Geometric Flow Grounding: A Unified Manifold Decoupling Framework for Dynamics Discovery and Verification

Modeling complex dynamics from observational data is fundamental to scientific discovery and artificial intelligence. However, existing approaches are often plagued by the entanglement of static state representations and instantaneous motion, leading to accumulated errors and off-manifold hallucinations where predicted trajectories violate intrinsic geometric constraints. To address this, we propose Geometric Flow Grounding, a unified framework that enforces dynamic evolution strictly along the tangent bundle of the learned data manifold via a differentiable Neural Tangent Projection Layer. By geometrically decoupling state representation from tangential dynamics, our method generalizes across diverse data regimes. In scientific discovery, GFG reduces numerical aliasing and improves long-horizon stability in sparse dynamical systems, while recovering interpretable gene regulatory motifs from single-cell data. For trustworthy AI, the projection residual provides a zero-shot metric for deepfake video detection by revealing inconsistencies with the implicit flow of pre-trained world models. Our results establish manifold-constrained projection as a universal operator for both discovering natural laws and verifying synthetic content. Code will be available at \url{https://github.com/yuchang97/GFG-public}

Many scientific and AI tasks require understanding how systems change over time, such as how cells develop, how physical objects move, or how video frames evolve. Existing models often learn these changes in a high-dimensional space without respecting the hidden geometric structure of real data. As a result, their predictions can drift into unrealistic states or mix different possible motions together. We introduce Geometric Flow Grounding, a framework that keeps learned dynamics on the data’s underlying geometric surface. It separates a system’s current state from its direction of movement, then projects each predicted movement onto locally valid directions. This helps prevent impossible trajectories and makes distinct motion patterns easier to identify. Across physics, biology, and video analysis, our approach improves equation discovery from sparse observations, produces more plausible cell-state transitions, and helps detect AI-generated videos with unrealistic motion. These results show that geometric constraints can make dynamic models more reliable and interpretable.