Delta-Learning approach combined with the cluster Gutzwiller approximation for strongly correlated bosonic systems

The cluster Gutzwiller method is widely used to study the strongly correlated bosonic systems, owing to its ability to provide a more precise description of quantum fluctuations. However, its utility is limited by the exponential increase in computational complexity as the cluster size grows. To overcome this limitation, we propose an artificial intelligence-based method known as $\Delta$-Learning. This approach constructs a predictive model by learning the discrepancies between lower-precision (small cluster sizes) and high-precision (large cluster sizes) implementations of the cluster Gutzwiller method, requiring only a small number of training samples. Using this predictive model, we can effectively forecast the outcomes of high-precision methods with high accuracy. Applied to various Bose-Hubbard models, the $\Delta$-Learning method effectively predicts phase diagrams while significantly reducing the computational resources and time. Furthermore, we have compared the predictive accuracy of $\Delta$-Learning with other direct learning methods and found that $\Delta$-Learning exhibits superior performance in scenarios with limited training data. Therefore, when combined with the cluster Gutzwiller approximation, the $\Delta$-Learning approach offers a computationally efficient and accurate method for studying phase transitions in large, complex bosonic systems.