Tropical cyclones (TCs) are responsible for large-scale loss of life and property, and billions of dollars worth of damage annually. Forecasting is typically accomplished through the use of dynamical models which directly integrate the governing equations of motion. These forecasts, and especially intensity forecasts, rely on accurate initial conditions obtained by assimilating real-time observational data. However, current methods used to generate these initial conditions are computationally expensive and limited, and TC intensity forecasts have much room for improvement. Here, we show that a Physics-Informed Neural Network can provide a promising alternative for data assimilation for TC initial conditions. Using synthetic training data sparsely sampled from hurricanes simulated in a forecast model, a PINN is able to accurately reconstruct full 2- and 3-dimensional wind and pressure fields of a TC. We also demonstrate how a set of sparse, real-time observations, can be used to accurately reconstruct Hurricane Ida. Our results demonstrate how recent advances in computational capabilities and deep learning can be applied to TC data assimilation, offering an accurate and efficient complement to existing data assimilation methods.