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arxiv: 2605.03080 · v1 · submitted 2026-05-04 · 🧮 math.NA · cs.NA· physics.comp-ph· stat.CO

High-Dimensional Enhanced Sampling via Regularized Path-Dependent McKean--Vlasov Dynamics using Tensor Density Approximation

Liyao Lyu , Siyu Guo , Huan Lei This is my paper

Pith reviewed 2026-05-08 17:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-phstat.CO
keywords enhanced samplingMcKean-Vlasov dynamicscollective variablestensor approximationGibbs measuresadaptive biasinghigh-dimensional samplingmolecular dynamics
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The pith

A regularized path-dependent McKean-Vlasov dynamics with tensor-approximated densities enables scalable enhanced sampling up to 64 collective variable dimensions.

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

This paper develops a new stochastic dynamics approach to sample high-dimensional Gibbs measures whose energy landscapes contain many metastable states. Standard adaptive biasing methods lose scalability once the number of collective variables exceeds modest sizes because they require costly operations over the full CV domain. The authors replace the usual variational regularization of the Wasserstein functional with a direct regularization inside the McKean-Vlasov drift term and replace the instantaneous particle law with a weighted average over the simulation path. They further approximate the resulting history-averaged CV marginal density by an optimization-free functional hierarchical tensor representation. If the construction works, it yields a density-based biasing scheme that remains practical for molecular systems whose collective-variable spaces reach dimension 64.

Core claim

The central claim is that a suitably regularized and path-dependent McKean-Vlasov dynamics, whose CV marginal density is represented in functional hierarchical tensor format, is well-posed under appropriate assumptions on the underlying potentials and measures and furnishes a numerically stable, scalable adaptive biasing method for high-dimensional enhanced sampling.

What carries the argument

The regularized path-dependent McKean-Vlasov drift that directly regularizes the CV marginal density and replaces the instantaneous law by a weighted path-history measure, together with its functional hierarchical tensor approximation.

If this is right

  • Adaptive biasing becomes feasible without an outer convolution over the full CV domain.
  • Statistical stability improves in the low-replica regime because the drift uses a path-history measure rather than the instantaneous empirical law.
  • Well-posedness of the resulting stochastic dynamics holds under the stated assumptions on potentials and measures.
  • Numerical tests on benchmark potentials and molecular systems confirm effectiveness up to CV dimension 64.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tensor-based density representation might be reused inside other mean-field sampling or optimization algorithms that require high-dimensional marginals.
  • Accuracy of the tensor format could be tested by deliberately increasing CV dimension beyond 64 while monitoring bias in the recovered free-energy surface.
  • The path-dependent regularization might be combined with machine-learned collective variables whose dimension is also high.

Load-bearing premise

The history-averaged CV marginal density can be represented accurately by the functional hierarchical tensor format without introducing uncontrolled bias, and the potentials and measures obey the conditions needed for well-posedness of the dynamics.

What would settle it

A controlled numerical experiment in which the tensor approximation error grows with dimension and produces visible trapping in metastable states for a 64-dimensional CV problem.

Figures

Figures reproduced from arXiv: 2605.03080 by Huan Lei, Liyao Lyu, Siyu Guo.

