Thermodynamic fidelity of generative models for Ising system

Machine learning has become a central technique for modeling in science and engineering, either complementing or as surrogates to physics-based models. Significant efforts have recently been devoted to models capable of predicting field quantities, but the limitations of current state-of-the-art models in describing complex physics are not well understood. We characterize the ability of generative diffusion models and generative adversarial networks (GANs) to describe the Ising model. We find diffusion models trained using equilibrium configurations obtained using Metropolis Monte Carlo for a range of temperatures around the critical temperature that can capture average thermodynamic variables across the phase transformation and extrapolate to higher and lower temperatures. The model also captures the overall trends of physical properties associated with fluctuations (specific heat and susceptibility) except at the non-ergodic low temperatures and non-trivial scale-free correlations at the critical temperature, albeit with some difference in the critical exponent compared to Monte Carlo simulations. GANs perform more poorly on thermodynamic properties and are susceptible to mode collapse without careful training. This investigation highlights the potential and limitations of generative models in capturing the complex phenomena associated with certain physical systems.