A research team from the Hong Kong University of Science and Technology (HKUST), led by Prof. Pan Ding, Associate Professor in the Departments of Physics and Chemistry, and Dr. Li Shuo-Hui, Research Assistant Professor in the Department of Physics, has developed a novel deep learning-based method for efficient sampling of the Boltzmann distribution across a continuous temperature range. The findings were published in Physical Review Letters. The Boltzmann distribution is a foundational concept in statistical mechanics, describing the probability distribution of particle states in a system at thermal equilibrium. Accurate sampling from this distribution is essential for studying complex phenomena such as phase transitions, chemical reactions, and biomolecular structural changes. However, traditional methods like molecular dynamics (MD) and Markov chain Monte Carlo (MCMC) simulations often face significant computational challenges, especially when high energy barriers are present, leading to slow convergence and high resource demands. To address these limitations, Dr. Li and the team introduced the variational temperature-differentiable (VaTD) method—a general framework built on deep generative models. VaTD is compatible with various tractable density models, including autoregressive networks and normalizing flows. The key innovation lies in its ability to learn the Boltzmann distribution across a continuous temperature range in a single training process. By leveraging automatic differentiation, VaTD can efficiently compute both first- and second-order derivatives of thermodynamic quantities with respect to temperature. This capability enables the approximation of an analytical partition function, a critical step in thermodynamic analysis. Under optimal conditions, the method theoretically guarantees an unbiased representation of the Boltzmann distribution. A major advantage of VaTD is its ability to integrate over a continuous temperature range, which helps systems overcome energy barriers more effectively and reduces sampling bias. Unlike many existing generative models that depend on large pre-existing datasets generated from MD or Monte Carlo simulations, VaTD requires only the system’s potential energy function, making it more flexible and less reliant on prior simulations. The method was rigorously tested using classical models in statistical physics, including the Ising model and the XY model, where it demonstrated high accuracy and computational efficiency. The results confirm that VaTD can deliver reliable thermodynamic predictions with significantly reduced computational cost compared to traditional approaches. Prof. Pan Ding highlighted the broader impact of the work, stating, “This breakthrough opens new avenues for investigating complex statistical systems, with promising applications across physics, chemistry, materials science, and life sciences.” The VaTD framework represents a significant step forward in combining machine learning with statistical mechanics, offering a powerful new tool for simulating and understanding equilibrium behavior in intricate systems.