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Purpose: To develop a machine learning model to classify the severity grades of pulmonary edema on chest radiographs. Materials and Methods: In this retrospective study, 369,071 chest radiographs and associated radiology reports from 64,581 (mean age, 51.71; 54.51% women) patients from the MIMIC-CXR chest radiograph dataset were included. This dataset was split into patients with and without congestive heart failure (CHF). Pulmonary edema severity labels from the associated radiology reports were extracted from patients with CHF as four different ordinal levels: 0, no edema; 1, vascular congestion; 2, interstitial edema; and 3, alveolar edema. Deep learning models were developed using two approaches: a semi-supervised model using a variational autoencoder and a pre-trained supervised learning model using a dense neural network. Receiver operating characteristic curve analysis was performed on both models. Results: The area under the receiver operating characteristic curve (AUC) for differentiating alveolar edema from no edema was 0.99 for the semi-supervised model and 0.87 for the pre-trained models. Performance of the algorithm was inversely related to the difficulty in categorizing milder states of pulmonary edema (shown as AUCs for semi-supervised model and pre-trained model, respectively): 2 versus 0, 0.88 and 0.81; 1 versus 0, 0.79 and 0.66; 3 versus 1, 0.93 and 0.82; 2 versus 1, 0.69 and 0.73; and, 3 versus 2, 0.88 and 0.63. Conclusion: Deep learning models were trained on a large chest radiograph dataset and could grade the severity of pulmonary edema on chest radiographs with high performance.