We introduce a Lattice-Boltzmann-driven kinetic physics-informed neural network (K-PINN) for predictive modeling of droplet dynamics on structured surfaces, in which the discrete Boltzmann-BGK equation is incorporated into the learning framework. Different from traditional PINNs that are restricted by macroscopic continuum equations, the K-PINN framework is built on the mesoscopic kinetic level, in which the essential Lattice-Boltzmann physics is preserved in the data-efficient neural network. The K-PINN has been successfully employed for modeling non-trivial droplet phenomena such as contact pinning, anisotropic spreading, and capillary hysteresis on substrates of different morphologies, ranging from random roughness to periodic pillar structures. Moreover, strict physical consistency, such as mass conservation within 1.5%, is ensured in the K-PINN framework. Furthermore, the U-Net-based encoder-decoder structure of the K-PINN results in a 50-75% reduction in error compared to traditional neural networks, achieving almost perfect agreement with high-resolution Lattice-Boltzmann simulations $L_2$ ~ 0.021-0.026, $R^2$ ~ 0.999. Robust convergence of the K-PINN to diverse surface morphologies is ensured through curriculum learning and adaptive two-phase optimization. Upon convergence, the K-PINN can perform real-time prediction with over $10^4$ evaluations per second. Through the combination of kinetic theory and physics-informed learning, this work establishes a new paradigm for fast, physically consistent modeling of multiphase flows on complex surfaces.

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