Causal emergence in complex systems

Focusing on complex systems: a quantitative description is provided for emergence, causality, and causal emergence．The relationship between causality and emergence is elucidated． Two quantitative methods, coarse-graining and information decomposition, are reviewed for characterizing causal emergence． Two causal emergence identification methods based on information decomposition and neural information compression are presented． Basic principles, advantages, disadvantages, and relevant applications of each method are detailed． The framework of causal emergence is defined based on coarse-graining method, which can be applied to discrete dynamical systems． The use of information decomposition method in solving problems relies on redundant information． The proposed indicators based on information decomposition can identify causal emergence in data and find a sufficient condition． The data-driven neural information compression method is scalable and applicable to continuous dynamical systems． It can automatically extract coarse-grained functions at different levels and construct hierarchical dynamics, and identify causal emergence in different types of dynamical systems． Existing methods are improved and applied to identify more complex systems, automate learning groups, and address challenges in general dynamics of large-scale models and other related problems．