VESPER: Online Learning of Free Energies with a Multi-Tempered Variational Objective

Jay L. Kaplan; Gustavo R. Pérez-Lemus; Mengjian Hua; Juan J. de Pablo; Eric Vanden-Eijnden

physics.chem-phOct 30, 2026

VESPER: Online Learning of Free Energies with a Multi-Tempered Variational Objective

Authors: Jay L. Kaplan, Gustavo R. Pérez-Lemus, Mengjian Hua, Juan J. de Pablo, Eric Vanden-Eijnden

Organizations: Courant Institute School of Mathematics, Computing, and Data Science, New York University, New York, NY 10012, USA · Department of Chemical and Biomolecular Engineering, Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA · Department of Physics, New York University, New York, NY 10012, USA · Machine Learning Lab, Capital Fund Management

Abstract

Computing molecular free-energy landscapes over collective variables (CVs) is central to understanding molecular mechanisms, but poses a difficult learning problem without training data available a priori. Samples of the marginal CV distribution must be generated by molecular dynamics (MD), whose trajectories can remain trapped in metastable states and rarely cross the free-energy barriers governing conformational changes or chemical reactions. We introduce VESPER, which couples the generation of these samples to neural-network-driven free-energy learning: MD supplies configurations for learning, while the evolving model biases the MD to accelerate exploration and generate further training data. VESPER minimizes a single variational objective that is convex in the free energy and can combine several tempering levels, each flattening the free-energy landscape to promote barrier crossing. Every level has the same physical free energy as its minimizer, allowing data from all levels to train one shared model without reweighting. Learning uses the difference in mean parameter gradients between biased MD configurations and auxiliary walkers that explore the flattened model in CV space. We prove conditional convergence of the coupled overdamped population dynamics without replica exchange under log-Sobolev and Polyak-Lojasiewicz assumptions. VESPER matches or improves on established enhanced-sampling methods on alanine dipeptide and yields self-consistent free-energy reconstructions from three to six CVs on alanine tetrapeptide. For methane dehydrogenation on Ni(511) with a machine-learned interatomic potential, it recovers two-CV Boltzmann marginals and a reactant-to-saddle free-energy difference along a three-CV path, both qualitatively consistent with the lower-dimensional reference. It also samples RMSD-defined folded and unfolded regions of chignolin while biasing seventeen CVs.

Figures & tables

Figure 1: VESPER couples sampling and learning. Each tempering level $k$ carries two walker populations, and all levels share one model $G_{\theta}$ . Left: physical walkers evolve in configuration space under $U(x)-(1-\alpha_{k})G_{\theta}(s(x))$ and are mapped to CV space by $s(x)$ . Replica exchange swaps configurations between adjacent levels. Right: auxiliary walkers evolve directly in CV space under $\alpha_{k}G_{\theta}(s)$ . The two populations supply the averages in equation 10 , whose $c_{k}$ -weighted difference updates $\theta$ . The updated model then sets the forces on both populations. At convergence, both populations at level $k$ sample the same flattened landscape $\alpha_{k}G_{\star}$ . During learning, the instantaneous target for the physical CVs is $Q_{G_{\theta},k}\propto p\,e^{(\beta-\hat{\beta}_{k})G_{\theta}}$ .

Figure 2: Analytic landscapes. (A) Full-evaluation-domain RMSE in $k_{B}T$ and (B) dimensionless $D_{\mathrm{KL}}(\rho\,\|\,\rho_{\theta})$ against reduced time on Müller–Brown. Curves show means for single levels and a five-level ladder with (L-RE) and without (L-NRE) exchange, with $1{,}000$ walkers per population. (C) Reference and learned surfaces at $t=80$ on a common $0$ – $20\,k_{B}T$ scale. Red and blue dots are physical and auxiliary walkers. (D–F) Four-well landscape in $d=2$ – $100$ coordinates, with fixed hidden width and primary walker budget: (D) final SWA full-coordinate NRMSE, (E) $D_{\mathrm{KL}}(\rho\,\|\,\rho_{\theta})$ , and (F) reduced time to sustained EMA NRMSE below $0.25$ . Points and error bars show means and sample SDs. Open circles in (D,E) mark values above the plotted range or non-finite divergences. Open markers in (F) indicate at least one run without a sustained crossing.

Figure 3: Alanine dipeptide. (A) RMSE against the reference surface [ Zubieta Rico et al., 2025 ] over 0–10 ns per replica, averaged over three independent runs. VESPER curves are solid and baselines dashed. Gray shading marks errors above the chemical accuracy of 1 kcal mol -1 (4.184 kJ mol -1 ). L-RE, L-NRE: four-level ladders with and without exchange; OPES: on-the-fly probability enhanced sampling [ Invernizzi & Parrinello, 2020 ] ; MetaD: metadynamics [ Laio & Parrinello, 2002 ] ; VES: neural variationally enhanced sampling [ Valsson & Parrinello, 2014 , Bonati et al., 2019 ] ; TAMD: temperature-accelerated MD [ Maragliano & Vanden-Eijnden, 2006 , Bajpai et al., 2023 ] ; CFF: combined force-frequency sampling [ Sevgen et al., 2020 ] ; S-CFF: its SIREN variant [ Zubieta Rico et al., 2025 ] . (B) Reference (top left), and VESPER surfaces for $\alpha=1$ (top right), a single level at $\alpha=0.01$ (bottom left) and the exchange ladder (bottom right), on a common scale. Red and blue dots are physical and auxiliary walkers. (C) ADP with $\phi,\psi$ labeled [ Humphrey et al., 1996 ] . (D) $\phi$ of the physical walkers over 1–10 ns, colored by the learned free energy.

