Deep learning method enables efficient Boltzmann distribution sampling across a continuous temperature range

A research team has developed a novel direct sampling method based on deep generative models. Their method enables efficient sampling of the Boltzmann distribution across a continuous temperature range. The findings have been published in Physical Review Letters. The team was led by Prof. Pan Ding, Associate Professor from the Departments of Physics and Chemistry, and Dr. Li Shuo-Hui, Research Assistant Professor from the Department of Physics at the Hong Kong University of Science and Technology (HKUST).

The Boltzmann distribution is one of the most important distributions in statistical mechanics for systems in thermal equilibrium. Sampling from it is crucial for understanding complex systems, such as phase transitions, chemical reactions, and biomolecular conformations. However, efficiently and accurately computing thermodynamic quantities for such systems has long been a major challenge in the field.

Traditional numerical methods in statistical mechanics, including molecular dynamics (MD) and Markov chain Monte Carlo (MCMC) sampling, require extensive simulation time to obtain ensemble averages when the system’s energy barrier is high, leading to significant computational costs.

Inspired by recent advances in deep generative models, Dr. Li and colleagues proposed a general framework—the variational temperature-differentiable (VaTD) method—applicable to any tractable density generative model, such as autoregressive models and normalizing flows.

VaTD can learn the Boltzmann distribution across a continuous temperature range, with first- and second-order derivatives of thermodynamic quantities with respect to temperature conveniently obtained through automatic differentiation. This effectively approximates an analytical partition function.

Under optimal conditions, the model theoretically guarantees an unbiased Boltzmann distribution. More importantly, integrating over a continuous temperature range helps overcome energy barriers, thereby reducing bias in simulations.

Unlike predominant generative models in statistical mechanics, VaTD requires only the potential energy of the system and does not rely on pre-generated datasets from MD or Monte Carlo simulations.

The team validated the method’s accuracy and efficiency through numerical experiments on classical statistical physics models, including the Ising model and the XY model.

Prof. Pan remarked, “This breakthrough paves the way for studying novel phenomena in complex statistical systems, with potential applications in physics, chemistry, materials science, and life sciences.”