Self-supervised reservoir computing with spatial-temporal encoding for identifying critical transitions.

Anticipating critical transitions and identifying bifurcation types in complex systems remains a major challenge due to high dimensionality and limited labeled data. In this study, we propose spatial-to-temporal auto reservoir computing, a self-supervised approach of reservoir computing designed to detect early warning signals of critical transitions and identify the corresponding bifurcation types, including transcritical, period-doubling, and Neimark-Sacker bifurcations. Grounded on Takens' embedding theorem, it performs spatial-to-temporal information transformation via a reservoir structure, by encoding high-dimensional spatial data into the temporal dynamics of a single representative variable. This ultralow one-dimensional representation is obtained in a self-supervised and analytical manner, making it particularly suited for critical transition analyses in time-varying, high-dimensional systems. In addition, based on the Poincaré recurrence principle, the proposed method captures the structural information of the local phase space by constructing a spatial neighborhood network centered at each input state to enhance the robustness. The proposed method is validated on synthetic models and real-world datasets across multiple domains including paleoclimate, ecology and physiology, consistently achieving high accuracy and robustness under varying noise levels and parameter choices.