

What would settle it

Run repeated independent enhanced-sampling simulations on a system with a known free energy surface (for example a double well with a narrow barrier), apply the same trajectory/window ablation, and compute the empirical coverage of the reported ±1σ credible intervals. If the intervals contain the true profile in substantially fewer than about 68% of reconstructions in the low-data cells, the central claim that the uncertainty tracks real error is falsified.

Watch

Extended reading notes

Core claim

The central claim is that predictive uncertainty in Gaussian-process free energy reconstruction is trustworthy only when the sources of model uncertainty — the GP hyperparameters and the observation noise — are marginalized over rather than fixed in advance or collapsed to a maximum-a-posteriori point. The evidence is a data-ablation study on the R9 peptide–membrane free energy profile and a metadynamics benchmark on phenol–membrane permeation. In both settings the hierarchical GP's predictive standard deviation evolves with the amount and quality of the simulation data and tracks the reconstruction RMSE (Pearson correlation 0.95 in the metadynamics case), whereas a fixed-hyperparameter GP g

Load-bearing premise

The free energy profile is modeled as a zero-mean Gaussian process with a stationary squared-exponential covariance, meaning a single correlation length applies across the whole collective-variable domain; if the true profile has different smoothness around barriers versus basins, the inferred noise will absorb the mismatch and the reported uncertainties may not track true error.

Editorial extensions

If this is right

  • A hyperparameter-sampled GP provides a usable convergence signal: its credible intervals narrow as trajectory length and window count increase, letting users distinguish a converged profile from an under-sampled one.
  • Fixed-hyperparameter GP uncertainty stays essentially constant across the ablation grid, so it cannot indicate whether additional sampling improved the reconstruction.
  • Umbrella integration with block averaging is underconfident in moderate-data regimes and collapses abruptly once data are nearly complete, making it a misleading convergence diagnostic.
  • Optimizing hyperparameters rather than sampling them underdisperses the predictive uncertainty by about 18% on average in the low-data setting, and full hyperposterior propagation corrects the overconfidence.
  • The framework transfers from umbrella sampling to extended Lagrangian metadynamics, where the hierarchical GP's average uncertainty tracks RMSE (correlation 0.95), suggesting it applies to enhanced-sampling methods generally.

Reading between the lines

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

  • If the demonstrated convergence signal holds beyond these two systems, the method could serve as a stopping rule in automated adaptive sampling, halting simulations once credible-interval widths fall below a target — a step the paper motivates but does not itself implement.
  • The stationary squared-exponential kernel is the premise most likely to fail first: real free energy surfaces may have different correlation lengths near barriers than in basins, and a non-stationary kernel comparison on a profile with known narrow barriers would reveal whether inferred noise absorbs the mismatch.
  • The 'only approach whose uncertainty adapted' claim is made relative to the three baselines tested; a sharper test would pit the hierarchical GP against other adaptive UQ schemes on the same ablation grid.
  • The leave-one-out-versus-marginal-likelihood finding suggests that for uncertainty-focused tasks, predictive objectives should be preferred over fit-based objectives; this lesson plausibly transfers to other Gaussian-process applications in computational chemistry.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper develops a hierarchical Bayesian Gaussian process (GP) framework for uncertainty quantification in free energy calculations. A zero-mean GP with squared-exponential kernel models the free energy profile from joint histogram and derivative observations, using a leave-one-out pseudo-likelihood and Hamiltonian Monte Carlo (HMC-NUTS) to sample the hyperposterior of kernel length scale, amplitude, and noise hyperparameters. Predictive uncertainty is obtained by propagating hyperparameter samples through the GP predictive equations, thereby including hyperparameter uncertainty. The method is applied to umbrella sampling data for an R9 peptide–membrane system and to extended Lagrangian metadynamics data for phenol–membrane interactions, with data-ablation studies comparing the proposed method against umbrella integration with block averaging, a fixed-hyperparameter GP, and a MAP-optimized GP. The central claim is that the hierarchical GP is the only approach whose uncertainty estimates consistently adapt to the amount and quality of simulation data, as assessed by visual heatmaps of average RMSE and average predictive standard deviation and by a Pearson correlation in the metadynamics case.

