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Mathematics
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Advanced Machine Learning Research in Complex System
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Advanced Machine Learning Research in Complex System

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 31 October 2026 | Viewed by 4964

Special Issue Editors

Dr. Duxin Chen

E-Mail Website
Guest Editor
School of Mathematics, Southeast University, Jiulonghu Campus, Nanjing 211189, China
Interests: complex systems; complex networks; control mechanisms; data analysis; model training; artificial intelligence
Special Issues, Collections and Topics in MDPI journals
Dr. Mengli Wei

E-Mail Website
Guest Editor
School of Mathematics, Southeast University, Nanjing 210096, China
Interests: distributed algorithms for complex networks; privacy protection for machine learning

Special Issue Information

Dear Colleagues,

Advanced machine learning is increasingly central to the understanding, modeling, and management of complex systems composed of many interacting components that exhibit nonlinear, uncertain, and emergent behaviors. Such systems arise in domains including biological and social networks, cyber–physical infrastructures, energy and transportation systems, and large-scale industrial and information platforms. This Special Issue, entitled “Advanced Machine Learning Research in Complex System,” aims to provide a focused venue for methodological advances at this interface. We particularly welcome contributions that develop new learning paradigms, models, and algorithms tailored to complex systems—such as graph-and network-based learning, representation learning for dynamical systems, causal and structure-learning methods, scalable optimization and inference schemes, and robust or safe reinforcement learning—together with accompanying theoretical analysis. Of special interest are works that rigorously address fundamental challenges including high dimensionality, multi-scale and spatio-temporal dependencies, distribution shifts, partial observability, robustness to uncertainty, safety constraints, and interpretability of learned models and policies. While application studies are welcome, they should primarily serve to validate and elucidate methodological innovations in real-world complex systems. By bringing together researchers and practitioners from machine learning, systems science, and domain-specific fields, this Special Issue seeks to advance advanced machine learning as a key methodological enabler for the analysis, prediction, control, and governance of complex systems.

Dr. Duxin Chen
Dr. Mengli Wei
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • complex systems
  • machine learning
  • distributed control and optimization
  • decentralized learning
  • privacy protection of machine learning in complex systems
  • time series analysis in complex systems
  • applications of complex systems

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Published Papers (5 papers)

