The boundary element method (BEM) has been widely used as an effective numerical approach for wave analysis in infinite domains. Since the BEM requires discretization only along the boundary of the analysis domain, preprocessing and postprocessing can be relatively straightforward. However, the BEM formulation becomes extremely complex for anisotropic elastodynamic problems, since the fundamental solutions required in the BEM cannot be expressed in a closed form. Moreover, the numerical evaluation of the fundamental solutions requires much computational time, which significantly increases the overall computational cost of the BEM. In recent years, Physics-Informed Neural Networks (PINNs) have been developed, and research in this field has been actively conducted. Applications of PINNs to wave propagation problems have been reported in several studies. However, applications to anisotropic elastic wave problems are still limited. Therefore, in this study, we develop a numerical method for 2-D pure anti-plane anisotropic elastic wave scattering problems using PINNs, in which only the boundary of the analysis domain is discritized, similar to the BEM. The proposed PINNs satisfy not only the governing equation and physical boundary conditions, but also the radiation condition at infinity, as in the BEM. Numerical results are shown to validate the proposed method and are compared with those obtained by the BEM.