Significance. If the central claim is established, the method could provide a practically useful convergence signal for enhanced-sampling free energy calculations, with direct implications for automated workflows and for avoiding wasted simulation effort. The paper contains several strengths: the GP machinery is standard and correctly assembled; the HMC diagnostics (Section C) are thorough, including R-hat, ESS, and divergence counts across multiple data regimes; a hyperprior sensitivity analysis (Section B.2) is reported; and code and data are made available on GitHub and Zenodo. The authors also explicitly acknowledge the key limitation of using a stationary covariance kernel. However, the headline validation is currently incomplete: the umbrella-sampling evidence is qualitative, the metadynamics evidence rests on a single aggregate correlation, and no coverage or calibration analysis is reported. These gaps are load-bearing for the claim that the uncertainty estimates 'track' reconstruction errors, and they can be addressed within the manuscript's scope by adding pointwise calibration diagnostics and by making the model-selection procedure more transparent.

major comments (4)
  1. [§3.2, Fig. 3] The central claim that hierarchical GP uncertainty 'consistently adapted to the amount and quality of the available simulation data' is supported only by visual comparison of averaged RMSE heatmaps and averaged predictive standard deviation heatmaps. No pointwise calibration, empirical coverage, or standardized-residual analysis is reported. Because the method outputs a full predictive covariance, the paper should quantify, for example, the frequency with which the WHAM reference falls inside the 68% or 95% credible intervals as a function of CV position and data-ablation condition, or report z-scores (f_i - μ_i)/σ_i. Without such diagnostics, the observed correlation between global average SD and average RMSE does not establish that the intervals are locally trustworthy, especially near barriers where GP smoothness assumptions may fail.
  2. [§3.4, Fig. 5] The metadynamics support for the headline claim is a single Pearson correlation coefficient ρ=0.95 between average predictive standard deviation and average RMSE, computed over a small number of trajectory-length aggregates (the text lists 1% and then 'remaining trajectory lengths', but the figure appears to show five points). This is not a sufficient quantitative basis for the claim. The correlation has no associated uncertainty or significance test, it averages over all CV positions, and it does not assess whether the predictive intervals have correct coverage. The authors should report the number of points, a pointwise analysis, and preferably a calibration curve or coverage statistic.
  3. [§2.4, SI Sections D and F.2] There is a circularity concern between model selection and evaluation. The LOO pseudo-likelihood was selected over the standard marginal likelihood (SI Section D) because it 'more consistently tracked the observed accuracy' on the same RMSE-vs-SD ablation grids used to evaluate the method, and the metadynamics discretization parameters n_h=120 and n_d=40 were chosen partly to keep inferred noise hyperparameters 'of comparable magnitude' and to avoid unstable reconstructions on those same benchmarks. This does not invalidate the method, but it means the reported 'tracking' is partly a result of tuning the objective and preprocessing to the benchmark. To support the general claim, the authors should either report the selection procedure explicitly as exploratory, or validate the method on an independent system with pre-specified settings.
  4. [§2.3, Eq. (23); Discussion, §4] The stationary squared-exponential kernel and scalar inferred noise hyperparameters are acknowledged in the Discussion to be a potential source of model mismatch: 'there is no physical reason why a free energy profile should exhibit uniform correlation structure throughout an entire CV domain.' This is more than a conceptual limitation; it directly threatens the UQ claim. If local length scales vary near barriers or basins, the inferred scalar noise and length scale can absorb the mismatch, producing intervals that are locally over- or under-confident even if the average SD correlates with average RMSE. The paper should test this sensitivity, for example by comparing with a non-stationary kernel on at least one benchmark, or by reporting pointwise coverage in data-rich regions where hyperparameters are well identified. The current aggregate evaluation cannot detect such local failures.
minor comments (6)
  1. [§1, Introduction] Typo: 'mulistate Bennet acceptance ratio' should be 'multistate Bennett acceptance ratio'.
  2. [§2.1, Eq. (2)] The notation P_w(ξ_i) is used both for a histogram estimate and, later, for probability in the SI noise formulas. Clarifying this would avoid confusion.
  3. [§3.1, Table 1] The table caption says 'LOO MAP hyperparameter estimates' but the text refers to 'generalized hyperposterior MAP'. Please make the terminology consistent.
  4. [SI Section E] Typo: 'hyperparamter' should be 'hyperparameter'.
  5. [§3.2, Fig. 3] The heatmap color bars are on a logarithmic scale and the white threshold is defined as RMSE=5 kJ/mol / SD=5 kJ/mol. It would be helpful to state explicitly in the caption that the white regions denote values below the threshold, to avoid ambiguity.
  6. [§7, Data Availability] The GitHub and Zenodo links are a welcome addition. Please state the software version used (e.g., Pyro version) and any relevant dependencies, or point to an environment file, to improve reproducibility.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the LOO objective was selected for the uncertainty-tracking behavior later reported as the main result, while the fixed-GP/UI comparisons remain independent.

