Efficient thermal management is vital in compact, high-power electronics. We present a physics-informed neural network (PINN) framework, implemented with NVIDIA PhysicsNeMo, to model and optimize steady incompressible flow and heat transfer in a 2D channel with periodic rectangular fins. A hybrid multi-network design decouples momentum (Flow_net for u,v,p) and energy (Heat_net for T), coupling them through PDE residuals, interface continuity, and domain-integral conservation constraints. Residual-based adoptive sampling, SDF reweighting, and adaptive loss balancing concentrate learning in high-gradient regions (fin wakes and thermal boundary layers). For Re=100 air flow, the PINN predictions agree closely with high-fidelity CFD benchmarks: relative L2 errors are <9% overall (pressure ≈0.7%, temperature ≈0.3%). Once trained, the surrogate enables near-instantaneous evaluation across 60 geometries, transforming a 360-h CFD sweep into ~15h of one-time training (>24× reduction). Parametric studies reveal a regime shift from conduction-dominated transport (narrow gaps) to convection-enhanced heat transfer (intermediate gaps). The optimal spacing is 0.15 cm, yielding Nu≈25.1at a moderate ∆p≈9.5 Pa, with a peak performance evaluation criterion PEC≈ 1.60 under constant pumping-power normalization. Flow visualizations attribute the performance to jet impingement between fins, shear-layer roll-up, and sustained vortex-induced mixing that renew near-wall fluid while avoiding excessive hydraulic penalties. The results demonstrate that hybrid, conservation-aware PINNs provide accurate, data-efficient surrogates for multi-parameter thermo-fluid optimization, offering practical speedups for heat-sink design while preserving first principles fidelity.