Nonlinear dissipative stochastic systems are widespread but face limitations from three main issues in traditional paradigms: symplectic geometry’s incompatibility with dissipation and stochasticity, classical stochastic analysis’s failure to capture high-order statistical information (missing path dependence), and underuse of high-dimensional observational data through empirical/semi-empirical parameterisations. To overcome these, we introduce a data-driven contact geometry paradigm that reverses the conventional “model-to-data” approach to “data→probability→geometry→dynamics.” Based on stochastic vector bundles, contact geometry, and the least constraint theorem, this framework encodes the evolution of probability, both state-driven distributional changes and distribution-driven state dependencies, into geometric structures. Starting from observational data, we construct infinite-order jet bundles to preserve complete statistical information, derive the system dynamics through variational principles, and inherently incorporate dissipation and path dependence. When considering water cycle dynamics, this paradigm enables parameter-free system reconstruction and stable long-term predictions grounded in global invariants, without relying on phenomenological parameterisations. It offers a unified first-principles framework for characterising nonlinear dissipative stochastic systems.

In 2000, the concept of the “Three Yellow River Initiative” (Prototype, Model, Digital) established a comprehensive framework for Yellow River study and management. After two decades, the limitations of these isolated modelling approaches have become increasingly apparent, particularly the insurmountable scaling challenges of physical models and the inherent simplification gaps of digital simulations that cannot capture the nonlinear, multi-scale water-sediment dynamics of the river. This paper introduces the Hi-Fi (High-Fidelity) Yellow River paradigm, an intelligent, closed-loop governance system that leverages artificial intelligence to enable instantaneous scenario manifold switching for water management based on real-time observations from the Prototype Yellow River. We formalize this paradigm using contact geometry: the state space of river dynamics forms a contact manifold $(\mathcal{E},\mathrm{\Theta })$, where the transition from model prediction to observed reality manifests as a transversal transition between scenario manifolds along $\mathrm{k}\mathrm{e}\mathrm{r}(\mathrm{d}\mathrm{\Theta })$. This mathematical framework rigorously captures the core of high-fidelity research—grounding governance decisions in real-world scenarios rather than simplified representations—while providing kinematic constraints for adaptive water-sediment regulation strategies. The framework unifies the “Three Yellow River Initiative” into a single reality-responsive intelligent system, transforming static prediction-based governance into dynamic, scenario-driven decision-making for the Yellow River basin.

Blue water, which includes freshwater in reservoirs, lakes, rivers, and aquifers, plays a crucial role in sustaining life on Earth and serves as a direct water resource for human societies. The distribution of blue water resources shows spatiotemporal variations, making it important to comprehensively understand the distribution patterns and influencing factors of blue water resources in river basins. In this study, the normalized runoff was analyzed using methods including stationarity testing, correlation analysis, and variance analysis to explore its spatiotemporal patterns and the relationships with river structure. The results indicate that the normalized runoff time series show stationarity at a global scale and show correlations with river network density and the distance to the nearest downstream sink, but the strength of these correlations varies across different regions on different continents. The normalized runoff also shows variations among river order. Further research could examine predicting blue water availability given information on river network structure and climate.

Green water, temporarily stored in the unsaturated zone of the soil or on the top of vegetation, plays an important role in the natural water cycle and water resource management. Unlike surface water flows, green water recycling has no obvious boundary restrictions and is characterized by long-distance transport. Thus, green water from specific basins can support precipitation not only within the basin but also in neighboring basins and even distant areas. However, water scarcity is typically viewed as a local problem for which management measures at the basin scale seem sufficient without fully considering the interdependence of green water among basins. Limited knowledge of green water recycling could make hydrological water assessments inaccurate. Here, we quantify the global green water recycling network, which links green water as a moisture supply to precipitation and thus contributes to local and remote freshwater resources. Our results show that large basins such as the Congo (green water recycling ratio: 56.6%) are comparatively independent, but still rely to some extent on moisture supplied by other basins (contributing 21.4% of precipitation for the Congo), contradicting the conventional view of basin independence in water management. This illustration of basin interdependence highlights the need for global water governance in the context of the severe global water crisis, and this global green water recycling network provides the basis for a comprehensive assessment of water resources and scarcity under global change.

The hydrosphere, a concept stemming from geology and geography, is conventionally recognized as a layer of water on the Earth. Despite its frequent mention in literature, systematic studies on this subject are rare yet. Presented in this paper is our recent perspective on the hydrosphere, providing a comprehensive overview of the research focused on the functions of water in global cycling. We defined the hydrosphere as a dynamical system comprising water materials in solid, liquid, or vapor states, which cyclically interact on a global scale, forming a complex network in alignment with the inherent nature of water and its cycling process. The challenges associated with studying the hydrosphere are discussed, emphasizing the significance of understanding its dynamics, topology, and infordynamics. Furthermore, potential approaches to achieve a thorough comprehension of the hydrosphere are discussed.