  1. fitted input called prediction [Section 2.4 and Supporting Information Section D, used as evidence for Section 5]
    "In the present work, the LOO objective also produced more informative uncertainty estimates than the standard GP marginal likelihood (Supporting Information Section D). ... Hyperposteriors constructed using the LOO pseudo-likelihood produced uncertainty estimates that more consistently tracked the observed accuracy of the reconstructed free energy surfaces across the data-ablation study."

    The main conclusion (Section 5) is that hierarchical GP with LOO is "the only approach whose uncertainty estimates consistently adapted" to data quantity/quality. That claim is supported by the data-ablation grid (Fig. 3). But the same grid was used in SI D to choose LOO over marginal likelihood on the grounds that LOO "more consistently tracked the observed accuracy" (RMSE vs SD). Therefore the LOO-vs-LML component of the result is not an independent prediction; it is the selection criterion applied to the same dataset. The comparisons against fixed-GP and UI baselines were not tuned in this way and retain independent content.

full rationale

The derivation chain is mostly self-contained. The GP predictive equations are standard (Rasmussen & Williams), the window-offset marginalization is credited to O'Hagan, the moment propagation (Eqs. 26-28) follows from the law of total variance, and the external R9 umbrella-sampling and phenol-metadynamics datasets provide test data not fitted to the target claim. The main circularity is the model-selection/evaluation loop: LOO was chosen over the marginal likelihood because it made predictive SD track RMSE better on the same T×W ablation grid that is later presented as evidence of the headline claim. This makes the LOO-specific component partly by construction. However, the structural comparisons—fixed GP with a constant SD vs hierarchical GP with data-dependent SD, and HMC hyperposterior propagation vs MAP—are not reduced to the evaluation metric, so the central claim has independent content. The self-citations (e.g., [24] for LOO in computational chemistry) support a peripheral modeling choice and are not load-bearing. No uniqueness theorem or ansatz is imported from the authors' prior work. Overall partial circularity, score 4.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The framework adds no new physical entities. It does rely on four data-fitted hyperparameters and two hand-tuned metadynamics discretization choices, plus the assumptions listed above. The core contribution is a statistical procedure, not a new law or particle.

free parameters (6)
  • GP length scale ℓ = MAP 0.48 nm; posterior median 0.0605 nm (7-window, 25%); fixed at π/2 in baseline
    Kernel hyperparameter governing smoothness; sampled from log-normal hyperprior and strongly affects UQ. Also hand-set in Appendix B.1 for fixed GP.
  • GP kernel amplitude w = MAP 7.51 kJ/mol; baseline fixed at 4.184√10 kJ/mol
    Amplitude of prior function variability; inferred from data.
  • Histogram noise hyperparameter σ_f = MAP 2.18 kJ/mol
    Diagonal noise assigned to histogram-derived free-energy observations; inferred, not fixed from simulation statistics (Section 2.3, Table 1).
  • Derivative noise hyperparameter σ_d = MAP 4.94 kJ/mol/nm
    Noise assigned to mean-force/derivative observations; inferred (Table 1).
  • Metadynamics histogram bins n_h = 120
    Discretization chosen after sensitivity scan; values below ~100 degrade predictions, above ~200 computationally prohibitive (Section F.2).
  • Metadynamics derivative windows n_d = 40
    Chosen so inferred σ_d and σ_f remain comparable across trajectory lengths; tuned on the same data used for validation (Section F.2).
assumptions (7)
  • domain assumption Free energy profile is a zero-mean GP with stationary squared-exponential kernel (Eq. 23)
    Assumes C∞ smoothness and location-independent correlation; acknowledged as a limitation in the Discussion.
  • domain assumption Histogram and derivative observations are Gaussian with independent diagonal noise (Eqs. 12, 21)
    Approximates log-histogram values and mean forces as Gaussian observations; neglects multinomial covariance between bins of the same umbrella window.
  • domain assumption LOO pseudo-likelihood is an appropriate objective for hyperparameter inference (Section 2.4)
    The sampled object is a pseudo-posterior; the method's UQ validity depends on LOO predictive performance being the right target.
  • domain assumption Hyperpriors on ℓ, w, σf, σd are weakly informative and not driving results (Appendix B)
    Sensitivity check only varies the length-scale prior; other hyperpriors are fixed broad log-normals.
  • domain assumption Full-data WHAM profile is an adequate reference for measuring reconstruction error
    All RMSE comparisons treat the full-dataset WHAM result as ground truth (Section 2.5).
  • standard math Standard GP predictive equations and total expectation/variance formulas (Eqs. 8-9, 13-14, 26-28)
    Background from Rasmussen & Williams [21], O'Hagan [20], Lalchand & Rasmussen [26].
  • standard math HMC-NUTS convergence (Rhat<1.01, ESS>200) implies representative hyperposterior samples
    Standard MCMC diagnostics; paper reports thorough assessment in Appendix C.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uncertainty Quantification for Free Energy Calculations by Generalized Hierarchical Bayesian Inference." pith.science (2026). https://pith.science/paper/UMFSPCNP

