The ice thickness of the world’s glaciers is mostly unmeasured and physics-based models to reconstruct ice thickness can not always deliver accurate estimates. In this study, we use deep learning paired with physical knowledge to generate ice thickness estimates for all glaciers of Spitsbergen, Barentsøya, and Edgeøya in Svalbard. We incorporate mass conservation and other physically derived conditions into a neural network to predict plausible ice thicknesses even for glaciers without any in situ ice thickness measurements. With a glacier-wise cross validation scheme we evaluate the performance of the physics-informed neural network. The results of the experiments let us identify several challenges and opportunities that affect the model’s performance in a real-world setting.