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Research

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17 pages, 2621 KB  
Article
Traffic Flow Prediction in Complex Transportation Networks via a Spatiotemporal Causal–Trend Network
by Xingyu Feng, Lina Sheng, Linglong Zhu, Yishan Feng, Chen Wei, Xudong Xiao and Haochen Wang
Mathematics 2026, 14(3), 443; https://doi.org/10.3390/math14030443 - 27 Jan 2026
Cited by 1 | Viewed by 735
Abstract
Traffic systems are quintessential complex systems, characterized by nonlinear interactions, multiscale dynamics, and emergent spatiotemporal patterns over complex networks. These properties make traffic prediction highly challenging, as it requires jointly modeling stable global topology and time-varying local dependencies. Existing graph neural networks often [...] Read more.
Traffic systems are quintessential complex systems, characterized by nonlinear interactions, multiscale dynamics, and emergent spatiotemporal patterns over complex networks. These properties make traffic prediction highly challenging, as it requires jointly modeling stable global topology and time-varying local dependencies. Existing graph neural networks often rely on predefined or static learnable graphs, overlooking hidden dynamic structures, while most RNN- or CNN-based approaches struggle with long-range temporal dependencies. This paper proposes a Spatiotemporal Causal–Trend Network (SCTN) tailored to complex transportation networks. First, we introduce a dual-path adaptive graph learning scheme: a static graph that captures global, topology-aligned dependencies of the complex network, and a dynamic graph that adapts to localized, time-varying interactions. Second, we design a Gated Temporal Attention Module (GTAM) with a causal–trend attention mechanism that integrates 1D and causal convolutions to reinforce temporal causality and local trend awareness while maintaining long-range attention. Extensive experiments on two real-world PeMS traffic flow datasets demonstrate that SCTN consistently achieves superior accuracy compared to strong baselines, reducing by 3.5–4.5% over the best-performing existing methods, highlighting its effectiveness for modeling the intrinsic complexity of urban traffic systems. Full article
(This article belongs to the Special Issue Advanced Machine Learning Research in Complex System)
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24 pages, 11970 KB  
Article
Data-Driven Probabilistic Wind Power Forecasting and Dispatch with Alternating Direction Method of Multipliers over Complex Networks
by Lina Sheng, Nan Fu, Juntao Mou, Linglong Zhu and Jinan Zhou
Mathematics 2026, 14(1), 112; https://doi.org/10.3390/math14010112 - 28 Dec 2025
Cited by 1 | Viewed by 593
Abstract
This paper proposes a privacy-preserving framework that couples probabilistic wind power forecasting with decentralized anomaly detection in complex power networks. We first design an adaptive federated learning (FL) scheme to produce probabilistic forecasts for multiple geographically distributed wind farms while keeping their raw [...] Read more.
This paper proposes a privacy-preserving framework that couples probabilistic wind power forecasting with decentralized anomaly detection in complex power networks. We first design an adaptive federated learning (FL) scheme to produce probabilistic forecasts for multiple geographically distributed wind farms while keeping their raw data local. In this scheme, an artificial neural network with quantile regression is trained collaboratively across sites to provide calibrated prediction intervals for wind power outputs. These forecasts are then embedded into an alternating direction method of multipliers (ADMM)-based load-side dispatch and anomaly detection model for decentralized power systems with plug-and-play industrial users. Each monitoring node uses local measurements and neighbor communication to solve a distributed economic dispatch problem, detect abnormal load behaviors, and maintain network consistency without a central coordinator. Experiments on the GEFCom 2014 wind power dataset show that the proposed FL-based probabilistic forecasting method outperforms persistence, local training, and standard FL in RMSE and MAE across multiple horizons. Simulations on IEEE 14-bus and 30-bus systems further verify fast convergence, accurate anomaly localization, and robust operation, indicating the effectiveness of the integrated forecasting–dispatch framework for smart industrial grids with high wind penetration. Full article
(This article belongs to the Special Issue Advanced Machine Learning Research in Complex System)
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22 pages, 602 KB  
Article
Projection-Free Decentralized Federated Learning with Privacy Guarantees in Complex Systems
by Chen Zhang, Yan Wang, Lin Yuan, Chaopeng Wang, Peishuo Li and Hong Wang
Mathematics 2026, 14(1), 81; https://doi.org/10.3390/math14010081 - 25 Dec 2025
Viewed by 697
Abstract
Constrained decentralized federated learning (DFL) has garnered significant attention for its decentralized approach, primarily tackling distributed constrained optimization problems in complex systems. In complex scenarios, the computational overhead of projecting onto constraint sets can be substantial. Moreover, in distributed data storage, transmitted model [...] Read more.
Constrained decentralized federated learning (DFL) has garnered significant attention for its decentralized approach, primarily tackling distributed constrained optimization problems in complex systems. In complex scenarios, the computational overhead of projecting onto constraint sets can be substantial. Moreover, in distributed data storage, transmitted model parameters or gradients may contain privacy-sensitive data, risking privacy breaches. To tackle these issues, this paper presents a Differentially Private Decentralized Online Projection-Free Optimization (DPDOPFO) federated learning algorithm. DPDOPFO adopts a Frank–Wolfe (FW) methodology, avoiding projection operations in each iteration. It utilizes gradient information to guide updates, reducing computational costs for high-dimensional problems. Integrated with differential privacy techniques, DPDOPFO ensures individual privacy protection. Additionally, it employs gradient tracking techniques, enhancing model performance and robustness. Theoretical analysis demonstrates DPDOPFO’s sublinear convergence and ϵ-differential privacy. Simulation experiments validate its effectiveness in federated learning problems. Full article
(This article belongs to the Special Issue Advanced Machine Learning Research in Complex System)
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Review

