This position paper presents a vision for self-driving particle accelerators that operate autonomously with minimal human intervention. We propose that future facilities be designed through artificial intelligence (AI) co-design, where AI jointly optimizes the accelerator lattice, diagnostics, and science application from inception to maximize performance while enabling autonomous operation. Rather than retrofitting AI onto human-centric systems, we envision facilities designed from the ground up as AI-native platforms. We outline nine critical research thrusts spanning agentic control architectures, knowledge integration, adaptive learning, digital twins, health monitoring, safety frameworks, modular hardware design, multimodal data fusion, and cross-domain collaboration. This roadmap aims to guide the accelerator community toward a future where AI-driven design and operation deliver unprecedented science output and reliability.