Nonlinear Modeling Method Research Articles

A Method for Solving Ill-Conditioned Nonlinear Least Squares Problems and Its Application in Image Distortion Correction Using Self-Calibration

In this study, the ill-conditioning of the iterative method for nonlinear models is discussed. Due to the effectiveness of ridge estimation for ill-conditioned problems and the lack of a combination of the H-K formula with the iterative method, the improvement of the LM algorithm is studied in this paper. Considering the LM algorithm for ill-conditioned nonlinear least squares, an improved LM algorithm based on the H-K formula is proposed for image distortion correction using self-calibration. Three finite difference methods are used to approximate the Jacobian matrix, and the H-K formula is used to calculate the damping factor in each iteration. The Brown model, quadratic polynomial model and Fourier model are applied to the self-calibration, and the improved LM algorithm is used to solve the model parameters. In the simulation experiment of space resection of a single image, we evaluate the performance of the LM algorithm based on the gain ratio (LMh) and the improved LM algorithm based on the H-K formula (LMHK), and the accuracy of different models and algorithms is compared. A ridge trace analysis is carried out on the damping factor to illustrate the effects of the improved algorithm in handling ill-conditioning. In the second experiment, the improved algorithm is applied to measure the diameter of a coin using a single camera. The experimental results show that the improved LM algorithm can reach the same or higher accuracy as the LMh algorithm, and it can weaken the ill-conditioning to a certain extent and enhance the stability of the solution. Meanwhile, the applicability of the improved LM algorithm in self-calibration is verified.

A new global damage index for steel moment resisting frames based on two steps pushover analyses

This paper proposes a new global damage index (DI) to evaluate and quantify the overall damage state induced by near-fault ground motions to the steel structures. DIs are conventional tools that evaluate and quantify the level of damage caused by earthquake excitement. The proposed damage index is based on two steps pushover analyses, which use roof residual drift ratio and maximum base shear before and after the earthquake. Moreover, the proposed damage index was compared with several well-known damage indices. Six steel moment resisting frames from 6- to 21-story frames were used for the evaluation. The strength and stiffness degradation were applied to all models as well. The incremental dynamic analysis was performed on the frames for 36 near-fault ground motions (a total number of 3312 nonlinear time history analyses) in the OpenSEES platform. The results show that the proposed DI can evaluate the damage properly in the nonlinear zone and collapse area in any nonlinear modeling methods. In addition, the merits and drawbacks of the proposed and other evaluated DIs were also discussed, and some of those drawbacks are being studied by the authors to be overcome in future works. Moreover, the efficiency of the proposed DI is also compared with the other studied DIs and shows that the proposed DI is significantly more efficient. The selected OpenSEES example files for the present study are shared through the link provided in the text.

Failure mechanisms and seismic fragility analysis of overhead transmission lines incorporating pile-soil-structure interaction

Overhead transmission lines (OTLs) are responsible for long-distance electric power transmission from power plants to end-users, inevitably passing through seismically active regions. A thorough literature review regarding seismic design and assessment of various structures underscores the significance of incorporating soil-structure interaction (SSI) and depth-varying spatial effect (DVSE). However, previous studies typically assumed that OTLs are fixed directly to the ground, and depth-varying ground motions (DVGMs) are oversimplified as uniform excitations, which may yield questionable structural performance estimations. To address this concern, this paper focuses on investigating the SSI effect on the seismic response and failure mechanism of an OTL subjected to DVGMs. Prior to this investigation, two commonly used methods for modeling the SSI effect in software ABAQUS are compared, with a nonlinear spring-based modeling method deems comparable in accuracy yet superior in efficiency. Subsequently, a refined finite element model of an OTL-soil-pile system is developed utilizing this simplified modeling method, and a group of three-dimensional ground motions varying within layered soils is stochastically synthesized. The influences of SSI and DVSE are systematically explored in terms of the seismic responses, failure mechanisms, and fragility probabilities of the OTL. The results expose the potential of the SSI effect to weaken the seismic capacity of the OTL and increase the risk of failure, while disregarding the DVSE is likely to lead to overestimated outcomes. This research is expected to raise awareness of considering SSI and DVSE in the seismic design and analysis of OTLs.

