Glacier ice thickness is difficult to measure directly but is essential for glacier evolution modelling. In this work, we employ a novel approach combining physical knowledge and data-driven machine learning to estimate the ice thickness of multiple glaciers in Spitsbergen, Barentsøya, and Edgeøya in Svalbard. We identify challenges for the physics-aware machine learning model and opportunities for improving the accuracy and physical consistency that would also apply to other geophysical tasks.