Complex Nonlinear Correlations Research Articles

Fossil Fuel CO2 Emissions and Economic Growth in the Visegrád Region: A Study Based on the Environmental Kuznets Curve Hypothesis

The relationship between fossil fuel CO2 emissions and economic growth in the Visegrád (V4) countries (Czechia, Hungary, Poland, and Slovakia) is examined through the lens of the environmental Kuznets curve (EKC) hypothesis. Employing the modified environmental Kuznets curve (MEKC) hypothesis, time-series data from 2010 to 2022 were analyzed. The methodology encompasses a range of econometric techniques, including temporal, comparative, correlational, and regression analyses, to unravel the intricate relationship between economic development (measured by GDP per capita) and environmental pollution (CO2 emissions). Results reveal a complex nonlinear correlation between GDP per capita and CO2 emissions in the V4 countries, following an inverted U-shaped pattern. Specifically, Czechia and Hungary exhibited peak emissions at approximately USD 5000 and USD 4500 GDP per capita, respectively, with corresponding emission levels of 1.15 and 0.64 metric tons. In contrast, Slovakia’s emissions decreased after its GDP per capita exceeded USD 5000 and carbon dioxide emissions reached 0.15 metric tons. However, Poland’s data deviate from the MEKC pattern, exhibiting a consistent rise in CO2 emissions across all levels of GDP per capita. The study highlights that the power industry is the largest source of CO2 emissions in all four countries, contributing 88.09% of total emissions. The transportation and industrial combustion sectors account for about 2.12% and 1.28% of annual emissions, respectively. GDP–CO2 emission correlations vary across the V4 countries. While Czechia exhibits a positive correlation of 0.35, Hungary (−0.37), Poland (−0.21), and Slovakia (−0.11) display negative relationships. Notably, Poland experiences the most significant increase in CO2 emissions from both road transport and air traffic. The conclusions drawn from this study provide a robust foundation for developing tailored environmental policies that support sustainable growth in the Visegrád region and other transitioning economies.

Small-batch product quality prediction using a novel discrete Choquet fuzzy grey model with complex interaction information

With manufacturing focused on multi-variety and small-batch intelligent production, the challenge is to guarantee and ensure good product quality. This paper proposes a novel small-batch product quality prediction approach considering complex interaction information. In this paper, first, to handle the complex nonlinear correlations among the manufacturing process variables, a kernel principal components analysis is used to extract the major features from the high-dimensional data. Second, we combine the discrete multivariate grey model and generalized discrete Choquet fuzzy integral to propose a discrete Choquet fuzzy grey model so as to capture the complex interaction information among the process variables. Next, an adjoint sensitivity analysis is applied to characterize the changes in the process variables on the predicted product quality index. We conduct an experiment on six semiconductor products made in China to validate the proposed model against a library of nine other methods. Our model yields an average reduction of 11.02% on MAPE, 20.47% on MSE, 13.56% on STD, and an average increase of 46.37% on EVS for six types of products. Our results inform that the proposed model is suitable for predicting manufacturing product quality when the process variables interact significantly, and we can prioritize the process variables for remedial action accordingly.

A novel procedure for cable damage identification of cable-stayed bridge using particle swarm optimization and machine learning

The cables are crucial components in the ensuring safety of the stayed-cable bridges. The early identification and quantification of cable damage based on the inherent structural health monitoring (SHM) system is a priority to prevent disasters. In this study, a procedure is proposed to identify the cable damage in the cable-stayed bridges using the particle swarm optimization (PSO) and the categorical gradient boosting (CatBoost) algorithm. The PSO-based finite element model updating method is implemented to establish the baseline model while a practical advanced analysis program is used to generate simulation data. As an efficient and up-to-date machine learning (ML) algorithm, CatBoost is utilized to capture the complex nonlinear correlations between the vibration characteristics and the cable damages. A case study of a benchmark bridge where cable damage has been identified is considered to evaluate the efficiency of the proposed procedure. The fivefold cross-validation and grid search methods are used to find the optimal model. The accuracy of the proposed cable damage identification model using CatBoost is also verified through the comparison with three existing ML methods: random forest, decision tree, and extreme gradient boosting. The identification results of both simulation and real cases of cable damage demonstrate that the proposed procedure is a novel and powerful approach for cable damage identification of the cable-stayed bridge using measurement data of the existing SHM system.

