Center-involved diabetic macular edema (ci-DME) is a major cause of vision loss. Although the gold standard for diagnosis involves 3D imaging, 2D imaging by fundus photography is usually used in screening settings, resulting in high false-positive and false-negative calls. To address this, we train a deep learning model to predict ci-DME from fundus photographs, with an ROC-AUC of 0.89 (95% CI: 0.87-0.91), corresponding to 85% sensitivity at 80% specificity. In comparison, retinal specialists have similar sensitivities (82-85%), but only half the specificity (45-50%, p < 0.001). Our model can also detect the presence of intraretinal fluid (AUC: 0.81; 95% CI: 0.81-0.86) and subretinal fluid (AUC 0.88; 95% CI: 0.85-0.91). Using deep learning to make predictions via simple 2D images without sophisticated 3D-imaging equipment and with better than specialist performance, has broad relevance to many other applications in medical imaging.