Modern natural language systems rely on large language models, whose sheer size makes full fine-tuning costly in computation, graphics processing unit (GPU) memory, and storage. Low-rank adaptation (LoRA) sidesteps most of that cost. It keeps the pre-trained weights frozen and captures each task-specific change as the product of two smaller matrices, so adapting a model reduces to a low-rank decomposition. This review covers LoRA and its main variants and pays particular attention to the linear algebra behind them. It first explains why the low intrinsic dimension of fine-tuning makes low-rank updates effective, then compares the major variants: quantized LoRA (QLoRA), quantization-aware LoRA (QA-LoRA), adaptive low-rank adaptation (AdaLoRA), sparse low-rank adaptation (SoRA), and weight-decomposed low-rank adaptation (DoRA). Across published studies, these methods come close to full fine-tuning accuracy while updating well under one percent of a model's parameters in some settings. For reference, LoRA cuts the trainable parameter count of Generative Pre-trained Transformer 3 (GPT-3) by four orders of magnitude, and QLoRA brings a 65-billion-parameter model within the memory of one 48 GB card. Open problems remain in choosing the rank, comparing results across studies, limiting quantization loss, and combining multiple adapters without interference. Ultimately, an established piece of linear algebra, approximating high-dimensional objects in low-dimensional subspaces, is what keeps the adaptation of very large models affordable.

Accurate short-term wind power prediction is a critical foundation for improving the accommodation capacity and dispatching security of power systems. Aiming at the problems of multicollinearity in traditional models and large prediction errors under extreme fluctuations, this paper proposes a prediction method combining multi-feature fusion ridge regression with extreme value theory. First, multiple ridge regression branches are constructed, and multi-dimensional electrical features such as voltage and reactive power are introduced. L2 regularization is applied to alleviate variable multicollinearity, and a difference strategy is adopted to adapt to the inertial characteristics of short-term power. Second, the optimal backbone model for different prediction horizons is adaptively matched based on validation set errors, and weighted fusion with the persistence model is performed to output point prediction results. Finally, the POT-GPD model is used to fit the tail of errors, and a 95% probabilistic prediction interval is constructed. Experiments show that the average error of the proposed model across multiple horizons outperforms comparison models, and the prediction interval coverage rate exceeds 95%, which balances both conventional prediction accuracy and early warning capability for extreme wind power fluctuation risks.

Data-driven modelling of nonlinear dynamical systems based on machine learning is reviewed in this paper. Nonlinear dynamics refer to all sorts of complicated and irregular phenomena in mathematics, physics, engineering and life that are not well explained by linear models. Based on the accumulation of observation data from sensors, experiments and high-fidelity simulations, machine learning has recently begun to be applied in system identification, state estimation and the discovery of governing equations. This paper systematically presents the basic ideas of dynamical systems and then introduces several data-driven methods, such as Sparse Identification of Nonlinear Dynamics (SINDy), Neural Ordinary Differential Equations (Neural ODEs) and Koopman operator theory. Based on the above research, a number of methods have been proposed to solve the problems of noise, irregular sampling, etc., in high-dimensional chaos, and their respective advantages are introduced below. At present, the problems of poor data quality, partial observability and lack of interpretability in deep learning models are well-known. In the future, Physics-Informed Machine Learning, uncertainty quantification and robust model methods will be applied in engineering. Data science and applied mathematics will be combined in this paper to offer a comprehensive introduction to the problems and models of complex systems.

Community detection is a key problem in complex-network analysis: densely connected groups may correspond to social circles, scientific fields, biological modules, or functional subsystems. This review considers how spectral graph methods translate a network into matrix form and then use eigenvalues and eigenvectors to uncover this structure. After introducing complex networks, graph matrices, community structure, and evaluation criteria, it develops the main ideas behind unnormalized and normalized graph Laplacians, the Fiedler vector, spectral embedding, normalized cut, and modularity-based eigenvector methods. Evidence from benchmark and real-world studies is then used to compare these classical techniques with regularized, non-backtracking, local, overlapping, neural-embedding, and randomized multi-layer extensions. Spectral methods remain competitive and mathematically interpretable. Their performance, however, depends on sparsity, degree heterogeneity, community overlap, network scale, and the choice of evaluation criteria. Future work is likely to combine sparse linear algebra and randomized eigensolvers with dynamic, multi-layer, and interpretable graph-learning models. By connecting core linear algebra with practical structure discovery, the review clarifies both the continuing value and the limits of the spectral perspective.

