Three-World Hierarchy for General Neural-Network-in-the-Loop Stochastic Dynamical System

The remarkable success of neural networks, both in theory and practice, has led to their increasing integration with the physical world, making physically interactive world models a tangible reality. The interaction between neural networks and the physical world is managed through human-crafted algorithms, reinforcement learning, and, more recently, world models. However, the standard neural network workflow-defining the network, training with data, and deploying to the real world-faces significant challenges when dealing with these complex, stochastic dynamical systems in real-world settings. These challenges include hallucination, out-of-distribution issues, and long-tail events. This manuscript proposes a novel hierarchical framework composed of three distinct levels of "worlds" to comprehensively describe general neural-network-in-the-loop stochastic dynamical systems: the data world, model world, and real world. Furthermore, to quantify the divergence between these multi-modal worlds, we introduce a new distance measurement called Fréchet World Distance (FWD). FWD generalizes the conventional Fréchet distance to accommodate dynamic and multi-modal settings, providing a crucial tool for analyzing and optimizing the interaction between neural networks and the physical environment.