Koopman Spectral Dynamics for Interpretable and Predictive Semantic Drift in Language Models

The emergence of dynamic mode decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear partial differential equation (PDE) presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis, and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in the Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions. As far as we are aware the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them, and it presents a nice example in which (i) the Koopman modes are linearly dependent and so they cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions, and (ii) the Koopman eigenvalues are highly degenerate, which means that computed Koopman modes become initial-condition-dependent. As a way of illustration, we discuss the form of the Koopman expansion with various initial conditions, and we assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.

Data-driven spectral decomposition and forecasting of ergodic dynamical systems

The accurate prediction of subway passenger flow is of great significance for the intelligent management of urban rail transit systems. However, traditional static network prediction models struggle to capture the dynamic correlations of subway passenger flow, and the graph structure generated by the DTW algorithm may overlook the differences between passenger flows. Therefore, this paper proposes a GAT-LSTM-Attention prediction model based on a graph structure constructed from the differences in passenger flow. Firstly, a graph structure is constructed based on the differences in peak and valley values between stations, so that the prediction of a station is not affected by stations that exhibit significant differences. Secondly, the periodic characteristics and weather features that may affect the passenger flow prediction are extracted and fused. Finally, the prediction is carried out by the GAT-LSTM-Attention. Through the dual attention mechanism of GAT and Attention, the model is capable of effectively capturing spatial features and key information. The experimental results show that the proposed model has lower MAE and RMSE error values compared with the baseline models. In the 30-minute and 60-minute prediction of outbound and inbound passenger flow, the proposed model achieves the higher R² values of 0.99, 0.99, 0.98, and 0.97. And the proposed graph structure based on passenger flow difference has lower MAE and RMSE error values than conventional methods for constructing graphical structures. This study provides a valuable method for passenger flow prediction in urban rail transit systems.

The stock price is one of the most studied time series data because it is deemed to be profitable doing so, however stock price data is still difficult to predict because it is non-linear, non-parametric, non-stationary, and chaotic. One of the methods that most recently used to predict stock price data is deep learning. Although deep learning has a good performance to solve various problems, deep learning must be trained using a lot of data or this method will experience overfitting. This paper proposes a scheme to train a classifier model for predicting stock price time series data using augmented time-series data generated using GAN. Evaluation shows that the classifier model trained using augmented data has better performance on the AMZN dataset of 24.47% and 30.27% lower RMSE and MAE respectively compared to just using the real data and FB dataset of 15.84% and 13.88% lower RMSE and MAE respectively compared to just using the real data, but for the GOOG dataset it does not show a significant change in RMSE that is 0.52% lower and even the MAE value is increased slightly by 2.62% compared to just using the real data

Spectral decomposition of dynamical systems is a popular methodology to investigate the fundamental qualitative and quantitative properties of these systems and their solutions. In this chapter, we consider a class of nonlinear cooperative protocols, which consist of multiple agents that are coupled together via an undirected state-dependent graph. We develop a representation of the system solution by decomposing the nonlinear system utilizing ideas from the Koopman operator theory and its spectral analysis. We use recent results on the extensions of the well-known Hartman theorem for hyperbolic systems to establish a connection between the original nonlinear dynamics and the linearized dynamics in terms of Koopman spectral properties. The expected value of the output energy of the nonlinear protocol, which is related to the notions of coherence and robustness in dynamical networks, is evaluated and characterized in terms of Koopman eigenvalues, eigenfunctions, and modes. Spectral representation of the performance measure enables us to develop algorithmic methods to assess the performance of this class of nonlinear dynamical networks as a function of their graph topology. Finally, we propose a scalable computational method for approximation of the components of the Koopman mode decomposition, which is necessary to evaluate the systemic performance measure of the nonlinear dynamic network.