Customer churn prediction is important across telecommunications, subscription platforms, retail, banking, and digital services because retaining existing customers is often less costly than acquiring new ones. This paper provides a review of customer churn prediction using logistic regression and behavioral features, aiming to demonstrate how statistical modeling can support customer risk identification and behavioral data interpretation. The review first clarifies the basic concepts of churn, behavioral features, and binary classification, then introduces the mathematical principles, modeling procedures, and evaluation metrics of logistic regression. Through a comparative analysis of four representative empirical studies across banking, telecommunications, retail, and digital subscription services, the paper finds that the applicability of logistic regression depends heavily on feature engineering quality and the degree of linearity in business contexts. Key challenges identified include inconsistent churn definitions, varying data quality, the confounding of correlation with causation, and privacy constraints on data usage. Future research directions such as dynamic behavioral modeling, causal inference, and privacy-preserving analytics are discussed to further enhance predictive reliability and interpretability. This review contributes to understanding how fundamental regression methods can serve as interpretable baseline models in business data science, offering a reference for both researchers and practitioners in customer retention and risk management.

Map navigation and graph-based path planning are the foundations of intelligent transportation systems today, supporting all kinds of applications in daily life, logistics and emergency rescue. A spatially weighted graph of the real road network is built in the system, and then the best path is selected based on the weights of these factors. This paper will introduce the main methods of the study and first present road-network models that combine static data from sources such as OpenStreetMap with dynamic Global Positioning System (GPS) trajectory data to construct and update weighted directed graphs. Next, this paper will introduce the basic and advanced shortest-path algorithms for static and dynamic urban environments, such as Dijkstra's algorithm, A* algorithm, etc., and compare their strengths and weaknesses. The following are the steps of the whole navigation workflow, as well as data preprocessing, multi-source fusion and execution of multi-objective path search. Lastly, this paper will also discuss the current deficiencies and problems in the above methods, such as missing data, algorithmic complexity of large-scale networks, trade-offs in multi-objective optimisation, GPS errors and user privacy, etc. Graph theory, spatial data science and optimisation algorithms have been combined to provide the theoretical basis for intelligent path planning in this paper and offer a reference for future research on lightweight, high-efficiency dynamic navigation.

In order to utilize solar energy more effectively and alleviate energy difficulties, this paper proposes the use of a metasurface to construct a solar absorber. The materials used include a highly reflective TiN substrate, a SiO 2 dielectric layer, and a TiN hexagonal-prism ring pattern layer. Through simulation calculations using COMSOL software, it is shown that for the wavelength range of 300—1000 nm, the average absorption is 90.8%, and the absorption rate reaches 99.985% near 925 nm. Judging from the electric field distribution, the absorbed energy likely comes from surface plasmon polaritons or cavity resonance phenomena. After examining the absorption conditions under different parameters, it is found that within the studied range, the height of the pattern layer and the width of the hollow quadrangular prism have little effect on the absorption rate, and the absorption performance is stable. The designed structure itself is uncomplicated, capable of handling visible light and part of the infrared wavelengths, and has practical prospects for solar energy conversion.

Throughout the propagation process, high-speed satellite communication signals are significantly influenced by both special and general relativistic effects. These relativistic effects, including Doppler shifts, gravitational redshifts, and time dilation, cause frequency deviations and reception time errors, affecting communication accuracy and the positioning performance of navigation systems. In high-speed satellite links, relativistic effects directly contribute to carrier frequency offsets and timing errors, which can degrade synchronization and decoding performance. Therefore, accurate modeling and effective compensation of relativistic effects are required to reduce communication errors and boost system reliability. In particular, relativistic Doppler effects pose challenges for high-dynamic satellite links due to their direct influence on carrier frequency stability and timing synchronization. This paper reviews theoretical analyses, frequency shift assessments, and engineering correction methods for relativistic Doppler effects, focusing on their impact on signal synchronization and decoding accuracy. The results demonstrate that, although relativistic frequency shifts are small in magnitude, they cannot be neglected in high-precision communication and navigation systems; and effective correction methods, like satellite-side pre-compensation and ground-side post-correction, can mitigate these errors and improve system accuracy.