Figure 1
Figure 1. Figure 1: Benchmark of the 2D Müller potential. (a) Contours of the reference potential (18). (b)–(e) Time series of the 𝑥 coordinate for unbiased dynamics and for biased dynamics with 𝛼 = 13, 16, and 20; the numbers in the panel titles report the corresponding counts of inter-basin transitions over the same simulation horizon. (f) Reweighted FES obtained after freezing the bias. (g) Pointwise difference between the… view at source ↗
Figure 2
Figure 2. Figure 2: Benchmark of the Ala2 molecule with 2 CVs. (a) Molecular structure of alanine dipeptide. (b) Estimated marginal density of 𝜙 obtained from trajectories generated with different bias strengths. (c)–(f) Time series of the dihedral angle 𝜙 for unbiased MD and for biased dynamics with 𝛼 = 0.5, 1.0, and 2.0, respectively. Increasing 𝛼 broadens the sampled angular range and produces more frequent transitions bet… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of the marginal free-energy surfaces 𝐹(𝜙) (top row) and 𝐹(𝜓) (bottom row) for Ala2 after the bias is frozen at iteration 20. Results are averaged over 32 independent production runs. Columns correspond to production lengths 𝑡 = 500, 1000, 2000, and 4000 ps. Solid curves denote the mean FES, and shaded bands indicate the variance across runs. Larger 𝛼 leads to faster convergence and lower statis… view at source ↗
Figure 4
Figure 4. Figure 4: ACE-(ALA)2 -NME benchmark with 3 CVs. (a) Molecular structure of the system. The remaining panels show time series of three representative torsions under unbiased MD and biased dynamics. Columns correspond to the unbiased trajectory and to biased simulations with 𝛼 = 1.0 and 𝛼 = 2.0, while rows correspond to 𝜙1 (top), 𝜙2 (middle), and 𝜓1 (bottom). The adaptive bias produces substantially more frequent tran… view at source ↗
Figure 5
Figure 5. Figure 5: One-dimensional marginal free-energy profiles for ACE-(ALA)2 -NME at different production lengths. Panels (a)–(d) show 𝐹(𝜙1 ), 𝐹(𝜓1 ), 𝐹(𝜙2 ), and 𝐹(𝜓2 ), respectively. Curves correspond to production lengths of 1000, 2000, 4000, and 8000 ps, and the shaded bands indicate the associated variance across independent runs. The principal features of the FES are already recovered after about 4000 ps. 3.4. Prote… view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional marginal free-energy surfaces for representative pairs of torsional coordinates in ACE-(ALA)2 - NME. Panels (a)–(d) correspond to (𝜙1 , 𝜓1 ), (𝜙2 , 𝜓2 ), (𝜙1 , 𝜙2 ), and (𝜓1 , 𝜓2 ), respectively. The surfaces resolve the dominant metastable basins and the principal transition channels in the reduced CV space. width 𝜎 = 0.1, and maximal internal bond dimension 𝑟 = 15. The Softplus regulariza… view at source ↗
Figure 7
Figure 7. Figure 7: s-(1)-phenylethyl peptoid benchmark with 3 CVs. (a) Molecular structure of s1pe. (b) Estimated marginal density of 𝜔 obtained from trajectories generated with different bias strengths after 50,000 ps. (c)–(f) Time series of 𝜔 for unbiased MD and for biased dynamics with 𝛼 = 2.0, 4.0, and 8.0, respectively. Larger 𝛼 yields more frequent transitions and broader exploration of the torsional space. −2.5 0.0 2.… view at source ↗
Figure 8
Figure 8. Figure 8: Reweighted one-dimensional marginal free-energy profiles for s-(1)-phenylethyl peptoid after freezing the bias at 50,000 ps. Panels (a)–(c) show 𝐹(𝜔), 𝐹(𝜙), and 𝐹(𝜓), respectively. Solid curves denote the mean over 32 independent production runs, and shaded bands indicate the corresponding variance. complementary view consistent with previous studies of Chignolin folding [27] and reveals an extended diagon… view at source ↗
Figure 9
Figure 9. Figure 9: Peptoid trimer benchmark with 9 CVs. (a) Molecular structure of (s1pe)3 . (b)–(j) Representative time series of the nine torsional CVs 𝜔1 , 𝜙1 , 𝜓1 , 𝜔2 , 𝜙2 , 𝜓2 , 𝜔3 , 𝜙3 , and 𝜓3 during biased sampling. Frequent transitions are observed along all coordinates, indicating sustained exploration of the high-dimensional torsional landscape. for folding simulations. Compared with the 10-residue Chignolin, thi… view at source ↗
Figure 10
Figure 10. Figure 10: Reweighted one-dimensional marginal free-energy profiles for the peptoid trimer (s1pe)3 . Panels (b)–(j) correspond to 𝐹(𝜔1 ), 𝐹(𝜙1 ), 𝐹(𝜓1 ), 𝐹(𝜔2 ), 𝐹(𝜙2 ), 𝐹(𝜓2 ), 𝐹(𝜔3 ), 𝐹(𝜙3 ), and 𝐹(𝜓3 ), respectively. The multimodal structure of these marginals confirms that the method can recover informative free-energy profiles even when the bias is built in a nine-dimensional CV space. L. Lyu, S. Guo and H. Lei… view at source ↗
Figure 11
Figure 11. Figure 11: Configurational exploration of Chignolin (1UAO) with 16 CVs, measured by the radius of gyration 𝑅𝑔 . (a) 𝑅𝑔 trajectory of a single walker colored by iteration number, with the 50-ps running average shown in black. (b) 𝑅𝑔 distribution across all walkers as a function of iteration; shaded bands show the 10–90th and 25–75th percentile ranges, and the solid line shows the median. (c, d) Within-iteration 𝑅𝑔 tr… view at source ↗
Figure 12
Figure 12. Figure 12: Free energy landscape of Chignolin (1UAO) estimated by the proposed method. (a) Native 𝛽-hairpin structure. (b) Two-dimensional FES projected onto RMSD𝐶𝛼 vs. 𝑅𝑔 , revealing four metastable states: the folded state (F, low RMSD and compact), an intermediate (I), and two unfolded states (𝑈1 , 𝑈2 ) of increasing radius of gyration. (c) FES projected onto the cross-strand contact distances 𝑑1 (atoms 31–92) vs… view at source ↗
Figure 13
Figure 13. Figure 13: Free energy landscape of the villin headpiece subdomain (1VII) with 64 CVs. (a) Native three-helix bundle structure. (b) Two-dimensional FES projected onto 𝑅𝑔 vs. end-to-end distance, revealing four metastable states: the folded state (F), an intermediate (I), and two unfolded states (𝑈1 , 𝑈2 ) with progressively larger end-to-end distances. (c) FES projected onto RMSD𝐶𝛼 vs. end-to-end distance, providing… view at source ↗
read the original abstract

