Traveltimes are essential for seismic applications ranging from imaging to tomography. Traveltime computations in anisotropic media, which are better representative of the true Earth, require solving the anisotropic eikonal equation. Numerical techniques to solve the anisotropic eikonal equation are known to suffer from instability and increased computational cost compared to the isotropic case. Here, we employ the emerging paradigm of physics-informed neural networks to solve the anisotropic qP-wave eikonal equation. By minimizing a loss function formed by imposing the validity of the eikonal equation, we train a neural network to produce traveltime solutions that are consistent with the underlying partial differential equation. We observe considerably higher accuracy compared to the first-order finite-difference solution using the fast sweeping method. We also show that once the network is trained for a particular source location in a given anisotropic model, the traveltimes for a new source location and/or an updated model can be computed much more efficiently using the pre-trained network. This feature is particularly attractive as it can speed up seismic imaging and inversion applications significantly.