Extended Dynamic Mode Decomposition with Invertible Dictionary Learning

We propose a Koopman operator-based surrogate model for propagating parameter uncertainties in power system nonlinear dynamic simulations. First, we augment a priori known state-space model by reformulating parameters deemed uncertain as pseudo-state variables. Then, we apply the Koopman operator theory to the resulting state-space model and obtain a linear dynamical system model. This transformation allows us to analyze the evolution of the system dynamics through its Koopman eigenfunctions, eigenvalues, and modes. Of particular importance for this letter, the obtained linear dynamical system is a surrogate that enables the evaluation of parameter uncertainties by simply perturbing the initial conditions of the Koopman eigenfunctions associated with the pseudo-state variables. Simulations carried out on the New England test system reveal the excellent performance of the proposed method in terms of accuracy and computational efficiency.

Development of offset-free Koopman Lyapunov-based model predictive control and mathematical analysis for zero steady-state offset condition considering influence of Lyapunov constraints on equilibrium point

Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This letter describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm is constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.

Proper forecasting of remaining useful life (RUL) of lithium-ion batteries is critical for safety, operational stability, and cost-effectiveness in modern electric vehicles. Because battery degradation arises from complex electrochemical and thermal processes, data-driven approaches have become a practical alternative to physics-based modeling. This study develops and critically compares supervised machine learning models, including Random Forest, Gradient Boosting, Linear Regression, Ridge Regression, K-Nearest Neighbors, Deep Neural Networks (DNN), and Convolutional Neural Networks (CNN), for cycle-level RUL estimation using a publicly available lithium-ion battery dataset. The dataset comprises engineered voltage-time features, capacity, and charge–discharge indicators, which are preprocessed using Min–Max normalization to support stable training and fair model comparison. Among the evaluated methods, the Random Forest regressor achieved the best overall performance, reaching an R2 of 0.9998 with lower MAE and RMSE than both traditional regressors and deep learning architectures. The results indicate that ensemble tree-based models are especially effective at capturing nonlinear degradation behavior when trained on engineered tabular features rather than raw time-series signals. While the DNN and CNN achieved high R2 values, their error metrics were comparatively larger, suggesting that deep architectures may benefit from higher-resolution time-series inputs or alternative sequence modeling formulations. Finally, the top-performing model was integrated into an end-to-end prediction system using FastAPI, SQLite, and a responsive web dashboard to demonstrate real-world applicability. The deployment allows estimating RUL in real time and at lowlatency and provides stakeholders with an accessible tool for battery health monitoring. Overall, the study shows that ensemble learning techniques can deliver strong performance for battery condition monitoring and provides a deployable architecture to practical RUL prediction in electric-vehicle battery management systems.

There is an ongoing debate on Instagram’s capabilities in sales generation as a social media-based marketing tool amongst business executives. Although an abundance of research was done to determine the impact of social media marketing on sales contribution, many of these studies have utilized purchase intention as a proxy for an actual purchase. This then creates a gap as purchase intention is merely the likelihood of an actual purchase and thus may be inaccurate when used to measure actual sales. This study then aims to investigate the impact of Instagram marketing on sales in the fashion domain via a data analytical approach to narrow the existing gap. A CRISP-DM framework is adopted, which includes a descriptive and predictive approach, in achieving the data mining goal of determining the impact of Instagram marketing on sales using Instagram and sales data of fashion retail in Klang Valley, Malaysia. The implementation of a data analytical approach in investigating the impact of Instagram marketing on sales was able to achieve all research objectives in determining the ability of Instagram marketing in influencing sales. In this line, both the XGBoost and LSTM models were able to predict sales using Instagram marketing factors, whereas the LSTM model performed better with lower MAE and RMSE values. In future, studies can be conducted with more Instagram features with additional modeling techniques to gain better results than those obtained in this study.

Optimal sensor placement is an important yet unsolved problem in control theory. In biological organisms, genetic activity is often highly nonlinear, making it difficult to design libraries of promoters to act as reporters of the cell state. We make use of the Koopman observability gramian to develop an algorithm for optimal sensor (or reporter) placement for discrete time nonlinear dynamical systems to ease the difficulty of design of the promoter library. This ease is enabled due to the fact that the Koopman operator represents the evolution of a nonlinear system linearly by lifting the states to an infinite-dimensional space of observables. The Koopman framework ideally demands high temporal resolution, but data in biology are often sampled sparsely in time. Therefore we compute what we call the temporally fine-grained Koopman operator from the temporally coarse-grained Koopman operator, the latter of which is identified from the sparse data. The optimal placement of sensors then corresponds to maximizing the observability of the fine-grained system. We demonstrate the algorithm on a simulation example of a circadian oscillator.