Sampling from high-dimensional Gibbs measures poses a challenge when the energy landscape consists of multiple metastable states. Enhanced-sampling methods mitigate this difficulty by introducing adaptive biasing potentials to facilitate the exploration along prescribed collective variables (CVs), but their scalability is often limited by the dimension of the CV space. Motivated by the Wasserstein-gradient-flow interpretation of adaptive biasing, we propose a regularized path-dependent McKean--Vlasov formulation for high-dimensional enhanced sampling. The formulation replaces the variational regularization of the Wasserstein functional by a direct regularization of the CV marginal density in the McKean--Vlasov drift, avoiding the outer convolution over the CV domain. Furthermore, it replaces the instantaneous law by a weighted path-history measure to improve statistical stability in the small-replica regime. We establish well-posedness of the resulting regularized and path-dependent stochastic dynamics under suitable assumptions. For numerical realization, the history-averaged CV marginal density is approximated using an optimization-free functional hierarchical tensor representation, leading to a scalable density-based adaptive biasing scheme. Numerical experiments on benchmark potentials and molecular systems demonstrate the effectiveness of the proposed method for sampling problems with CV dimensions up to 64.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a regularized path-dependent McKean-Vlasov formulation for enhanced sampling from high-dimensional Gibbs measures. It replaces variational regularization of the Wasserstein functional with direct regularization of the CV marginal density in the drift term and substitutes the instantaneous law with a weighted path-history measure for improved stability in the small-replica regime. Well-posedness of the resulting dynamics is established under suitable assumptions on the potentials and measures. The history-averaged CV marginal density is then approximated via an optimization-free functional hierarchical tensor representation, yielding a scalable density-based adaptive biasing scheme. Numerical experiments on benchmark potentials and molecular systems are reported to demonstrate effectiveness for collective-variable dimensions up to 64.

Significance. If the well-posedness result and the unbiased tensor approximation hold with controllable error, the work provides a technically coherent route to high-dimensional enhanced sampling that avoids outer convolutions and optimization steps. The combination of path-dependent regularization with hierarchical tensor formats addresses a recognized scalability bottleneck in adaptive biasing methods and could enable new applications in molecular dynamics. The optimization-free nature of the tensor representation is a concrete strength for reproducibility and computational efficiency.