Predictions of multi-scale vortex-induced vibrations based on a multi-fidelity data assimilation method

This paper presents a data assimilation method based on the POD-DeepONet structure to fuse two types of fidelity data from vortex-induced vibration (VIV) problems. The data is mainly focused on amplitude response in different oncoming flow cases from cross-flow (CF) and inline (IL) directions for new flow speed predictions. The low-fidelity data in this paper is calculated from a semi-empirical code called DAVIV, while the high-fidelity data is measured from laboratory experiments. For a complex nonlinear correlation in amplitude and phase error between low and high-fidelity data, the POD-DeepONet structure receives better accuracy and more stable predictions than Neural Network with few training cases. It can successfully reconstruct the amplitude response along the marine riser and capture the changing trend with the oncoming flow speed. The POD-DeepONet is then applied to predict the VIV cross-flow and inline responses with different datasets. The prediction of mean square error (MSE) shows an exponential decline trend with the increasing case number for training. With the exponentially fitted MSE formula, the required case number under the expected error can be quickly obtained, which may provide a reference for efficient and high-fidelity VIV response prediction in engineering.

Quality design based on kernel trick and Bayesian semiparametric model for multi-response processes with complex correlations

ABSTRACT Processes or products are typically complex systems with numerous interrelated procedures and interdependent components. This results in complex relationships between responses and input factors, as well as complex nonlinear correlations among multiple responses. If the two types of complex correlations in the quality design cannot be properly dealt with, it will affect the prediction accuracy of the response surface model, as well as the accuracy and reliability of the recommended optimal solutions. In this paper, we combine kernel trick-based kernel principal component analysis, spline-based Bayesian semiparametric additive model, and normal boundary intersection-based evolutionary algorithm to address these two types of complex correlations. The effectiveness of the proposed method in modeling and optimisation is validated through a simulation study and a case study. The results show that the proposed Bayesian semiparametric additive model can better describe the process relationships compared to least squares regression, random forest regression, and support vector basis regression, and the proposed multi-objective optimisation method performs well on several indicators mentioned in the paper.

Estimating the relationships between initial constituent molar ratios and physical–mechanical properties of RCFP-GBFS based geopolymers

Recycling waste materials for geopolymer synthesis offers a sustainable solution for reducing environmental impact from construction. In this study, the recycled concrete fine powder (RCFP) and granulated blast furnace slag (GBFS) were used to synthesize geopolymers, and the effects on physical performance, compressive strength, and microstructure were investigated. Building upon the existing results, the quantitative relationships between initial constituent molar ratios (Si/Al, Na/Al, and Si/Ca) and macro-properties of the geopolymers were explored and the influence mechanisms were analyzed. The findings demonstrate that the incorporation of GBFS significantly reduced setting time and increased compressive strength. Additionally, the blending of GBFS effectively enhanced the geopolymerization degree and generated more mixed gelling products of C-(A)-S-H, N-A-S-H and (C, N)-A-S-H, resulting in a refined internal structure of the matrix. The results revealed that the relationships between initial constituent molar ratios and macro-properties were not straightforward and involved multiple models. The initial and final setting times showed complex non-linear correlations with initial component molar ratios, while strong linear correlations were observed between initial component molar ratios and both flowability and compressive strength. A non-linear relationship was found between the setting time and fluidity. The outcomes contribute to the understanding of the interplay between composition, property, and structure in geopolymer materials and provide guidance for the composition design of geopolymers.

