Mingzhe Yang

Reductionism has underpinned modern science since the 17th-century Scientific Revolution. This methodology decomposes systems into minimal units and deduces wholes from parts, succeeding across physical, life and social sciences. However, reductionism reveals limits as the scientific frontier shifts from identifying building blocks to understanding how components generate collective behavior. We possess massive data and precise local equations but fail to predict cell fates, financial crises or cognitive emergence in neural networks. Complex systems science—a mid-20th-century discipline exploring the structures, behaviors, evolution and laws of complex systems—integrates with diverse fields to transform scientific paradigms. Rapid AI development triggers this paradigm shift. AI algorithms handle massive data and complex systems, while AI systems themselves constitute complex systems whose progress depends on complex systems science. In this NSR Forum, we convene five researchers to discuss the essence of complex systems science and its integration with other disciplines to support a pivotal scientific paradigm shift in the AI era. Tingting Gao Postdoc Researcher, Network Science Institute, Northeastern University, USA Wei Lin Professor, School of Mathematical Sciences and Research Institute of Intelligent Complex Systems, Fudan University, China Yu Liu Associate Professor, Department of Systems Science, Beijing Normal University at Zhuhai, China Jiang Zhang Professor, School of Systems Science, Beijing Normal University, China Lei Guo (Chair) Professor, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China

Causal emergence (CE) based on effective information (EI) demonstrates that macrostates can exhibit stronger causal effects than microstates in dynamics. However, the identification of CE and the maximization of EI both rely on coarse-graining strategies, which is a key challenge. A recently proposed CE framework based on approximate dynamical reversibility, utilizing singular-value decomposition (SVD), is independent of coarse-graining. Still, it is limited to transition probability matrices in discrete states. To address this, this article proposes a CE quantification framework for Gaussian iterative systems, based on approximate dynamical reversibility derived from the SVD of inverse covariance matrices in forward and backward dynamics. The positive correlation between SVD-based and EI-based CE, along with the equivalence condition, is given analytically. After that, we provide precise coarse-graining strategies directly from singular-value spectra and orthogonal matrices. This new framework can be applied to any dynamical system with continuous states and Gaussian noise, such as autoregressive growth models, Markov-Gaussian systems, and even SIR modeling using neural networks. Numerical simulations on typical cases validate our theory and offer a new approach to studying the CE phenomenon, emphasizing noise and covariance over dynamical functions in both known models and machine learning.

Living systems maintain structural and functional stability while adapting to environmental changes, a capability independent of specific system‑environment states. Existing frameworks, such as self‑organization theory and free energy principles, cannot measure system‑environment interaction at the causal level. In this article, we propose a new causal indicator, Flexibility, to measure a system’s ability to respond to its environment. We construct this indicator based on information theory and interventional operations from causal inference, which implies the indicator depends only on the dynamical causal mechanism. We show this indicator satisfies the axiom system of the partial information decomposition (PID) framework and decomposes into two components, Expansiveness and Introversion, which correspond to different strategic tendencies for environmental adaptation. This decomposition reveals that Flexibility depends on the entanglement between system‑environment variables and noise magnitude. Through experiments on cellular automata (CA), random Boolean networks, and real gene regulatory networks (GRNs), we validate that the indicator identifies the most complex and computationally capable CA (Langton’s parameter at 0.5), while demonstrating that feedback loops carrying important biological functions in GRNs exhibit the highest flexibility. We also find that flexibility peaks at a moderate level of dynamical noise. Furthermore, we combine this framework with machine learning techniques to demonstrate its applicability when the underlying dynamics are unknown.

Quantifying emergence and modeling emergent dynamics in a data-driven manner for complex dynamical systems is challenging due to the fact that emergent behaviors cannot be directly captured by micro-level observational data. Thus, it is crucial to develop a framework to identify emergent phenomena and capture emergent dynamics at the macro-level using available data. Inspired by the theory of causal emergence (CE), this paper introduces a machine learning framework to learn macro-dynamics in an emergent latent space and quantify the degree of CE. The framework maximizes effective information, resulting in a macro-dynamics model with enhanced causal effects. Experimental results on simulated and real data demonstrate the effectiveness of the proposed framework. It quantifies degrees of CE effectively under various conditions and reveals distinct influences of different noise types. It can learn a one-dimensional coarse-grained macro-state from functional magnetic resonance imaging data to represent complex neural activities during movie clip viewing. Furthermore, improved generalization to different test environments is observed across all simulation data.