@misc{pith2026260722338,
  author       = {Pith},
  title        = {Pith review of: Uncertainty Quantification for Free Energy Calculations by Generalized Hierarchical Bayesian Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMFSPCNP}},
  note         = {Machine review of arXiv:2607.22338}
}
read the original abstract

Free energy calculations are routinely used to study molecular processes inaccessible to unbiased molecular dynamics, but their utility ultimately depends on knowing when and how much their predictions can be trusted. Uncertainty estimation is therefore essential for distinguishing genuine physical features of a free energy profile from artifacts arising from limited simulation data or inadequate sampling. Gaussian processes have emerged as a powerful framework for reconstructing free energy profiles together with predictive uncertainties. However, existing implementations typically condition on fixed hyperparameters and observation noise, preventing predictive uncertainties from adapting to the information content of the simulation data. Here, we develop a generalized hierarchical Gaussian process framework that accounts for these neglected sources of uncertainty. Applications to umbrella sampling and extended Lagrangian metadynamics of peptide-lipid membrane interactions demonstrate that the resulting uncertainty estimates track reconstruction errors across a wide range of sampling and data conditions.

Figures

Figures reproduced from arXiv: 2607.22338 by the authors.

Figure 1
Figure 1. Flowchart summarizing the workflow of the generalized hierarchical Bayesian framework for free energy reconstruction. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Comparison between fixed and inferred noise hyperparameter models. (a) Corner plot of fixed noise hyperposterior with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Data ablation heatmap grids. Columns represent di [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Barrier-height distributions for three PMF estimation [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Metadynamics GP-based free energy reconstruction [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Length-scale hyperprior sensitivity analysis. (a) Poste [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: NUTS chain tracking for the super-hard data case (3 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Mean RMSE and SD ablation grid heat maps compar [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Percentage difference (∆) in mean RMSE and stan￾dard deviation between propagated hyperposteriors and MAP estimation. Blue indicates a decrease in the plotted quantity and red indicates an increase. We note that, for the systems considered in this work, the addi￾tional…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

53 extracted references · 1 canonical work pages

  1. [1]

    M.; Valleau, J

    Torrie, G. M.; Valleau, J. P. Monte Carlo free energy esti- mates using non-Boltzmann sampling: Application to the sub-critical Lennard-Jones fluid.Chem. Phys. Lett.1974, 28, 578–581

    1974

  2. [2]

    M.; Bouzida, D.; Swend- sen, R

    Kumar, S.; Rosenberg, J. M.; Bouzida, D.; Swend- sen, R. H.; Kollman, P. A. The weighted histogram analy- sis method for free-energy calculations on biomolecules.J. Comput. Chem.1992,13, 1011–1021

    1992

  3. [3]

    Umbrella integration

    Kästner, J.; Thiel, W. Bridging the gap between thermody- namic integration and umbrella sampling provides a novel analysis method: “Umbrella integration”.J. Chem. Phys. 2005,123, 144104

    2005

  4. [4]

    Escaping free-energy minima

    Laio, A.; Parrinello, M. Escaping free-energy minima. Proc. Natl. Acad. Sci. U.S.A.2002,99, 12562–12566

    2002

  5. [5]

    Laio, A.; Gervasio, F. L. Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science.Rep. Prog. Phys.2008,71, 126601

    2008

  6. [6]

    Shirts, M.; Chodera, J. D. Statistically optimal analysis of samples from multiple equilibrium states.J. Chem. Phys. 2008,129, 124105

