Community extraction for multi-layer networks is a fundamental problem. Community extraction methods for multi-layer networks typically rely on a single-layer network, weights for each layer, and an additional clustering step. Current methods usually utilize average or empirical weights for each layer, which may not be reasonable. To address the challenge of assigning better weights to the layers, this paper proposes a Spectral Clustering Community Detection (SCCD) optimization model based on a unified similarity matrix. Specifically, the unified similarity matrix is defined via an optimization model that uses a weighted combination of similarity matrices from each layer, where the layer weights are learned under simplex constraints. A rank constraint is further imposed on the Laplacian matrix of the unified similarity matrix, which intrinsically leads to the division of the nodes into the desired number of clusters. Then, the SCCD optimization model is proposed, and its ability to generate community partitions is shown. An alternating minimization algorithm with simplex projections and closed-form updates is developed to solve the model, and its convergence is proven. Finally, numerical experiments are conducted on both synthetic and real-world multi-layer networks. Compared with several methods reported in the references, our method improves average NMI by 6.78% and ARI by 6.50% over the strongest baselines on the AUCs, UCI mfeat, Wikipedia, and Primary school networks—see Table III.

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.

Maritime networks constitute a critical pillar of the global economic system and reflect intricate trade relationships between nations. Meanwhile, the application of complex network theory has increasingly highlighted the intrinsic link between node influence and dependencies among non-adjacent nodes. However, a central challenge lies in uncovering latent dependencies—mediated by intermediaries—between non-directly connected nations, which is crucial for the precise modeling of maritime networks and the analysis of indirect influence. In this study, construct and optimize a national-level multi-order network (MON) by mining multi-order dependency rules through advanced network construction methods. Empirical analysis of real-world ship navigation trajectory data (August 2018 and August 2023) reveals increasingly interconnected global dependencies. The proposed method successfully quantifies multi-order influence, identifies strategic hubs, tracks the evolution of influence, and offers actionable insights into geopolitical shifts and strategies for trade resilience.

Modeling complex systems using standard neural ordinary differential equations (NODEs) often faces some essential challenges, including high computational costs and susceptibility to local optima. To address these challenges, we propose a simulation-free framework, called Fourier NODEs (FNODEs), that effectively trains NODEs by directly matching the target vector field based on Fourier analysis. Specifically, we employ the Fourier analysis to estimate temporal and potential high-order spatial gradients from noisy observational data. We then incorporate the estimated spatial gradients as additional inputs to a neural network. Furthermore, we utilize the estimated temporal gradient as the optimization objective for the output of the neural network. Later, the trained neural network generates more data points through an ODE solver without participating in the computational graph, facilitating more accurate estimations of gradients based on Fourier analysis. These two steps form a positive feedback loop, enabling accurate dynamics modeling in our framework. Consequently, our approach outperforms state-of-the-art methods in terms of training time, dynamics prediction, and robustness. Finally, we demonstrate the superior performance of our framework using a number of representative complex systems.

Abstract This article presents a new DIRECT-type SCABALL (scattering balls) algorithm with a new partition method for derivation-free optimization problems. It does not focus on dividing the region of interest into specific geometric shapes, but rather scatters several balls to cover it. In SCABALL, several potential optimal regions are selected at each iteration, and they are covered by smaller balls sequentially. In this way, the SCABALL ensures the everywhere dense convergence. The center points and radii of the scattered balls significantly influence the efficiency of SCABALL; therefore, the minimax designs are used in the initial and sequential stages to obtain better coverage. The SCABALL parameters, including the number of balls and their radii, were analyzed by numerical investigation. We provided the empirical choices for those parameters and found that the balls’ radii can be contracted to balance efficiency and global convergence. Numerical experiments show that the SCABALL algorithm is locally biased and robust.

Surrogate models are widely used to mimic complex systems to reduce the high experimental cost. As the system becomes high-dimensional and complex, there is an increasing demand for building relatively simplified surrogate models that capture key variables and represent complex interactions. This study proposes the annealing combinable Gaussian process, an integrated solution for identifying key variables and constructing the high-precision surrogate model. Firstly, to identify redundant variables, this study optimises variables selection with a modified simulated annealing algorithm over the complete model space. This process is called the outer loop. Secondly, to improve the model accuracy and structure, this study constructs a non-parametric Gaussian process model with additive or multiplicative kernel, effectively extracting high-order interactions. Simultaneously, a Markov chain is proposed to sample the model space. The conditional entropy is used as the scoring rule for the simulated annealing algorithm of the outer loop. It is named the inner loop. This study also discusses the rationality of conditional entropy as a criterion. The annealing combinable Gaussian process performs well in various application scenarios, including regression and classification problems. Finally, our method is implemented with the particle-in-cell simulation to find out the key physics parameters.

Sensitivity analysis plays an important role in analyzing most influential factors of models. There are few studies on sensitivity analysis (SA) of continuous time models. By using the concept of information flow, this paper presents a new sensitivity analysis method for dynamical models. By combining the direction and magnitude of information transfer, the method builds a novel framework for SA, with the rigorous mathematical and statistic theories on information flow. This method is effective and feasible for both static systems and dynamical systems including continuous time models and discrete time series. The new method is applied to an example of linear dynamical models and the Lorenz system, and the results indicate that the method could find out the most influential variables, which conforms with entropy method, and it also give more detailed information for system analysis.