Reconstruction and prediction for nonlinear dissipative stochastic systems: A data-driven contact geometry paradigm [version 1]

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.