Detection in high fidelity of tipping points, the emergence of which is often induced by invisible changes in internal structures or/and external interferences, is paramountly beneficial to understanding and predicting complex dynamical systems (CDSs). Detection approaches, which have been fruitfully developed from several perspectives (e.g., statistics, dynamics, and machine learning), have their own advantages but still encounter difficulties in the face of high-dimensional, fluctuating datasets. Here, using the reservoir computing (RC), a recently notable, resource-conserving machine learning method for reconstructing and predicting CDSs, we articulate a model-free framework to accomplish the detection only using the time series observationally recorded from the underlying unknown CDSs. Specifically, we encode the information of the CDS in consecutive time durations of finite length into the weights of the readout layer in an RC, and then we use the learned weights as the dynamical features and establish a mapping from these features to the system’s changes. Our designed framework can not only efficiently detect the changing positions of the system but also accurately predict the intensity change as the intensity information is available in the training data. We demonstrate the efficacy of our supervised framework using the dataset produced by representative physical, biological, and real-world systems, showing that our framework outperforms those traditional methods on the short-term data produced by the time-varying or/and noise-perturbed systems. We believe that our framework, on one hand, complements the major functions of the notable RC intelligent machine and, on the other hand, becomes one of the indispensable methods for deciphering complex systems.

Locating the source that undergoes a diffusion-like process is a fundamental and challenging problem in complex network, which can help inhibit the outbreak of epidemics among humans, suppress the spread of rumors on the Internet, prevent cascading failures of power grids, etc. However, our ability to accurately locate the diffusion source is strictly limited by incomplete information of nodes and inevitable randomness of diffusion process. In this paper, we propose an efficient optimization approach via maximum likelihood estimation to locate the diffusion source in complex networks with limited observations. By modeling the informed times of the observers, we derive an optimal source localization solution for arbitrary trees and then extend it to general graphs via proper approximations. The numerical analyses on synthetic networks and real networks all indicate that our method is superior to several benchmark methods in terms of the average localization accuracy, high-precision localization and approximate area localization. In addition, low computational cost enables our method to be widely applied for the source localization problem in large-scale networks. We believe that our work can provide valuable insights on the interplay between information diffusion and source localization in complex networks.

Simultaneous outbreaks of contagion are a great threat against human life, resulting in great panic in society. It is urgent for us to find an efficient multiple sources localization method with the aim of studying its pathogenic mechanism and minimizing its harm. However, our ability to locate multiple sources is strictly limited by incomplete information about nodes and the inescapable randomness of the propagation process. In this paper, we present a valid approach, namely the Potential Concentration Label method, which helps locate multiple sources of contagion faster and more accurately in complex networks under the SIR(Susceptible-Infected-Recovered) model. Through label assignment in each node, our aim is to find the nodes with maximal value after several iterations. The experiments demonstrate that the accuracy of our multiple sources localization method is high enough. With the number of sources increasing, the accuracy of our method declines gradually. However, the accuracy remains at a slight fluctuation when average degree and network scale make a change. Moreover, our method still keeps a high multiple sources localization accuracy with noise of various intensities, which shows its strong anti-noise ability. I believe that our method provides a new perspective for accurate and fast multi-sources localization in complex networks.

Epidemic source localization is one of the most meaningful areas of research in complex networks, which helps solve the problem of infectious disease spread. Limited by incomplete information of nodes and inevitable randomness of the spread process, locating the epidemic source becomes a little difficult. In this paper, we propose an efficient algorithm via Bayesian Estimation to locate the epidemic source and find the initial time in complex networks with sparse observers. By modeling the infected time of observers, we put forward a valid epidemic source localization method for tree network and further extend it to the general network via maximum spanning tree. The numerical analyses in synthetic networks and empirical networks show that our algorithm has a higher source localization accuracy than other comparison algorithms. In particular, when the randomness of the spread path enhances, our algorithm has a better performance. We believe that our method can provide an effective reference for epidemic spread and source localization in complex networks.

Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neural network via a gradient flow matching strategy; subsequently, fine-tuning this network using Neural ODEs; and finally, constructing a general neural network to capture residuals. In this process, we apply the SNODEs framework to partial differential equation systems through Fourier analysis, achieving resolution-invariant modeling. Moreover, this framework integrates the strengths of symbolism and connectionism, boasting a universal approximation theorem while significantly enhancing interpretability and extrapolation capabilities relative to state-of-the-art baseline methods. We demonstrate this through experiments on several representative complex systems. Therefore, our framework can be further applied to a wide range of scientific problems, such as system bifurcation and control, reconstruction and forecasting, as well as the discovery of new equations.

Human mobility data are indispensable in modeling large-scale epidemics, especially in predicting the spatial spread of diseases and in evaluating spatial heterogeneity intervention strategies. However, statistical data that can accurately describe large-scale population migration are often difficult to obtain. We propose an algorithm model based on the network science approach, which estimates the travel flow data in mainland China by transforming location big data and airline operation data into network structure information. In addition, we established a simplified deterministic SEIR (Susceptible-Exposed-Infectious-Recovered)-metapopulation model to verify the effectiveness of the estimated travel flow data in the study of predicting epidemic spread. The results show that individual travel distance in mainland China is mainly within 100 km. There is far more travel between prefectures within the same province than across provinces. The epidemic spatial spread model incorporating estimated travel data accurately predicts the spread of COVID-19 in mainland China. The results suggest that there are far more travelers than usual during the Spring Festival in mainland China, and the number of travelers from Wuhan mainly determines the number of confirmed cases of COVID-19 in each prefecture.