When completed, the PHANGS-HST project will provide a census of roughly 50,000 compact star clusters and associations, as well as human morphological classifications for roughly 20,000 of those objects. These large numbers motivated the development of a more objective and repeatable method to help perform source classifications. In this paper we consider the results for five PHANGS-HST galaxies (NGC 628, NGC 1433, NGC 1566, NGC 3351, NGC 3627) using classifications from two convolutional neural network architectures (RESNET and VGG) trained using deep transfer learning techniques. The results are compared to classifications performed by humans. The primary result is that the neural network classifications are comparable in quality to the human classifications with typical agreement around 70 to 80$%$ for Class 1 clusters (symmetric, centrally concentrated) and 40 to 70$%$ for Class 2 clusters (asymmetric, centrally concentrated). If Class 1 and 2 are considered together the agreement is 82 $pm$ 3$%$. Dependencies on magnitudes, crowding, and background surface brightness are examined. A detailed description of the criteria and methodology used for the human classifications is included along with an examination of systematic differences between PHANGS-HST and LEGUS. The distribution of data points in a colour-colour diagram is used as a figure of merit to further test the relative performances of the different methods. The effects on science results (e.g., determinations of mass and age functions) of using different cluster classification methods are examined and found to be minimal.
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