    2008

  7. [7]

    Accelerated weight his- togram method for exploring free energy landscapes.J

    Lindahl, V .; Lidmar, J.; Hess, B. Accelerated weight his- togram method for exploring free energy landscapes.J. Chem. Phys.2014,141, 044110

    2014

  8. [8]

    C.; Henin, J.; Lelievre, T.; Poho- rille, A.; Chipot, C

    Comer, J.; Gumbart, J. C.; Henin, J.; Lelievre, T.; Poho- rille, A.; Chipot, C. The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid to Ask.J. Phys. Chem. B2015,119, 1129–1151

Show all 53 references
  1. [9]

    York, D. M. Modern Alchemical Free Energy Methods for Drug Discovery Explained.ACS Phys. Chem. Au2023,3, 478–491

  2. [10]

    Analysis of the statistical error in umbrella sampling simulations by umbrella integration.J

    Kästner, J.; Thiel, W. Analysis of the statistical error in umbrella sampling simulations by umbrella integration.J. Chem. Phys.2006,124, 234106

    2006

  3. [11]

    Convergence and error estimation in free energy calculations using the weighted histogram analysis method.J

    Zhu, F.; Hummer, G. Convergence and error estimation in free energy calculations using the weighted histogram analysis method.J. Comput. Chem.2012,33, 453–465

    2012

  4. [12]

    Free Energy Surface Reconstruction from Umbrella Samples Using Gaussian Process Regression.J

    Stecher, T.; Bernstein, N.; Csányi, G. Free Energy Surface Reconstruction from Umbrella Samples Using Gaussian Process Regression.J. Chem. Theory Comput.2014,10, 4079–4097

    2014

  5. [13]

    Exploration, Sam- pling, And Reconstruction of Free Energy Surfaces with Gaussian Process Regression.J

    Mones, L.; Bernstein, N.; Csányi, G. Exploration, Sam- pling, And Reconstruction of Free Energy Surfaces with Gaussian Process Regression.J. Chem. Theory Comput. 2016,12, 5100–5110

    2016

  6. [14]

    Bayesian Multistate Bennett Acceptance Ratio Methods.J

    Ding, X. Bayesian Multistate Bennett Acceptance Ratio Methods.J. Chem. Theory Comput.2024,20, 1878–1888

    2024

  7. [15]

    Dai, J.; Krems, R. V . Interpolation and extrapolation of global potential energy surfaces for polyatomic systems by Gaussian processes with composite kernels.J. Chem. Theory Comput.2020,16, 1386–1395

    2020

  8. [16]

    Sugisawa, H.; Ida, T.; Krems, R. V . Gaussian process model of 51-dimensional potential energy surface for proto- nated imidazole dimer.J. Chem. Phys.2020,153, 114101

    2020

  9. [17]

    K.; Ortíz, A

    Kempkes, E. K.; Ortíz, A. P. d. A. Bayesian umbrella quadrature accelerates free-energy calculations across di- verse molecular systems and processes.J. Chem. Theory Comput.2026,

    2026

  10. [18]

    Good Practices in Free-Energy Calculations.J

    Pohorille, A.; Jarzynski, C.; Chipot, C. Good Practices in Free-Energy Calculations.J. Phys. Chem. B2010,114, 10235–10253

  11. [19]

    Ferguson, A. L. BayesWHAM: A Bayesian approach for free energy estimation, reweighting, and uncertainty quan- tification in the weighted histogram analysis method.J. Comput. Chem.2017,38, 1583–1605

    2017

  12. [20]

    Curve Fitting and Optimal Design for Predic- tion.J

    O’Hagan, A. Curve Fitting and Optimal Design for Predic- tion.J. R. Stat. Soc. Ser . B1978,40, 1–24

  13. [21]

    E.; Williams, C

    Rasmussen, C. E.; Williams, C. K. I.Gaussian Processes for Machine Learning; MIT Press, 2006

    2006

  14. [22]

    Sundararajan, S.; Keerthi, S. S. Predictive approaches for choosing hyperparameters in gaussian processes.Neural Comput.2001,13, 1103–1118

    2001

  15. [23]

    P.; Kermode, J

    Bartók, A. P.; Kermode, J. R. Improved uncertainty quan- tification for Gaussian process regression based inter- atomic potentials.2022, arXiv:2206.08744