Figure 4: Alanine tetrapeptide: consistency across CV dimension. Two-dimensional Boltzmann marginals of (A) the three-CV and (B) the six-CV model, each from the mean surface of the four-level ladder with exchange. Columns: $(\phi_{1},\phi_{2})$ , $(\phi_{1},\phi_{3})$ , $(\phi_{2},\phi_{3})$ . (C) ATP with $\phi$ bonds in blue and $\psi$ bonds in red.

Figure 5: Methane on Ni(511): three-CV free energy. (A–C) Two-dimensional Boltzmann marginals in the planes (C–Ni, C–H coordination), ( $S_{1}$ , C–H coordination) and ( $S_{1}$ , C–Ni coordination). White line: bounded path on the mean surface obtained by a string-method calculation [ E et al., 2007 ] followed by constrained gradient-flow refinement, from reactant (1) through the saddle (3) to product (5). (D) Trajectory frames corresponding to the labeled points.

Figure 6: Chignolin with seventeen dihedral CVs (four independent runs, twelve levels from $\alpha=1$ to $1/18$ ). (A) Mean number of levels holding folded (C α RMSD below $0.20$ nm), intermediate (labeled Partial) or unfolded (above $0.40$ nm) configurations. Shading shows the minimum-to-maximum range across runs. (B) Mean folded fraction at each tempering level. In A and B, occupancies are averaged in $0.2$ ns bins and across the four runs. Both panels display the first $25$ ns per replica. (C) Free-energy projection at the end of the run from unbiased-level auxiliary walkers reconstructed on a fixed backbone, in C $\alpha$ radius of gyration and RMSD (dots: MD frames, red, and auxiliary walkers, blue, from all levels). (D) Dihedral marginal in $(\psi_{3},\psi_{4})$ from the same unbiased-level auxiliary walkers. Contours every $5$ kJ mol -1 .

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

System	$\alpha$	$K$	Physical	Auxiliary
ADP	$0.1,\,0.05,\,0.02,\,0.01$	4	1	$10^{4}$
ATP, $D=3,\,6$	$0.1,\,0.05,\,0.02,\,0.01$	4	1	$2\times 10^{4}$
Methane/Ni(511)	$0.05$	1	1	$6.5\times 10^{4}$
Chignolin	$18^{-k/11}$ , $k=0,\dots,11$	12	1	$10^{5}$

Appendix

Table 1: Levels per molecular system: tempering exponents

\alpha

, number of levels

K

, and walkers per level.

System	Method	$K$	ns/replica	aggregate ns	Runs
ADP	VESPER, $\alpha=0.01$ or $1$	1	10	10	3
ADP	VESPER ladders (L-RE, L-NRE)	4	10	40	3
ADP	Each baseline	1	10	10	3
ATP, $D=3,\,6$	VESPER, $\alpha=0.01$	1	10	10	3
ATP, $D=3,\,6$	VESPER ladders (L-RE, L-NRE)	4	10	40	3
Methane/Ni(511)	VESPER, $\alpha=0.05$	1	7.5	7.5	3

Appendix

Table 2: Replicas

K

, simulated time per replica at the readout, and aggregate MD per run (

K

times the time per replica).

System	Quantity	Value
ADP	RMSE, VESPER $\alpha=1$ (kJ mol -1 )	$13.27\pm 0.26$
ADP	RMSE, VESPER $\alpha=0.01$	$2.19\pm 0.02$
ADP	RMSE, L-RE / L-NRE	$2.02\pm 0.24$ / $2.05\pm 0.09$
ADP	RMSE, MetaD/VES/S-CFF	$2.61\pm 0.22$ / $4.72\pm 0.25$ / $2.04\pm 0.51$
ATP	$r^{2}$ , 3-CV vs 6-CV marginals, three planes	$0.951$ / $0.895$ / $0.890$
ATP	RMSE of the same planes (kJ mol -1 )	$5.5$ / $8.1$ / $7.5$

Appendix

Table 3: Values reported in the main text for the molecular systems. ADP: 10 ns per replica, SWA8 readout, three independent runs. ATP: 10 ns, SWA10, three independent runs, mean surfaces for the plane comparisons. Methane: 7.5 ns, EMA readout, three independent runs. Its

1.12\pm 0.25

eV entry is the route value on the mean surface with the sample SD evaluated at fixed route points. Chignolin: event counts use the first 25 ns per replica and the unbiased-level occupancy uses the full 26.2 ns trajectory, four independent runs. RMSEs are mean

\pm

sample SD over runs.

