Predicting the future contribution of the ice sheets to sea level rise over the next decades presents several challenges due to a poor understanding of critical boundary conditions, such as basal sliding. Traditional numerical models often rely on data assimilation methods to infer spatially variable friction coefficients by solving an inverse problem, given an empirical friction law. However, these approaches are not versatile, as they sometimes demand extensive code development efforts when integrating new physics into the model. Furthermore, this approach makes it difficult to handle sparse data effectively. To tackle these challenges, we propose a novel approach utilizing Physics-Informed Neural Networks (PINNs) to seamlessly integrate observational data and governing equations of ice flow into a unified loss function, facilitating the solution of both forward and inverse problems within the same framework. We illustrate the versatility of this approach by applying the framework to two-dimensional problems on the Helheim Glacier in southeast Greenland. By systematically concealing one variable (e.g. ice speed, ice thickness, etc.), we demonstrate the ability of PINNs to accurately reconstruct hidden information. Furthermore, we extend this application to address a challenging mixed inversion problem. We show how PINNs are capable of inferring the basal friction coefficient while simultaneously filling gaps in the sparsely observed ice thickness. This unified framework offers a promising avenue to enhance the predictive capabilities of ice sheet models, reducing uncertainties, and advancing our understanding of poorly constrained physical processes.