Accurate prediction of melt-pool evolution is essential for optimizing Laser Powder Bed Fusion (LPBF) processes but remains computationally demanding when based solely on transient finite element analysis (FEA). This work presents an adaptive hybrid Finite Element Analysis–Physics-Informed Neural Network (FEA–PINN) framework for rapid prediction of the transient thermal field and phase evolution during LPBF of Inconel 718. A three-dimensional transient conduction model with an enthalpy-based phase-change formulation is first employed to generate high-fidelity thermal data using a moving Gaussian volumetric heat source. These data are subsequently used to train a physics-informed neural network that simultaneously satisfies the governing heat-transfer equation, initial and boundary conditions, and phase-transition constraints. An adaptive refinement strategy combines localized mesh refinement, residual-based collocation-point enrichment, and on-demand local FEA correction to improve prediction accuracy while limiting computational cost. For the investigated LPBF process (laser power 300 W, scan speed 1000 mm s⁻¹, layer thickness 40 μm, beam diameter 80 μm), the proposed framework predicts the transient temperature field with a mean relative error of 1.48% compared with high-fidelity FEA. The corresponding errors in melt-pool width, depth, and length are 2.12%, 2.37%, and 2.41%, respectively, while the liquid-fraction prediction error remains below 2.1%. The predicted average melt-pool dimensions are approximately 128 μm (width), 67 μm (depth), and 245 μm (length), in agreement with reported experimental values for conduction-mode LPBF of Inconel 718. A conventional transient FEA simulation required approximately 27.4 h, whereas the trained hybrid FEA–PINN model completed the corresponding prediction in 15.8 min, representing a computational speedup of approximately 104× while preserving engineering-level accuracy. The proposed framework is applicable to conduction-mode LPBF, where evaporation, recoil pressure, keyhole formation, free-surface deformation, and individual powder-particle dynamics are neglected. The results demonstrate that adaptive FEA–PINN coupling provides an efficient surrogate for high-fidelity thermal simulations and offers a practical approach for rapid process optimization and digital-twin applications in metal additive manufacturing.