The present study examined the compressive mechanical properties and failure modes of Schwarz P and Schwarz G Triple Periodic Minimal Surface (TPMS) cellular structures using Finite Element (FE) simulations and experimental tests. A novel approach based on the initial damage parameter, which accounts for the initial defects induced by the Laser Powder Bed Fusion (LPBF) manufacturing process, was employed to model the compression behavior of the cellular structures. Compression tests were conducted on 3 Schwarz P and 3 Schwarz G specimens. A digital camera captured the deformation patterns of the specimens, revealing that shear bands formed in the Schwarz P specimens, while the Schwarz G specimens deformed uniformly. The FE simulations utilized modified material parameters according to the Continuum Damage Mechanics (CDM) framework. Moreover, an isotropic CDM model with micro-defect closure effect was implemented to investigate the failure mechanisms of the specimens. The FE simulation results indicated that the CDM model accurately predicted the mechanical behavior of the cellular structures fabricated by the LPBF process. The Scanning Electron Microscopy (SEM) images confirmed that the specimens had manufacturing defects such as micro-voids, pores, and surface roughness and that the crack initiation locations corresponded to the maximum damage locations of the FE simulations.