Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models

Rare events are central to the evolution of complex many-body systems, characterized as key transitional configurations on the free energy surface (FES). Conventional methods require adequate sampling of rare event transitions to obtain the FES, which is computationally very demanding. Here we introduce the variational free energy surface (VaFES), a dataset-free framework that directly models FESs using tractable-density generative models. Rare events can then be immediately identified from the FES with their configurations generated directly via one-shot sampling of generative models. By extending a coarse-grained collective variable (CV) into its reversible equivalent, VaFES constructs a latent space of intermediate representation in which the CVs explicitly occupy a subset of dimensions. This latent-space construction preserves the physical interpretability and transparent controllability of the CVs by design, while accommodating arbitrary CV formulations. The reversibility makes the system energy exactly accessible, enabling variational optimization of the FES without pre-generated simulation data. A single optimization yields a continuous, differentiable FES together with one-shot generation of rare-event configurations. Our method can reproduce the exact analytical solution for the bistable dimer potential and identify a chignolin native folded state in close alignment with the experimental NMR structure. Our approach thus establishes a scalable, systematic framework for advancing the study of complex statistical systems.

Deep generative modelling of canonical ensemble with differentiable thermal properties

We propose a variational modelling method with differentiable temperature for canonical ensembles. Using a deep generative model, the free energy is estimated and minimized simultaneously in a continuous temperature range. At optimal, this generative model is a Boltzmann distribution with temperature dependence. The training process requires no dataset, and works with arbitrary explicit density generative models. We applied our method to study the phase transitions (PT) in the Ising and XY models, and showed that the direct-sampling simulation of our model is as accurate as the Markov Chain Monte Carlo (MCMC) simulation, but more efficient. Moreover, our method can give thermodynamic quantities as differentiable functions of temperature akin to an analytical solution. The free energy aligns closely with the exact one to the second-order derivative, so this inclusion of temperature dependence enables the otherwise biased variational model to capture the subtle thermal effects at the PTs. These findings shed light on the direct simulation of physical systems using deep generative models