The maturity of 5G technology provides a guarantee for increasingly large communication networks, while the resources required for communication and computation are also increasing, and reasonable resource allocation can improve the efficiency of network communication and reduce the consumption of communication resources. Existing deep learning methods have been able to predict network traffic to a certain extent, so as to solve the communication efficiency and resource consumption problems in the field of integrated sensing, communication and computation (ISCC) through rational resource allocation. However, the following problems still exist: (1) The feature learning ability of the prediction model is insufficient, and the prediction accuracy needs to be improved. (2) Powerful and complex deep learning methods lead to an increase in the prediction cost of the model. To address these problems, this paper proposes a deep learning method based on a lightweight hybrid attention network. In order to capture the key features of 5G data more effectively, an efficient hybrid attention mechanism (EHA) is proposed. After this attention is applied to convolution, the key information can be well enhanced. We use depthwise separable convolution in feature extraction, which greatly improves the efficiency of lightweight convolution layer (LC) in feature extraction. Combined with the efficient hybrid attention mechanism (EHA), the proposed model has better lightweight properties. Experimental results show that the model proposed in this paper has lower RMSE and MAE values on the three datasets, as well as fewer parameters and computational effort compared to the baseline scheme.

Nowadays, the increasing number of vehicles and shortage of parking spaces have become an inescapable condition in big cities across the world. Car parking problem is not a new phenomenon, especially in a crowded city such as Dhaka, Bangladesh. Shortage of parking spaces leads to several problems such as road congestion, illegal parking on the streets, and fuel waste in searching for a free parking space. In order to overcome the parking problem, we develop a spatio-temporal based car parking system namely, SlotFinder. We collect the data of 408 buildings those have parking slots from seven different locations. We then cluster these data based on time and locations. Later, we train location wise vacant parking spaces by using stacked Long Short-Term Memory (LSTM) based on their temporal patterns. We also compare our technique with the baseline models and conduct an ablation analysis, which outperforms (lower RMSE and MAE of 0.29 and 0.24, respectively) than that of the previous approaches.

Duplex stainless steel (DSS) welding is critical for producing structures and components in a variety of industries. Because traditional optimization approaches cannot manage the complexity of welding, additional solutions must be investigated. This study addresses the need for a systematic exploration of the welding parameters affecting the quality and performance of DSS butt joints, specifically employing fiber laser welding. This study focuses on butt joints in UNS S32304 and 304L using a fiber laser. Welding process parameters were systematically varied, including laser power, scanning speed, beam diameter, and focal position. To efficiently examine the parameters in the design space, the response surface methodology (RSM) with Box–Behnken design (BBD) was used. The combination of the specific parameter values used in the 7th run (Laser Power: 1600 W, Scanning Speed: 2000 mm/min, Beam Diameter: 1.5 [Formula: see text]m, Focal Position: 30 mm) led to better yield point stress (YPS), maximum tensile stress (MTS), break stress (BS) and reduction in area (RA). ANOVA and lack of fit analysis confirmed the significance of the experiment. In particular, the selected welding parameters significantly influenced all responses. A hybrid machine learning method of deep neural network (DNN) and human eye vision algorithm (HEVA) was used for predicting weld quality during laser beam welding (LBW). The hybrid DNN-HEVA consistently outperformed other optimization methods (RSM-BBD, FNN, CNN, and RNN) across key welding quality parameters. High [Formula: see text]2 values (0.9922, 0.9962, 0.9983, and 0.9907) indicated a strong correlation between predicted and actual values for all calculated responses. Lower RMSE, MSE, and MAE values were also highlighted in the precision of predicted values to the experimental data.