Urban spontaneous plant richness in response to the 2D/3D building and green space patterns in a highly urbanized area

The impact of landscape patterns on urban plant diversity has received significant attention; however, previous studies have primarily focused on two-dimensional (2D) patterns. Limited investigations have been conducted to explore the effects of three-dimensional (3D) landscape patterns on urban plant diversity. Spontaneous plants are ideal objects for studying the response of plant diversity to urban environments due to their independence from intentional human inputs. Buildings, as a crucial symbol of urbanization, exhibit strong 3D characteristics in urban landscapes. Green spaces, the most important source of plant propagules in cities, are closely related to spontaneous plant diversity in urban areas. Therefore, we examined how 2D/3D building and green space patterns influence urban spontaneous plant species richness and compared the responses between different categories (native and non-native). We surveyed spontaneous plants in the built-up areas of Shenzhen, a highly urbanized region situated along China’s southern coast. Our survey recorded 278 species belonging to 208 genera and 77 families. Results from boosted regression tree models (BRT) indicated significant correlations between spontaneous plant species richness and various metrics, including building coverage ratio (BCR), mean building height (MAH), green space coverage ratio (GCR), edge density of green spaces (ED_G), spatial congestion degree (SCD), and floor area ratio (FAR). These metrics exhibited complex nonlinear correlations with the species richness of spontaneous plants. Furthermore, native and non-native species richness responded differently to building and green space patterns; while native species richness was equally affected by green space and building patterns, non-native species richness was more strongly impacted by building patterns than green space ones. Our study quantified the nonlinear relationships between urban spontaneous plant species richness and 2D/3D green space and building patterns, demonstrating that optimizing green space and building patterns can be a practical approach for managing and promoting spontaneous plant diversity in highly urbanized areas.

Smart modelling of dairy milk production with machine learning

Complex non-linear correlations between the predictors (features) and the independent variable can be modeled using machine learning techniques. This modeling strategy is clever because it uses machine learning to develop process predictions after the right model has been created. Therefore, the purpose of this work was to examine how to simulate dairy milk production using machine learning. The daily milk output in Iwo and its environs between May 26, 2021, and May 31, 2022, as measured in liters, was used. Five features in the data set were identified; the day of the week, month, year, season and day number. We tested a total of 14 different supervised learning (regression) machine learning techniques. 20% of the data were used for validation, while 80% were used to train these algorithms. The Bagged Tree gave the highest R - square value of 0.67 and the lowest RMSE of 20.26 among the 14 Machine Learning techniques taken into account. It is therefore recommended to be used in smart prediction of daily milk production in Iwo and its environs. Also, season was found to influence milk production in the study area with higher milk production in wet season than dry season(p<.05).

Autoencoder-Based Iterative Modeling and Multivariate Time-Series Subsequence Clustering Algorithm

This paper introduces an algorithm for the detection of change-points and the identification of the corresponding subsequences in transient multivariate time-series data (MTSD). The analysis of such data has become increasingly important due to growing availability in many industrial fields. Labeling, sorting or filtering highly transient measurement data for training Condition-based Maintenance (CbM) models is cumbersome and error-prone. For some applications it can be sufficient to filter measurements by simple thresholds or finding change-points based on changes in mean value and variation. But a robust diagnosis of a component within a component group for example, which has a complex non-linear correlation between multiple sensor values, a simple approach would not be feasible. No meaningful and coherent measurement data, which could be used for training a CbM model, would emerge. Therefore, we introduce an algorithm that uses a recurrent neural network (RNN) based Autoencoder (AE) which is iteratively trained on incoming data. The scoring function uses the reconstruction error and latent space information. A model of the identified subsequence is saved and used for recognition of repeating subsequences as well as fast offline clustering. For evaluation, we propose a new similarity measure based on the curvature for a more intuitive time-series subsequence clustering metric. A comparison with seven other state-of-the-art algorithms and eight datasets shows the capability and the increased performance of our algorithm to cluster MTSD online and offline in conjunction with mechatronic systems.

Complex ecological and socioeconomic impacts on medicinal plant diversity.