    2022 arXiv

  16. [24]

    L.; Sullivan, H

    Shanks, B. L.; Sullivan, H. W.; Shazed, A. R.; Hoepfner, M. P. Accelerated Bayesian inference for molec- ular simulations using local Gaussian process surrogate models.J. Chem. Theory Comput.2024,20, 3798–3808

    2024

  17. [25]

    P.; Jankowiak, M.; Obermeyer, F.; Pradhan, N.; Karaletsos, T.; Singh, R.; Szerlip, P.; Hors- fall, P.; Goodman, N

    Bingham, E.; Chen, J. P.; Jankowiak, M.; Obermeyer, F.; Pradhan, N.; Karaletsos, T.; Singh, R.; Szerlip, P.; Hors- fall, P.; Goodman, N. D. Pyro: Deep Universal Probabilis- tic Programming.J. Mach. Learn. Res.2019,20, 1–6

    2019

  18. [26]

    Lalchand, V .; Rasmussen, C. E. Approximate Inference for Fully Bayesian Gaussian Process Regression.PMLR2020, 1–12

  19. [27]

    L.; Morandi, M

    Baxová, K. L.; Morandi, M. I.; Scher, N.; Kula, P.; Tichacek, O.; Schachter, I.; Busko, P.; Zahrad- nik, J.; Vazdar, M.; Koikkara, J.; Allolio, C.; Avi- noam, O.; Jungwirth, P. Direct Membrane Penetration of Oligoarginines by Fluorescence and Cryo-electron Mi- croscopy Combine...

    2026 doi

  20. [28]

    W.; Cervenka, M.; Shanks, B

    Sullivan, H. W.; Cervenka, M.; Shanks, B. L.; Hoepfner, M. P. Physics-Informed Gaussian Process In- ference of Liquid Structure from Scattering Data.J. Phys. Chem. B2025,129, 11802–11815

  21. [29]

    Non-stationary Gaussian process regres- sion with Hamiltonian Monte Carlo.PMLR2016, 732– 740

    Heinonen, M.; Mannerström, H.; Rousu, J.; Kaski, S.; Lähdesmäki, H. Non-stationary Gaussian process regres- sion with Hamiltonian Monte Carlo.PMLR2016, 732– 740

  22. [30]

    L.; Sullivan, H

    Shanks, B. L.; Sullivan, H. W.; Hoepfner, M. P. Bayesian Analysis Reveals the Key to Extracting Pair Potentials Preprint– UncertaintyQuantification forFreeEnergyCalculations byGeneralizedHierarchicalBayesianInference17 from Neutron Scattering Data.J. Phys. Chem. Lett.2024, 15,...

    2024

  23. [31]

    E.; Chamorro, V

    Fan, S.; Mason, P. E.; Chamorro, V . C.; Shanks, B. L.; Martinez-Seara, H.; Jungwirth, P. Charge Scaling Force Field for Biologically Relevant Ions Utilizing a Global Optimization Method.J. Chem. Theory Comput.2025,21, 9023–9034

    2025

  24. [32]

    L.; Jungwirth, P.; Martinez-Seara, H

    Kostal, V .; Shanks, B. L.; Jungwirth, P.; Martinez-Seara, H. Bayesian Learning for Accurate and Robust Biomolecular Force Fields.J. Chem. Theory Comput.2026,22, 2652– 2663

    2026

  25. [33]

    P.; Payne, M

    Bartók, A. P.; Payne, M. C.; Kondor, R.; Csányi, G. Gaus- sian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons.Phys. Rev. Lett.2010, 104, 136403

    2010

  26. [34]

    L.; Potoff, J

    Shanks, B. L.; Potoff, J. J.; Hoepfner, M. P. Transferable Force Fields from Experimental Scattering Data with Ma- chine Learning Assisted Structure Refinement.J. Phys. Chem. Lett.2022,13, 11512–11520

    2022

  27. [35]

    L.; Sullivan, H

    Shanks, B. L.; Sullivan, H. W.; Jungwirth, P.; Hoepfner, M. P. Experimental evidence of quantum Drude oscillator behavior in liquids revealed with probabilistic iterative Boltzmann inversion.J. Chem. Phys.2025,162, 164501

    2025

  28. [36]

    G.; Im, W

    Jo, S.; Kim, T.; Iyer, V . G.; Im, W. CHARMM-GUI: A web- based graphical user interface for CHARMM.J. Comput. Chem.2008,29, 1859–1865