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27 pages, 1494 KB  
Review
A Survey on Missing Data Generation in Networks
by Qi Shao, Ruizhe Shi, Xiaoyu Zhang and Duxin Chen
Mathematics 2026, 14(2), 341; https://doi.org/10.3390/math14020341 - 20 Jan 2026
Viewed by 824
Abstract
The prevalence of massive, multi-scale, high-dimensional, and dynamic data sets resulting from advances in information and network communication technologies is frequently hampered by data incompleteness, a consequence of complex network structures and constrained sensor capabilities. The necessity of complete data for effective data [...] Read more.
The prevalence of massive, multi-scale, high-dimensional, and dynamic data sets resulting from advances in information and network communication technologies is frequently hampered by data incompleteness, a consequence of complex network structures and constrained sensor capabilities. The necessity of complete data for effective data analysis and mining mandates robust preprocessing techniques. This comprehensive survey systematically reviews missing value interpolation methodologies specifically tailored for time series flow network data, organizing them into four principal categories: classical statistical algorithms, matrix/tensor-based interpolation methods, nearest-neighbor-weighted methods, and deep learning generative models. We detail the evolution and technical underpinnings of diverse approaches, including mean imputation, the ARMA family, matrix factorization, KNN variants, and the latest deep generative paradigms such as GANs, VAEs, normalizing flows, autoregressive models, diffusion probabilistic models, causal generative models, and reinforcement learning generative models. By delineating the strengths and weaknesses across these categories, this survey establishes a structured foundation and offers a forward-looking perspective on state-of-the-art techniques for missing data generation and imputation in complex networks. Full article
(This article belongs to the Special Issue Advanced Machine Learning Research in Complex System)
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28 pages, 789 KB  
Review
An Overview of Spatiotemporal Network Forecasting: Current Research Status and Methodological Evolution
by Chenchen Yang, Wenbing Zhang and Yingjiang Zhou
Mathematics 2026, 14(1), 18; https://doi.org/10.3390/math14010018 - 21 Dec 2025
Viewed by 1682
Abstract
Time series and spatio-temporal forecasting are fundamental tasks for complex system modeling and intelligent decision-making, with broad applications in transportation, meteorology, finance, healthcare, and public safety. Compared with simple univariate time series, real-world spatio-temporal data exhibit rich temporal dynamics and intricate spatial interactions, [...] Read more.
Time series and spatio-temporal forecasting are fundamental tasks for complex system modeling and intelligent decision-making, with broad applications in transportation, meteorology, finance, healthcare, and public safety. Compared with simple univariate time series, real-world spatio-temporal data exhibit rich temporal dynamics and intricate spatial interactions, leading to heterogeneity, non-stationarity, and evolving topologies. Addressing these challenges requires modeling frameworks that can simultaneously capture temporal evolution, spatial correlations, and cross-domain regularities. This survey provides a comprehensive synthesis of forecasting methods, spanning statistical algorithms, traditional machine learning approaches, neural architectures, and recent generative and causal paradigms. We review the methodological evolution from classical linear models to deep learning–based temporal modules and emphasize the role of attention-based Transformers as general-purpose sequence architectures. In parallel, we distinguish these architectural advances from pre-trained foundation models for time series and spatio-temporal data (e.g., large models trained across diverse domains), which leverage self-supervised objectives and exhibit strong zero-/few-shot transfer capabilities. We organize the review along both data-type and architectural dimensions—single long-term time series, Euclidean-structured spatio-temporal data, and graph-structured spatio-temporal data—while also examining advanced paradigms such as diffusion models, causal modeling, multimodal-driven frameworks, and pre-trained foundation models. Through this taxonomy, we highlight common strengths and limitations across approaches, including issues of scalability, robustness, real-time efficiency, and interpretability. Finally, we summarize open challenges and future directions, with a particular focus on the joint evolution of graph-based, causal, diffusion, and foundation-model paradigms for next-generation spatio-temporal forecasting. Full article
(This article belongs to the Special Issue Advanced Machine Learning Research in Complex System)
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