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

提出了一种具有可微温度的变分建模方法，通过深度生成模型，在连续温度范围内同时估计和最小化自由能，应用于研究 Ising 模型和 XY 模型中的相变，并显示出相对于 Markov Chain Monte Carlo（MCMC）模拟更高效准确的直接采样模拟，此外，这种方法能够以温度的可微函数给出热力学量，与精确解的自由能二阶导数非常接近，因此这种温度依赖性的包含使得原来有偏差的变分模型能够捕捉到相变中的微妙热效应，这些发现为使用深度生成模型直接模拟物理系统提供了新的思路。