    2008

  29. [37]

    L.; Cheng, X.; Jo, S.; Rui, H.; Song, K

    Wu, E. L.; Cheng, X.; Jo, S.; Rui, H.; Song, K. C.; Dávila- Contreras, E. M.; Qi, Y .; Lee, J.; Monje-Galvan, V .; Ven- able, R. M.; Klauda, J. B.; Im, W. CHARMM-GUI Mem- brane Builder toward realistic biological membrane simu- lations.J. Comput. Chem.2014,35, 1997–2004

    2014

  30. [38]

    B.; Klauda, J

    Jo, S.; Lim, J. B.; Klauda, J. B.; Im, W. CHARMM-GUI Membrane Builder for Mixed Bilayers and Its Application to Yeast Membranes.Biophys. J.2009,97, 50–58

    2009

  31. [39]

    J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J

    Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers.SoftwareX 2015,1-2, 19–25

    2015

  32. [40]

    J.; Kutzner, C.; Hess, B.; Lindahl, E

    Páll, S.; Abraham, M. J.; Kutzner, C.; Hess, B.; Lindahl, E. Solving Software Challenges for Exascale; Springer Inter- national Publishing, 2015; pp 3–27

    2015

  33. [41]

    E.; Berendsen, H

    Van Der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark, A. E.; Berendsen, H. J. C. GROMACS: Fast, flexible, and free.J. Comput. Chem.2005,26, 1701–1718

    2005

  34. [42]

    Huang, J.; MacKerell Jr, A. D. CHARMM36 all-atom ad- ditive protein force field: Validation based on comparison to NMR data.J. Comput. Chem.2013, 2135–2145

    2013

  35. [43]

    Brooks, B. R. et al. CHARMM: The biomolecular simula- tion program.J. Comput. Chem.2009,30, 1545–1614

    2009

  36. [44]

    Lee, J. et al. CHARMM-GUI Input Generator for NAMD, GROMACS, AMBER, OpenMM, and CHARMM/OpenMM Simulations Using the CHARMM36 Additive Force Field.J. Chem. Theory Comput.2016,12, 405–413

    2016

  37. [45]

    L.; Lee, H

    Kim, S.; Lee, J.; Jo, S.; Brooks, C. L.; Lee, H. S.; Im, W. CHARMM-GUI ligand reader and modeler for CHARMM force field generation of small molecules: CHARMM-GUI Ligand Reader and Modeler for CHARMM Force Field Generation of Small Molecules.J. Comput. Chem.2017, 38, 1879–1886

    2017

  38. [46]

    Bonomi, M. et al. Promoting transparency and repro- ducibility in enhanced molecular simulations.Nat. Meth- ods2019,16, 670–673

  39. [47]

    A.; Bonomi, M.; Branduardi, D.; Camil- loni, C.; Bussi, G

    Tribello, G. A.; Bonomi, M.; Branduardi, D.; Camil- loni, C.; Bussi, G. PLUMED 2: New feathers for an old bird.Comput. Phys. Commun.2014,185, 604–613

    2014

  40. [48]

    Canonical sam- pling through velocity rescaling.J

    Bussi, G.; Donadio, D.; Parrinello, M. Canonical sam- pling through velocity rescaling.J. Chem. Phys.2007,126, 014101

    2007

  41. [49]

    Pressure control using stochastic cell rescaling.The J

    Bernetti, M.; Bussi, G. Pressure control using stochastic cell rescaling.The J. Chem. Phys.2020,153, 114107

    2020

  42. [50]

    Experiments

    Verlet, L. Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Phys. Rev.1967,159, 98–103

    1967

  43. [51]

    Particle mesh Ewald: AnNlog(N) method for Ewald sums in large systems.J

    Darden, T.; York, D.; Pedersen, L. Particle mesh Ewald: AnNlog(N) method for Ewald sums in large systems.J. Chem. Phys.1993,98, 10089–10092

    1993

  44. [52]

    Hess, B.; Bekker, H.; Berendsen, H. J. C.; Fraaije, J. G. E. M. LINCS: A linear constraint solver for molecular simulations.J. Comput. Chem.1997,18, 1463–1472

    1997

  45. [53]

    Miyamoto, S.; Kollman, P. A. Settle: An analytical version of the SHAKE and RATTLE algorithm for rigid water models.J. Comput. Chem.1992,13, 952–962

    1992

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.