Dynamic Learning: Mathematical Framework for Autonomous Knowledge Acquisition in Artificial Intelligence

Traditional artificial intelligence systems suffer from static learning paradigms that prevent adaptation after deployment, catastrophic forgetting when learning new tasks, and inability to autonomously identify learning opportunities. We present Dynamic Learning, a mathematical framework for continuous, autonomous knowledge acquisition in AI systems. Our framework introduces the Dynamic Learning Operator \(\Gamma \) that governs knowledge evolution through principled integration of new information while maintaining consistency. We establish convergence guarantees through contraction mapping analysis, prove stability against catastrophic forgetting, and derive information-theoretic bounds on learning capacity. The framework incorporates curiosity-driven exploration through an information-theoretic curiosity function and hierarchical memory consolidation. We prove that Dynamic Learning systems achieve near-optimal information acquisition rates while maintaining polynomial computational complexity. The system architecture comprises five interconnected components implementing autonomous learning through multi-loop control. Our theoretical analysis establishes PAC-learning bounds, regret guarantees for adaptive strategy selection, and robustness properties under adversarial conditions. Dynamic Learning provides rigorous mathematical foundations for creating AI systems capable of lifelong learning, establishing theoretical principles for next-generation adaptive AI architectures.