Medicinal plant diversity (MPD) is an important component of plant diversity. Over-collection based on medicinal and economic value has the potential to damage the stability of the regional ecosystem. It is important to understand the current distribution of MPD and the factors influencing it. However, it is still unclear whether environmental and socioeconomic conditions have an impact on their distribution. We selected the Inner Mongolia as a representative study area which covers a wide area, accounting for 12.29% of China's national land area and 0.79% of the world's land area. At the same time, the region is a long-standing traditional medicinal area for Mongolians in China. Therefore, the region is significantly influenced by changes in environmental factors and socio-economic factors. We used 9-years field survey of the distribution of medicinal plants in Inner Mongolia for assessing the distribution of MPD as influenced by environmental and socioeconomic activities by combining spatial analyses, species distribution models, and generalized additive models. The results from the spatial analysis show that the western region of Inner Mongolia is the main cold spot area of the MPD, and the central-eastern and northeastern regions of Inner Mongolia are the main hot spot areas of the MPD. At the same time, the distribution of cold spots and hot spots of MPD is more obvious at large spatial scales, and with the refinement of spatial scales, the cold spots in scattered areas are gradually revealed, which is indicative for the conservation and development of MPD at different spatial scales. Under the future climate change of shared socioeconomic pathways (SSP), areas with high habitat suitability for medicinal plants remain mainly dominated by the Yellow River, Yin Mountains, and Greater Khingan Range. Notably, the SSP245 development pathway remains the most significant concern in either long- or short-term development. The nonlinear relationship between the driving factors of MPD at different spatial scales shows that temperature, precipitation and socioeconomic development do have complex effects on MPD. The presence of a certain temperature, altitude, and precipitation range has an optimal facilitation effect on MPD, rather than a single facilitation effect. This complex nonlinear correlation provides a reference for further studies on plant diversity and sustainable development and management. In this study, the spatial distribution of medicinal plant resources and the extent to which they are driven by ecological and socioeconomic factors were analyzed through a macroscopic approach. This provides a reference for larger-scale studies on the environmental and socioeconomic influences on the distribution of plant resources.

Cardiac disease detection from ECG signal using discrete wavelet transform with machine learning method

ObjectivesCardiac disease is the leading cause of death worldwide. If a proper diagnosis is made early, cardiovascular problems can be prevented. The ECG test is a diagnostic method used on the screen for heart disease. Based on a combination of multi-field extraction and nonlinear analysis of ECG data, this paper presents a framework for automated detection of heart disease. The main aim of this study is to develop a model for future diagnosis of cardiac vascular disease using ECG analysis and symptom-based detection. MethodsDiscrete wavelet transform and Nonlinear Vector Decomposed Neural Network methods are used to predict Cardiac disease. Here is the discrete wavelet transform used for preprocessing to remove unwanted noise or artifacts. The neural network was fed with thirteen clinical features as input which was then trained using a non-linear vector decomposition of the presence or absence of heart disease. ResultsThe modules were implemented, trained, and tested using UCI and Physio net data repositories. The sensitivity, specificity and accuracy of this research work are 92.0%, 89.33% and 90.67% ConclusionsThe proposed approach can discover complex non-linear correlations between dependent and independent variables without requiring traditional statistical training. The suggested approach improves ECG classification accuracy, allowing for more accurate cardiac disease diagnosis. The accuracy of ECG categorization in identifying cardiac illness is far greater than these other approaches.

Spatiotemporal estimation of the PM2.5 concentration and human health risks combining the three-dimensional landscape pattern index and machine learning methods to optimize land use regression modeling in Shaanxi, China

