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Primordial black holes (PBHs) are a well-motivated candidate for dark matter and a potential probe of early universe physics through Hawking radiation. In the first part of this talk, I will discuss the cosmological propagation and decay of axion-like particles (ALPs) emitted from PBHs. By computing the resulting photon flux spectra from ALP decays, including the effects of boosted decay kinematics, we can place constraints on the PBH abundance. In the second part, I will introduce ongoing work aimed at making the inference of PBH mass distributions from emission spectra numerically tractable. This is achieved using a framework based on the Deep Operator Network, which learns the functional mapping between the PBH mass distribution and the resulting emission spectra. This framework enables the inversion from observed particle spectra to the underlying mass distribution and provides a neural-operator-based approach for PBH phenomenology.