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The source angle localization problem is studied based on scattering of elastic waves in two dimensions by a phononic array and the exceptional points of its band structure. Exceptional points are complex singularities of a parameterized eigen-spectrum, where two modes coalesce with identical mode shapes. These special points mark the qualitative transitions in the system behavior and have been proposed for sensing applications. The equi-frequency band structures are analyzed with focus on the angle-dependent modal behaviors. At the exceptional points and critical angles, the eigen-modes switch their energy characteristics and symmetry, leading to enhanced sensitivity as the scattering response of the medium is inherently angle-dependent. An artificial neural network is trained with randomly weighted and superposed eigen-modes to achieve deep learning of the angle-dependent dynamics. The trained algorithm can accurately classify the incident angle of an unknown scattering signal, with minimal sidelobe levels and suppressed main lobewidth. The neural network approach shows superior localization performance compared with standard delay-and-sum technique. The proposed application of the phononic array highlights the physical relevance of band topology and eigen-modes to a technological application, adds extra strength to the existing localization methods, and can be easily enhanced with the fast-growing data-driven techniques.