Grounded ice thickness plays a critical role in understanding the behavior and stability of ice sheets, particularly in polar regions such as Greenland, Antarctica, and the Canadian Arctic. Ice sheet dynamics are governed by complex interactions between ice flow, surface accumulation, and bedrock topography, making the accurate modeling of these processes essential for predicting long-term ice sheet behavior and their contributions to global sea level rise [14, 18]. In particular, the Shallow Ice Approximation (SIA) provides a framework for modeling grounded ice, where ice flow is driven by internal deformation and the base is often assumed to be frozen, constraining the ice thickness by bedrock topography [12, 15]. A key challenge in modeling grounded ice involves solving the partial differential equations (PDEs) that govern ice thickness evolution, while incorporating these constraints. This leads to a free boundary problem, where the ice thickness must remain non-negative and above the bedrock, giving rise to an obstacle problem [21, 3].

Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g., TensorFlow or PyTorch. Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here extend PINNs to solve obstacle-related PDEs which present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of the solution that lies above a given obstacle. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.