PM2.5 pollution endangers human health and urban sustainable development. Land use regression (LUR) is one of the most important methods to reveal the temporal and spatial heterogeneity of PM2.5, and the introduction of characteristic variables of geographical factors and the improvement of model construction methods are important research directions for its optimization. However, the complex non-linear correlation between PM2.5 and influencing indicators is always unrecognized by the traditional regression model. The two-dimensional landscape pattern index is difficult to reflect the real information of the surface, and the research accuracy cannot meet the requirements. As such, a novel integrated three-dimensional landscape pattern index (TDLPI) and machine learning extreme gradient boosting (XGBOOST) improved LUR model (LTX) are developed to estimate the spatiotemporal heterogeneity in the fine particle concentration in Shaanxi, China, and health risks of exposure and inhalation of PM2.5 were explored. The LTX model performed well with R2 = 0.88, RMSE of 8.73 μg/m3 and MAE of 5.85 μg/m3. Our findings suggest that integrated three-dimensional landscape pattern information and XGBOOST approaches can accurately estimate annual and seasonal variations of PM2.5 pollution The Guanzhong Plain and northern Shaanxi always feature high PM2.5 values, which exhibit similar distribution trends to those of the observed PM2.5 pollution. This study demonstrated the outstanding performance of the LTX model, which outperforms most models in past researches. On the whole, LTX approach is reliable and can improve the accuracy of pollutant concentration prediction. The health risks of human exposure to fine particles are relatively high in winter. Central part is a high health risk area, while northern area is low. Our study provides a new method for atmospheric pollutants assessing, which is important for LUR model optimization, high-precision PM2.5 pollution prediction and landscape pattern planning. These results can also contribute to human health exposure risks and future epidemiological studies of air pollution.

Temporal Hypergraph Attention Network for Silicon Content Prediction in Blast Furnace

Online dynamic prediction of the hot metal silicon content in the blast furnace ironmaking process is crucial for stabilizing the furnace condition and improving the molten iron quality. However, due to the complex nonlinear correlations and time-varying time lags between silicon content and process variables, silicon content prediction is a challenging task. To tackle the problem, we propose a novel silicon content prediction method, called temporal hypergraph attention network (T-HyperGAT), which is combined the hypergraph attention network (HyperGAT) and the gated recurrent unit (GRU) network. Specifically, the Hyper-GAT is used to capture the high-order correlations of input features and perform equal-dimensional feature transformation to maintain the temporality of input features, and the GRU network is used to capture the time-series characteristics of transformed input features to overcome the time-varying time lags of process variables. Then the T-HyperGAT model can capture high-order correlations and time-series characteristics from complex industrial data. The effectiveness of the proposed T-HyperGAT method is verified by actual blast furnace ironmaking process data from a blast furnace in China.

Application of Machine Learning Algorithm in China National Geohazard Susceptibility Assessment

Landslide susceptibility is an important activity in landslide hazard assessment. With the advancement of artificial intelligence, the machine learning algorithm is applied in the landslide susceptibility assessment, has recently gained immense attention due to the advantages of obtaining the insights of landslide events and conditioning parameters based on data mining, which is important when tackling the challenge of mapping landslide prone areas in regional scale due to the complex nonlinear correlations among landslides and parameters and uncertainties associated during parameters reclassification. Therefore, the machine learning algorithm has become a standard approach for modeling landslide susceptibility over large regions. In this study, the random forest method is applied to produce the China national landslide susceptibility mapping based on the national landslide database containing more than 300 thousand landslide events. Thirty different categories of conditioning parameters related to the development, triggering, and potentially vulnerable elements of the landslide were collected using a scale of approximately 1:1,000,000. Through the data mining process, lithology, faults, topography, soil erosion, precipitation, and human activities were found to be the top six important contribution factors to landslide susceptibility. The mapping results show the areas of four degrees of susceptibility from high to low are 101, 191, 337, and 331 thousand square kilometers, respectively. The receiver operating curve (ROC) and area under curve (AUC) value was calculated to 0.81, indicating that results are well satisfying and could guide landslide mitigation on the national level.

