Deep learning of phase transitions with minimal examples

Abstract: Over the past several years, there have been many studies demonstrating the ability of neural networks and deep learning methods to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature $T_{\mathrm{c}}$ behave analogously to an order parameter, and this analogy has been successfully used to locate $T_{\mathrm{c}}$ and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the 2-$d$ Ising model, when the network is trained on configurations at $T=0$ and $T=\infty$ only. We apply histogram reweighting to the neural network prediction and compare its capabilities when trained more conventionally at multiple temperatures. We find that the network trained on two temperatures is still able to identify $T_{\mathrm{c}}$ and $\nu$, while the extraction of $\gamma$ becomes more challenging.