major comments (2)
  1. [Well-posedness section] Well-posedness section: the result is stated to hold 'under suitable assumptions' on the potentials and measures, yet these assumptions are not listed explicitly nor shown to be satisfied by standard benchmark potentials. Because the central claim of a well-defined regularized path-dependent dynamics rests on this step, the assumptions and a sketch of the proof must be supplied.
  2. [Tensor approximation and numerical sections] Tensor approximation and numerical sections: the claim that the functional hierarchical tensor format represents the history-averaged CV marginal 'without introducing uncontrolled bias' is load-bearing for the scalability assertion up to dimension 64. No a-priori error bounds, convergence rates, or controlled numerical studies isolating the approximation error are referenced in the abstract; these must be added.
minor comments (2)
  1. [Abstract] Abstract: quantitative performance metrics (e.g., mean first-passage times, acceptance rates, or direct comparison tables against existing high-dimensional methods) are absent; adding one or two concrete numbers would strengthen the effectiveness claim.
  2. [Notation] Notation: the precise definitions of the regularization strength and the path-history weighting parameter, together with any selection procedure, should be stated once in a dedicated paragraph to avoid ambiguity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and agree that the requested clarifications will strengthen the presentation.

read point-by-point responses

  1. Referee: [Well-posedness section] Well-posedness section: the result is stated to hold 'under suitable assumptions' on the potentials and measures, yet these assumptions are not listed explicitly nor shown to be satisfied by standard benchmark potentials. Because the central claim of a well-defined regularized path-dependent dynamics rests on this step, the assumptions and a sketch of the proof must be supplied.

    Authors: We acknowledge that the assumptions were referenced only generically. In the revised manuscript we will explicitly list the required conditions (Lipschitz continuity and linear growth of the drift coefficients, boundedness and smoothness of the potentials, and appropriate integrability of the initial measure). We will also add a concise proof sketch in an appendix, outlining the fixed-point argument for the path-dependent McKean–Vlasov equation under these hypotheses. For each benchmark potential appearing in the numerical section we will verify that the assumptions hold (e.g., by confirming global Lipschitz constants and polynomial growth). revision: yes

  2. Referee: [Tensor approximation and numerical sections] Tensor approximation and numerical sections: the claim that the functional hierarchical tensor format represents the history-averaged CV marginal 'without introducing uncontrolled bias' is load-bearing for the scalability assertion up to dimension 64. No a-priori error bounds, convergence rates, or controlled numerical studies isolating the approximation error are referenced in the abstract; these must be added.

    Authors: We agree that explicit error control is necessary to support the dimension-64 claim. The functional hierarchical tensor format is known to converge in the appropriate Sobolev norm as the rank increases; we will insert the relevant a-priori bounds and convergence rates (with references to the hierarchical-tensor literature) into the tensor-approximation section. We will also add a short numerical subsection that isolates the approximation error by comparing tensor reconstructions against high-fidelity Monte-Carlo references on low-dimensional test cases and report the observed rates. The abstract will be updated to mention that the tensor error is controlled by rank selection. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified in the proposed formulation and numerical scheme

full rationale

The paper introduces a novel regularized path-dependent McKean-Vlasov dynamics for high-dimensional enhanced sampling, motivated by Wasserstein-gradient-flow but with direct regularization of the CV marginal density. Well-posedness is asserted under suitable assumptions on potentials and measures, without reducing to prior fitted results. The history-averaged density is approximated via functional hierarchical tensor format, presented as an optimization-free scalable method. Numerical experiments on benchmarks up to CV dimension 64 provide independent demonstration of effectiveness. No self-definitional steps, fitted inputs called predictions, or load-bearing self-citations are evident in the derivation chain. The central claims rest on the new construction and its numerical realization rather than circular reductions.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard existence/uniqueness assumptions for McKean-Vlasov equations and on the representability of the history-averaged marginal by hierarchical tensors; no explicit free parameters or new entities are named in the abstract.

free parameters (2)
  • regularization strength
    Appears in the direct regularization of the CV marginal density inside the McKean-Vlasov drift; concrete value not given in abstract.
  • path-history weight
    Controls the weighted path-history measure for statistical stability; not quantified in abstract.
axioms (1)
  • domain assumption Well-posedness of the regularized path-dependent dynamics under suitable assumptions on the energy landscape and measures
    Invoked to establish existence and uniqueness of the stochastic dynamics.

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