A dense, dilated 3D U-net was trained to predict the 5th and 95th percentile distributions of planning dose plan using the nominal dose and planning CT images. The data set comprised proton therapy plans for 582 H&N cancer patients. Ground truth percentile values were estimated for each patient through 600 dose recalculations, representing randomly sampled uncertainty scenarios. The comprehensive comparisons of different models were conducted for H&N cancer patients, considering those with and without a beam mask (BM) and diverse beam configurations, including varying beam angles, couch angles, and beam numbers. The performance of our model trained based on a mixture of patients with H&N and prostate cancer was also assessed in contrast with models trained based on data specific for patients with cancer at either site. The DL-based model's predictions of percentile dose distributions exhibited excellent agreement with the ground truth dose distributions. The average gamma index with 2 mm/2%, consistently exceeded 97% for both 5th and 95th percentile dose volumes. Mean dose-volume histogram error analysis revealed that predictions from the combined training set yielded mean errors and standard deviations that were generally similar to those in the specific patient training data sets. Our proposed DL-based method for evaluation of the robustness of proton therapy plans provides precise, rapid predictions of percentile dose for a given confidence level regardless of the beam arrangement and cancer site. This versatility positions our model as a valuable tool for evaluating the robustness of proton therapy across various cancer sites.