Tri-tert-butyl(n-alkyl)phosphonium Ionic Liquids: Structure, Properties and Application as Hybrid Catalyst Nanomaterials

A series of sterically hindered tri-tert-butyl(n-alkyl)phosphonium salts (n-CnH2n+1 with n = 1, 3, 5, 7, 9, 11, 13, 15, 17) was synthesized and systematically studied by 1H, 13C, 31P NMR spectroscopy, ESI-MS, single-crystal X-ray diffraction analysis and melting point measurement. Formation and stabilization palladium nanoparticles (PdNPs) were used to characterize the phosphonium ionic liquid (PIL) nanoscale interaction ability. The colloidal Pd in the PIL systems was described with TEM and DLS analyses and applied in the Suzuki cross-coupling reaction. The PILs were proven to be suitable stabilizers of PdNPs possessing high catalytic activity. The tri-tert-butyl(n-alkyl)phosphonium salts showed a complex nonlinear correlation of the structure–property relationship. The synthesized family of PILs has a broad variety of structural features, including hydrophobic and hydrophilic structures that are entirely expressed in the diversity of their properties

Landslide Susceptibility Prediction Using Sparse Feature Extraction and Machine Learning Models Based on GIS and Remote Sensing

Landslide susceptibility prediction (LSP) is a useful technology for landslide prevention. Due to the complex nonlinear correlations among environmental factors, traditional machine learning (ML) models have unsatisfactory LSP accuracies. In this letter, a sparse feature extraction network (SFE+) is proposed for LSP. First, the landslides and environmental factors are collected, and frequency ratios of environmental factors are calculated as the model inputs. Second, the input data are passed through the input layer with the dropout, and then, the features are passed through the hidden layers, that is, the k% lifetime sparsity layers. The hidden layers are employed to further sparse these factors to obtain the independent and redundant prediction features as much as possible. Finally, certain classifiers are used to realize the LSP in the study area. SFE-support vector machine (SVM), SFE-logistic regression (LR), and SFE-stochastic gradient descent (SGD) models are built. For comparison, principal component analysis (PCA)-SVM, PCA-LR, PCA-SGD, SVM, LR, and SGD models are also built for LSP in Shicheng County, China. Results show that the SFE-based ML models, especially the SFE-SVM, can effectively extract the sparse nonlinear features of environmental factors to improve LSP accuracies and have promising prospects for LSP.

ADST: Forecasting Metro Flow Using Attention-Based Deep Spatial-Temporal Networks with Multi-Task Learning

Passenger flow prediction has drawn increasing attention in the deep learning research field due to its great importance in traffic management and public safety. The major challenge of this essential task lies in multiple spatiotemporal correlations that exhibit complex non-linear correlations. Although both the spatial and temporal perspectives have been considered in modeling, most existing works have ignored complex temporal correlations or underlying spatial similarity. In this paper, we identify the unique spatiotemporal correlation of urban metro flow, and propose an attention-based deep spatiotemporal network with multi-task learning (ADST-Net) at a citywide level to predict the future flow from historical observations. ADST-Net uses three independent channels with the same structure to model the recent, daily-periodic and weekly-periodic complicated spatiotemporal correlations, respectively. Specifically, each channel uses the framework of residual networks, the rectified block and the multi-scale convolutions to mine spatiotemporal correlations. The residual networks can effectively overcome the gradient vanishing problem. The rectified block adopts an attentional mechanism to automatically reweigh measurements at different time intervals, and the multi-scale convolutions are used to extract explicit spatial relationships. ADST-Net also introduces an external embedding mechanism to extract the influence of external factors on flow prediction, such as weather conditions. Furthermore, we enforce multi-task learning to utilize transition passenger flow volume prediction as an auxiliary task during the training process for generalization. Through this model, we can not only capture the steady trend, but also the sudden changes of passenger flow. Extensive experimental results on two real-world traffic flow datasets demonstrate the obvious improvement and superior performance of our proposed algorithm compared with state-of-the-art baselines.

