Accurate forecasting of tsunami waves is critical to a functioning early warning system. While physical models of tsunami waves are well-understood and solvable on a computer, simulation of such models at high resolution is computationally expensive and time consuming. The primary source of practical wave-height data is supplied by the DART (Deep-ocean Assessment and Reporting of Tsunami) Network, a series of buoys that provide wave-height measurements throughout the ocean. The challenge we address in this work is in accurately estimating densely sampled wave-height fields given sparse measurements obtained from the DART network. We use an attention-based neural network designed for sparse sensing problems in the physical sciences, and test its reconstruction accuracy on realistic tsunami simulations with mixed initial conditions. Our experiments demonstrate a promising new tool for obtaining densely sampled observation networks for tsunami forecasting.