$\alpha$	RMSE	Walkers	RMSE
$0.05$	$0.109\pm 0.018$	$1$	$1.948\pm 0.749$
$0.1$	$0.098\pm 0.026$	$10$	$0.377\pm 0.160$
$0.25$	$0.104\pm 0.024$	$100$	$0.169\pm 0.082$
$0.5$	$0.093\pm 0.029$	$1000$	$0.104\pm 0.024$
$1$	$0.602\pm 0.494$	$10000$	$0.070\pm 0.017$

Appendix

Table 4: Müller–Brown errors at

t=320

: Boltzmann-weighted FES RMSE in

k_{B}T

of the parameter average, mean

\pm

sample SD over four independent runs, against the tempering exponent at 1000 walkers per population and against the walker count at

\alpha=0.25

. A flat surface scores

1.281

	NRMSE			$D_{\mathrm{KL}}$
$d$	L-RE	Best single ( $\alpha$ )	L-NRE	L-RE	Best single ( $\alpha$ )	L-NRE
$2$	$0.36\pm 0.20$	$0.28\pm 0.10\;(0.05)$	$0.38\pm 0.11$	$0.07\pm 0.07$	$0.05\pm 0.03\;(0.05)$	$0.07\pm 0.03$
$5$	$0.42\pm 0.10$	$0.26\pm 0.09\;(0.5)$	$0.41\pm 0.31$	$0.18\pm 0.07$	$0.08\pm 0.05\;(0.5)$	$0.22\pm 0.27$
$10$	$0.23\pm 0.07$	$0.16\pm 0.04\;(0.25)$	$0.22\pm 0.06$	$0.13\pm 0.07$	$0.06\pm 0.02\;(0.25)$	$0.12\pm 0.06$
$25$	$0.22\pm 0.06$	$0.20\pm 0.03\;(0.5)$	$0.16\pm 0.01$	$0.27\pm 0.13$	$0.24\pm 0.08\;(0.5)$	$0.18\pm 0.01$
$50$	$0.19\pm 0.01$	$0.24\pm 0.04\;(0.5)$	$0.17\pm 0.01$	$0.41\pm 0.04$	$0.60\pm 0.09\;(0.5)$	$0.36\pm 0.04$

Appendix

Table 5: Four-well dimensional scaling: mean

\pm

sample SD of the full-coordinate NRMSE and

D_{\mathrm{KL}}(\rho\,\|\,\rho_{\theta})

at the final SWA readout. Each best single level is selected by the lowest mean of the corresponding metric among the tested levels at that dimension. All primary arms use the same total walker budget.

	NRMSE		$D_{\mathrm{KL}}$
$d$	L-RE	L-NRE	L-RE	L-NRE
$2$	$0.14\pm 0.03$	$0.18\pm 0.06$	$0.010\pm 0.004$	$0.02\pm 0.01$
$5$	$0.26\pm 0.02$	$0.24\pm 0.04$	$0.08\pm 0.01$	$0.07\pm 0.02$
$10$	$0.27\pm 0.03$	$0.26\pm 0.02$	$0.16\pm 0.03$	$0.17\pm 0.03$
$25$	$0.25\pm 0.05$	$0.20\pm 0.05$	$0.40\pm 0.20$	$0.24\pm 0.13$
$50$	$0.20\pm 0.02$	$0.179\pm 0.001$	$0.47\pm 0.09$	$0.39\pm 0.01$

Appendix

Table 6: Larger-population controls for the four-well landscape: mean

\pm

sample SD with

1{,}000

walkers per level and population, four times the primary ladder budget.

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SPIBER: Reconstructing Free Energy Landscapes from Short, Unconverged Trajectories with Generative Flow Networks

Venkata Sai Sreyas Adury, Pratyush Tiwary

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Free Energy Calculation Molecular Dynamics Simulation

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Molecular systems have many degrees of freedom, but their metastable behavior can often be described by a few collective variables. Identifying these variables and estimating free energies along them from limited simulation data remains a challenging, important problem. Separate short trajectories may sample different metastable states without capturing transitions or establishing their relative equilibrium populations. For unbiased trajectories generated with the same Hamiltonian at a single temperature, alternate methods based on histogram reweighting cannot correct this imbalance. Here we present SPIBER, which combines the State Predictive Information Bottleneck (SPIB) with Generative Flow Networks (GFlowNets). SPIB uses deep learning to approximate slow degrees of freedom through a past-future information bottleneck, retaining information needed to predict future metastable states. We show that this compression limits conditional entropy variations in populated regions, allowing conditional mean potential energies, which are much easier to calculate, to be used to approximate free energy differences. Given sufficient local sampling to estimate these energies, they define the target distribution for GFlowNets, energy-based generative samplers that sample according to estimated thermodynamic stability rather than observed populations. For a particle in a radial double-well potential, for alanine dipeptide, and for the nine-residue peptide AIB9, SPIBER recovers free energy differences between sampled metastable states to within one thermal energy unit of reference values. The method combines collective-variable learning and free energy estimation in up to four latent dimensions, without requiring converged state populations or additional molecular dynamics simulations.

Abstract

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