Efficient Monte-Carlo sampling of metastable systems using non-local collective variable updates

TL;DR

The paper tackles metastability in Monte Carlo sampling for complex molecular systems by introducing a CV-guided MCMC framework that uses non-linear CVs and underdamped Langevin dynamics. It combines non-local CV proposals with constrained, Steinman-RATTLE steering and a work-based Metropolis acceptance, and proves reversibility via a Jarzynski–Crooks argument. The authors provide a concrete algorithm, a normalized parameterization, and demonstrate substantial performance gains—up to two orders of magnitude—across four model systems, including linear and non-linear CVs and high-dimensional CVs enabled by normalizing-flow proposals. This approach broadens the applicability of CV-based enhanced sampling to intermediate-dimensional CVs (tens to hundreds of variables) and to more realistic molecular systems, with practical implications for sampling efficiency and accuracy in complex energy landscapes.

Abstract

Monte-Carlo simulations are widely used to simulate complex molecular systems, but standard approaches suffer from metastability. Lately, the use of non-local proposal updates in a collective-variable (CV) space has been proposed in several works. Here, we generalize these approaches and explicitly spell out an algorithm for non-linear CVs and underdamped Langevin dynamics. We prove reversibility of the resulting scheme and demonstrate its performance on several numerical examples, observing a substantial performance increase compared to methods based on overdamped Langevin dynamics as considered previously. Advances in generative machine-learning-based proposal samplers now enable efficient sampling in CV spaces of intermediate dimensionality (tens to hundreds of variables), and our results extend their applicability toward more realistic molecular systems.

Figures (17)

• Figure 1: Performance of the algorithm for the Gaussian tunnel example in the special cases of deterministic ($\alpha_1 = 0$) and overdamped Langevin dynamics ($\alpha_1=1$). The top row shows the acceptance rate normalized by $K_T$ for a fixed jump from $z=0$ to $\widetilde{z}=b$ for different values of $K_T$, averaged over 10,000 random initializations of $x^\perp\sim\nu_\perp(x^\perp | z)$. The second row shows the inverse mode jump cost, averaged over 20 chains and 10,000 iterations, as a function of the number of steps $K_T(b)$ for a jump of distance $b$, which is related to the inverse of the dimensionless velocity $\widetilde{v}$ via $K_T(b) = \mathrm{ceil}(b/\widetilde{v})$. White space indicates that at least one chain never switched mode.
• Figure 2: Typical driven transition path from one mode center to another, $z=0$ to $z=b=10$, for overdamped ($\alpha_1=1)$ and deterministic ($\alpha_1=0$) dynamics for the optimal $\alpha_2$ and $K_T=b/\widetilde{v}$.
• Figure 3: Gaussian Tunnel marginals for the optimum performance values $\alpha_1=0$, $\alpha_2 = 0.67$ and $\widetilde{v}=0.2$, corresponding to $K_T(b)=50$. Results are obtained by running 8 independent chains for 200,000 iterations.
• Figure 4: Performance of the algorithm applied to the $\phi^4$ model, estimated from 10 chains over 20,000 iterations. The x-axis represents the number of steps for a steered trajectory of over the distance $2\overline{\phi}^*$ between the two modes. White space indicates that at least one chain never switched mode.
• Figure 5: Marginal distribution of the $\phi^4$ model obtained from running 8 chains for $10^4$ steps with $\alpha_1=0$, $\alpha_2=0.0014$ and $\widetilde{v}=1.2\times 10^{-3}$ (the optimal parameters obtained from optimization in \ref{['fig:phi_comparison']}). In this case, a biased GMM proposal with weights $0.3$ and $0.7$ was used.

Theorems & Definitions (20)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Lemma B.1
• proof
• Remark B.2
• Remark B.3
• Remark B.4