For physicians to make rapid clinical decisions for patients with congestive heart failure, the assessment of pulmonary edema severity in chest radiographs is vital. Although deep learning has shown promise in detecting the presence or absence or discrete grades of severity, of such edema, prediction of continuous-valued severity yet remains a challenge. Here, we propose PENet: Siamese convolutional neural networks to assess the continuous spectrum of severity of lung edema from chest radiographs. We present different modes of implementing this network and demonstrate that our best model outperforms that of earlier work (mean AUC of 0.91 over 0.87), while using only 1/16-th the dimension of input images and 1/69-th the size of training data, thus also saving expensive computation.