The cluster Gutzwiller method is widely used to study the strongly correlated bosonic systems, particularly in the context of ultracold bosons in optical lattices, 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 $\mathrm{\Delta }$ learning. This approach constructs a predictive model by learning the discrepancies between low-precision (small cluster sizes) and high-precision (large cluster sizes) implementations of the cluster Gutzwiller method, requiring only a small number of training samples. By utilizing this predictive model, we can effectively forecast the outcomes of high-precision methods with high accuracy. We demonstrate the effectiveness of $\mathrm{\Delta }$ learning in predicting phase diagrams across various Bose-Hubbard models, achieving a significant reduction in computational resources and time. Furthermore, we compare the predictive accuracy of $\mathrm{\Delta }$ learning with other direct learning methods and find that $\mathrm{\Delta }$ learning exhibits superior performance in scenarios with limited training data. Therefore, when combined with the cluster Gutzwiller approximation, the $\mathrm{\Delta }$-learning approach offers a computationally efficient and accurate framework for exploring phase transitions in nonstandard Bose-Hubbard models within large, complex optical lattices, paving the way for future research in this area.

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• Cold atoms & matter waves
• Cold gases in optical lattices
• Phase diagrams
• Quantum fluids & solids
• Strongly correlated systems
• GW method
• Machine learning