This paper demonstrates the effectiveness of machine learning-driven optimization for designing application-specific GaN tri-gate FinFETs in vertical power delivery systems. Conventional TCAD-based approaches are computationally intensive and insufficient for navigating the high-dimensional, nonlinear design space of advanced GaN devices. To address this, a physics-informed active learning framework is used to intelligently guide simulations, accelerating convergence while preserving accuracy. This ML-guided approach enables the discovery of optimal configurations by efficiently exploring key structural parameters -- most notably the GaN-to-AlGaN thickness ratio -- a long-standing focus of debate in device design. By systematically exploring key structural parameters, two optimized devices with aggressively scaled gate-to-drain lengths are identified. Single-fin, multi-channel simulations show that device~D2, with a thinner GaN channel relative to the AlGaN barrier, achieves higher drive current. However, in a 300-fin configuration, device~D1 outperforms device~D2 by delivering 3.3\,A at 0.49~ohm on-resistance -- approximately 2$\times$ better -- despite slightly higher parasitics. Both devices operate in a normally-off mode. Based on an application-specific figure of merit, device~D1 achieves 5\,pC$\cdot$ohm, demonstrating 2$\times$ greater switching efficiency than device~D2, while both designs outperform industrial benchmarks from different performance standpoints.