Global principal component analysis (PCA) has been successfully introduced for modeling distributed parameter systems (DPSs). In spite of the merits, this method is not feasible due to parameter variations and multiple operating domains. A novel multimode spatiotemporal modeling method based on the locally weighted PCA (LW-PCA) method is developed for large-scale highly nonlinear DPSs with parameter variations, by separating the original dataset into tractable subsets. This method implements the decomposition by making full use of the dependence among subset densities. First, the spatiotemporal snapshots are divided into multiple different Gaussian components by using a finite Gaussian mixture model (FGMM). Once the components are derived, a Bayesian inference strategy is then applied to calculate the posterior probabilities of each spatiotemporal snapshot belonging to each component, which will be regarded as the local weights of the LW-PCA method. Second, LW-PCA is adopted to calculate each locally weighted snapshot matrix, and the corresponding local spatial basis functions (SBFs) can be generated by the PCA method. Third, all the local temporal models are estimated using the extreme learning machine (ELM). Thus, the local spatiotemporal models can be produced with local SBFs and corresponding temporal model. Finally, the original system can be approximated using the sum form of each local spatiotemporal model. Unlike global PCA, which uses global SBFs to construct a global spatiotemporal model, LW-PCA approximates the original system by multiple local reduced SBFs. Numerical simulations verify the effectiveness of the developed multimode spatiotemporal model.

In this paper, deep extreme learning machines (DELM) based two-phase spatiotemporal modeling framework is proposed for DPSs. The modeling process consists of two phases. In phase I, DELM based model is constructed to approximate the spatiotemporal dynamics of DPSs. First, an unsupervised stacked ELM autoencoder (ELM-AE) is designed to create a multilayer ELM (ML-ELM), by which the high-dimensional spatiotemporal data can be transformed into low-dimensional time data. The corresponding parameters are optimized by a fast leave-one-out cross-validation (FLOO-CV) algorithm. Second, an ELM model is employed to disclose the temporal dynamic of the low-dimensional time data. Finally, the spatiotemporal model can be reconstructed by kernel ELM (K-ELM). In phase II, a Karhunen-Loeve (KL) based ELM model is constructed to compensate for modeling errors caused by reconstructed error or unknown nonlinear dynamics. The two-phase model can be obtained by integrating these two models. Experiments demonstrate the superior model performance.

Exploration of the green flexible job-shop scheduling problem is essential for enterprises aiming for sustainable practices, including energy conservation, emissions reduction, and enhanced economic and social benefits. While existing research has predominantly focused on carbon emissions or energy consumption as green scheduling objectives, this paper addresses the broader scope by incorporating the impact of variable energy prices on energy cost. Through the introduction of an energy cost model based on time-of-use electricity pricing, the study formulates a multi-objective optimization model for green flexible job-shop scheduling. The objectives include minimizing cost, reducing carbon emissions, and maximizing customer satisfaction. To prevent premature convergence and maintain population diversity, an enhanced genetic algorithm is employed for solving. The validation of the algorithm’s effectiveness is demonstrated through specific examples, providing decision results for optimal scheduling under various weight combinations. The research outcomes hold substantial practical value as they can significantly reduce energy expenses, lower carbon emissions, and elevate customer satisfaction while safeguarding production efficiency. This contributes to enhancing the market competitiveness and green brand image of businesses.

Complex nonlinear distributed parameter systems (DPSs) widely exist in real industrial thermal processes. Modeling of such systems often leads to the following challenges: strong nonlinearities, time-varying dynamics, and large operating range with multiple working points. Therefore, traditional single spatiotemporal model will become ill suited. Motivated by the idea of multimodeling, integration of finite Gaussian mixture model (FGMM) and principle component regression (PCR) based multiple spatiotemporal modeling is proposed in this paper for complex nonlinear DPSs. The main idea of the proposed method can be summarized as the following three parts: FGMM-based operating space separation, Karhunen-Loève based local spatiotemporal modeling, and PCR-based local spatiotemporal models integration. To evaluate the generalization bound of the proposed method, the Rademacher complexity is also developed here theoretically. Since multimodeling can reduce the nonlinear complexity, the proposed model has strong ability to track and handle the complex nonlinear dynamics. Simulations on a two-dimensional curing thermal process demonstrated the superior model performance of the proposed model.

In this article, we introduce a novel method that uses enhanced sequential multitask learning to predict the remaining useful life (RUL) of an aeroengine accurately and efficiently while evaluating its status. The method innovatively employs extreme gradient boosting to extract critical performance parameters and construct a robust health indicator representing performance degradation. To capture the time-series features of the health indicator, the study modifies the traditional multigate mixture-of-experts (MMoE) model and integrates it with the gated recurrent unit (GRU) network, creating a hybrid MMoE-GRU model. In addition, we propose a dynamic weight balancing method to optimize the tradeoff in the joint loss function for multitask learning. Extensive experiments on the new commercial modular aero-propulsion system simulation (N-CMAPSS) dataset demonstrate that the proposed method significantly outperforms the existing models, achieving lower error rates and higher accuracy in RUL prediction and health status evaluation. The technique has a root-mean-square error (RMSE) of 7.1%, 6.9%, 1.3%, 0.6%, 1.7%, and 1.5% lower than the long short-term memory (LSTM), GRU, sequence-based bottom information gated recurrent unit (SB-GRU), deep gated recurrent unit (DGRU), multi-scale and multi-task convolutional neural network (M²STCNN), and MMoE models. The average accuracy of the proposed method is 96.732%, which is 10.629%, 1.587%, and 2.499% higher than those of the LSTM, DGRU, and MMoE, respectively. The superiority of the proposed method is validated on the N-CMAPSS dataset.