MetaDNS: Enhancing Exploration in Discrete Neural Samplers via Well-Tempered Metadynamics

The rare-event sampling problem has long been the central limiting factor in molecular dynamics (MD), especially in biomolecular simulation. Recently, diffusion models such as BioEmu have emerged as powerful equilibrium samplers that generate independent samples from complex molecular distributions, eliminating the cost of sampling rare transition events. However, a sampling problem remains when computing observables that rely on states which are rare in equilibrium, for example folding free energies. Here, we introduce enhanced diffusion sampling, enabling efficient exploration of rare-event regions while preserving unbiased thermodynamic estimators. The key idea is to perform quantitatively accurate steering protocols to generate biased ensembles and subsequently recover equilibrium statistics via exact reweighting. We instantiate our framework in three algorithms: UmbrellaDiff (umbrella sampling with diffusion models), MetaDiff (a batchwise analogue for metadynamics), and $Δ$G-Diff (free-energy differences via tilted ensembles). Across toy systems, protein folding landscapes and folding free energies, our methods achieve fast, accurate, and scalable estimation of equilibrium properties within GPU-minutes to hours per system-closing the rare-event sampling gap that remained after the advent of diffusion-model equilibrium samplers.

The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DiMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.

Many proteins function through transitions between conformational states, yet rare states are rarely sampled by diffusion models trained on an equilibrium ensemble, demanding better sampling methods. We introduce METALICA, which implements Metadynamics on a pretrained diffusion model via Replica Exchange. It accumulates a bias potential along a Collective Variable, repels new samples from previous ones through biased sampling, and reweights samples onto the unbiased distribution. METALICA holds one replica per diffusion level, forming a Markov Chain that evolves through inter-replica communication and is refined in place as the bias grows. METALICA is the dual of sequential control, in which Sequential Monte Carlo parallelizes the sampler over a batch of particles. Parallelism over the levels of the diffusion-time schedule instead allows METALICA to generate samples from long chains, essential for the discovery of rare events, with accuracy set by run length rather than by the memory available. We validate on a bimodal target with known free energies, then apply METALICA to the unfolding of a protein. At a budget for which sequential control yields no unfolded structure, METALICA populates the basin and resolves a second free energy minimum.