An explicit methodology for manufacturing cost–tolerance modeling and optimization using the neural network integrated with the genetic algorithm

AbstractThe objective of the paper is to develop a new method to model the manufacturing cost–tolerance and to optimize the tolerance values along with its manufacturing cost. A cost–tolerance relation has a complex nonlinear correlation among them. The property of a neural network makes it possible to model the complex correlation, and the genetic algorithm (GA) is integrated with the best neural network model to optimize the tolerance values. The proposed method used three types of neural network models (multilayer perceptron, backpropagation network, and radial basis function). These network models were developed separately for prismatic and rotational parts. For the construction of network models, part size and tolerance values were used as input neurons. The reference manufacturing cost was assigned as the output neuron. The qualitative production data set was gathered in a workshop and partitioned into three files for training, testing, and validation, respectively. The architecture of the network model was identified based on the best regression coefficient and the root-mean-square-error value. The best network model was integrated into the GA, and the role of genetic operators was also studied. Finally, two case studies from the literature were demonstrated in order to validate the proposed method. A new methodology based on the neural network model enables the design and process planning engineers to propose an intelligent decision irrespective of their experience.

A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems

Currently the training of neural networks relies on data of comparable accuracy but in real applications only a very small set of high-fidelity data is available while inexpensive lower fidelity data may be plentiful. We propose a new composite neural network (NN) that can be trained based on multi-fidelity data. It is comprised of three NNs, with the first NN trained using the low-fidelity data and coupled to two high-fidelity NNs, one with activation functions and another one without, in order to discover and exploit nonlinear and linear correlations, respectively, between the low-fidelity and the high-fidelity data. We first demonstrate the accuracy of the new multi-fidelity NN for approximating some standard benchmark functions but also a 20-dimensional function that is not easy to approximate with other methods, e.g. Gaussian process regression. Subsequently, we extend the recently developed physics-informed neural networks (PINNs) to be trained with multi-fidelity data sets (MPINNs). MPINNs contain four fully-connected neural networks, where the first one approximates the low-fidelity data, while the second and third construct the correlation between the low- and high-fidelity data and produce the multi-fidelity approximation, which is then used in the last NN that encodes the partial differential equations (PDEs). Specifically, by decomposing the correlation into a linear and nonlinear part, the present model is capable of learning both the linear and complex nonlinear correlations between the low- and high-fidelity data adaptively. By training the MPINNs, we can: (1) obtain the correlation between the low- and high-fidelity data, (2) infer the quantities of interest based on a few scattered data, and (3) identify the unknown parameters in the PDEs. In particular, we employ the MPINNs to learn the hydraulic conductivity field for unsaturated flows as well as the reactive models for reactive transport. The results demonstrate that MPINNs can achieve relatively high accuracy based on a very small set of high-fidelity data. Despite the relatively low dimension and limited number of fidelities (two-fidelity levels) for the benchmark problems in the present study, the proposed model can be readily extended to very high-dimensional regression and classification problems involving multi-fidelity data.

Adaptive kriging-based efficient reliability method for structural systems with multiple failure modes and mixed variables

The reliability analysis of structural systems with multiple failure modes and mixed variables is a critical problem because of complex nonlinear correlations among failure modes (or components), huge computational burden of time-consuming implicit functions, and complex failure regions. In this paper, aleatory and epistemic uncertainties are considered simultaneously, and an efficient adaptive kriging-based reliability method is proposed for structural systems with multiple failure modes and mixed variables. Two new learning functions are developed as guidelines for selecting new training samples at each iteration. The proposed learning functions and corresponding stopping criteria are directly linked to system probability of failure; this allows the proposed method to select new training samples efficiently To determine the lower and upper bounds of system probability of failure, the limit-state functions in the entire uncertainty space of interest are accurately constructed while avoiding complicated nested optimizations. The proposed method has the following advantages: (1) the learning functions and stopping criteria are directly linked to system probability of failure, and the structure importance of components is also considered; (2) it requires fewer samples to achieve accurate results, and can be applied to small system probability of failure; (3) it is easy to use for extremely complex systems (e.g., bridge systems); (4) it can be applied to a system with multiple failure modes and mixed variables (e.g., mixture of random and p-box variables). The capabilities and efficiency of the proposed method are validated through four numerical examples; results show that it has high applicability and accuracy.