Researchers at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) have recently achieved a significant breakthrough in the modeling of complex systems. A research team led by Professor Rongling Wu has successfully integrated the Yau–Yau nonlinear filtering algorithm with stochastic network theory, resulting in a novel model capable of reconstructing the hidden internal interaction mechanisms of complex systems from highly noisy and chaotic data. This new framework, termed the Yau–Yau stochastic network, was reported in the paper “Statistical learning of stochastic complex systems via the Yau–Yau nonlinear filter,” which has been published in the international journal The Innovation (5-year Impact Factor: 40.2).

This work provides a new mathematical framework and a powerful analytical tool for addressing long-standing challenges in the analysis of complex systems across multiple domains, including life sciences, environmental ecology, and socio-economic systems. Shuyuan Xu, a jointly trained PhD student of BIMSA and the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, is the first author of the paper. Yu Wang, a jointly trained PhD student of BIMSA and the Institute of Statistics and Big Data, Renmin University of China, is the second author. Professor Rongling Wu serves as the corresponding author. BIMSA Assistant Researcher Ang Dong and postdoctoral researchers Shuang Wu and Yu Wang contributed to parts of the study. Professors Stephen Shing-Toung Yau and Shing-Tung Yau, the inventors of the Yau–Yau nonlinear filtering algorithm, provided important guidance on the overall design of the study and the interpretation of the algorithm.

Research Background: Noise Challenges and Real-World Demands in Complex Systems

From microscopic processes such as cellular signaling and neural circuits to macroscopic phenomena including ecosystem evolution, financial market fluctuations, and celestial dynamics, real-world systems are universally characterized by nonlinearity, uncertainty, and strong noise. The behavior of such systems is not fully deterministic, and the interactions among their internal components often undergo persistent random fluctuations driven by time evolution and environmental perturbations.

Most mainstream network modeling approaches are built upon deterministic frameworks, focusing on average interaction patterns or static topological structures. As a result, they generally fail to effectively characterize and quantify these intrinsic stochastic dynamics. Importantly, randomness is not merely background noise; rather, it often constitutes a fundamental mechanism through which systems maintain resilience, adaptability, and robust regulation. Consequently, there is an urgent need for new types of network models that can simultaneously capture nonlinear interactions, intrinsic stochasticity, and external noise, enabling a more fundamental understanding of complex system behavior as well as more accurate prediction and intervention.

Core Innovation: Reconstructing Stochastic Networks via the Yau–Yau Nonlinear Filter

To address these challenges, the research team introduced the Yau–Yau nonlinear filtering algorithm into complex system modeling. Originally developed by Professors Stephen Shing-Toung Yau and Shing-Tung Yau, this algorithm is distinguished by its ability to transform the Duncan–Mortensen–Zakai (DMZ) equation, which governs the evolution of conditional probability densities in stochastic systems, into a class of forward Kolmogorov (Fokker–Planck) equations that are more amenable to numerical computation through a series of elegant mathematical transformations.

Compared with classical Kalman filtering methods and their nonlinear extensions—such as the ensemble Kalman filter and the unscented Kalman filter—the Yau–Yau algorithm demonstrates superior estimation accuracy and numerical stability when dealing with systems exhibiting strong nonlinearity, high-dimensional state spaces, and significant observation noise.

Building on this foundation, the team developed the Yau–Yau stochastic network within the Idopnetwork analysis framework. While inheriting the strengths of traditional network approaches in revealing system structure, this model achieves a critical advance: it can dynamically and quantitatively disentangle and reconstruct interaction fluctuations and latent dependencies among system components that arise from stochastic perturbations, from noisy observational time series. This enables a more faithful and fine-grained mathematical description of the dynamic behavior of complex systems.

Systematic Validation: Evidence from Simulations and Experiments

The performance of the Yau–Yau stochastic network was rigorously validated through comprehensive numerical and experimental studies.

In simulation experiments, the researchers constructed a variety of representative nonlinear stochastic systems with different levels of complexity and noise intensity. The results demonstrated that the Yau–Yau algorithm delivered consistent and robust performance, with particularly notable improvements observed after orthogonalization of the observation data.

For real-world validation, the team designed and conducted a carefully controlled microbial ecology experiment. Escherichia coli, Staphylococcus aureus, and Pseudomonas aeruginosa were cultured together in a shared, controllable environment, forming an observable and simplified microbial ecosystem. Time-resolved abundance data for each species were continuously collected and analyzed using both traditional deterministic network models and the Yau–Yau stochastic network.

The deterministic models yielded only static, averaged competitive or cooperative relationships among species. In contrast, the Yau–Yau stochastic network successfully extracted the stochastic fluctuations embedded in the temporal abundance trajectories and accurately mapped them to dynamically evolving interaction strengths. This analysis suggests that microbial populations adaptively adjust their interaction strategies in response to microenvironmental changes such as nutrient depletion and metabolite accumulation. These results provide strong evidence of the model’s analytical power and application potential in real biological complex systems.

Broad Applications and Future Directions

The Yau–Yau stochastic network may offer a general analytical framework for addressing key challenges across multiple interdisciplinary frontiers:

• Systems and synthetic biology: enabling the analysis of stochastic dynamics in gene regulatory networks, metabolic networks, and microbial communities, thereby guiding the design of more robust synthetic biological systems.
• Neuroscience and brain disorders: facilitating the decoding of dynamically changing functional connectivity among neuronal populations from noisy signals such as EEG and fMRI, and offering new perspectives on cognition and neurological disease mechanisms.
• Environmental ecology and climate science: supporting the modeling and prediction of stochastic evolution and stability of ecological networks under extreme climate events.
• Finance and economic systems: allowing the identification of hidden stochastic pathways of risk contagion and the evolution of investor behavior networks amid market fluctuations.

Looking ahead, the research team plans to pursue several directions:

1. Expanding application scope: promoting applications of the model to major biomedical problems, including tumor microenvironments and gut microbiota–host interactions.

2. Enhancing model capability: developing higher-order stochastic interaction models that capture multi-node cooperative effects, and exploring their integration with GLMY theory and topological data analysis to achieve deeper insights into the relationship between network structure and function.

3. Improving methodological generality: investigating approaches for network construction using discontinuous, non-uniformly sampled, or even static data, thereby reducing dependence on stringent data acquisition conditions.

Paper link: https://www.cell.com/the-innovation/fulltext/S2666-6758(26)00014-7