Efficient Solutions with the LRPS Method for Non-Linear Fractional Order Tuberculosis Models

In this research article, we present a novel Non-Linear Fractional Order Tuberculosis mathematical model (NLFOTB) and introduce an efficient technique to obtain its solution. Fractional Order Models (FOMs) have garnered significant attention in contemporary research due to their widespread applicability. We address the challenge of solving the coupled Initial Value Problems (IVPs) associated with NLFOTB models by utilizing the groundbreaking LRPS method, which combines the RPS approach with the Laplace transform operator. This innovative approach generates approximate solutions in rapidly converging series forms, offering enhanced efficiency and reduced computational effort compared to conventional methods. Through the implementation of the LRPS method, we successfully derive an approximate solution for the NLFOTB model, contributing significantly to the field. Furthermore, our proposed approach demonstrates its efficacy in accurately capturing the dynamics of Tuberculosis (TB) through extensive computations and graphical representations, contributing to a deeper understanding of TB dynamics within a mathematical framework. Additionally, the LRPS method shows promise in tackling real world problems involving differential equations of various orders. Future investigations can extend the application of the LRPS method to explore other Fractional Order Models, further validating its effectiveness in a wide range of epidemic scenarios. Consequently, our study not only provides valuable insights into Tuberculosis dynamics but also introduces a powerful computational tool applicable to various practical problems in diverse disciplines, making a substantial contribution to the field of mathematical modeling and computation. HIGHLIGHTS · The comprehensive study highlights the LRPS strategy’s efficiency and accuracy in approximating solutions for fractional order equations. The research demonstrates its capability to predict compartmental behavior accurately within the specified range · The discussions presented in the article significantly contribute to the field of epidemiology by introducing and showcasing the LRPS approach’s effectiveness. As a valuable tool for investigating and validating epidemic models, the LRPS method offers improved efficiency and convenience, thereby enhancing the understanding of disease dynamics · The researchers anticipate that their findings will inspire further exploration and utilization of the LRPS technique in solving nonlinear models. This, in turn, is expected to contribute to advancements in the broader field of epidemiology, fostering continued innovation and development GRAPHICAL ABSTRACT

Anomaly detection using large-scale multimode industrial data: An integration method of nonstationary kernel and autoencoder

Kernel methods and neural networks (NNs) are two mainstream nonlinear data modeling methods and have been widely applied to industrial process monitoring. However, they both present imperfect properties, so the relevant applications are limited. On the one hand, kernels are not so reconstructable, scalable, and robust to hyperparameters that they suffer performance degradation for large-scale data modeling and monitoring. On the other hand, the high-dimensional parameter space of NNs that is sorted to parameter initialization presents severe anomaly detection performance inconsistency, which makes the industry cautious about using NNs. Motivated by these facts, we propose to integrate kernels and NNs, forming a new model structure that is scalable, reconstructable, and performance-consistent. Specifically, a novel autoencoder-based nonstationary pattern selection kernel (AE-NPSK) is proposed by (1) selecting from the training set the critical edges and interior data as the centers of the radial basis functions in the hidden layers and (2) adaptively adjusting the kernel width in the training procedure. Also, the new NN has strong performance consistency, which facilitates the search for optimal parameters. Finally, we test the performance of the proposed method on the challenging multimode processes. The results validate the efficacy of the proposed method.

Meltdose Tacrolimus Population Pharmacokinetics and Limited Sampling Strategy Evaluation in Elderly Kidney Transplant Recipients.

Meltdose tacrolimus (Envarsus®) has been marketed as a formulation achieving a more consistent tacrolimus exposure. Due to the narrow therapeutic window of tacrolimus, dose individualization is essential. Relaxation of the upper age limits for kidney transplantations has resulted in larger numbers of elderly patients receiving tacrolimus. However, due to the physiological changes caused by aging, the tacrolimus pharmacokinetics (PK) might be altered. The primary aim was to develop a population PK model in elderly kidney transplant recipients. Secondary aims were the development and evaluation of a limited sampling strategy (LSS) for AUC estimation. A total of 34 kidney transplant recipients aged ≥65 years, starting on meltdose tacrolimus directly after transplantation, were included. An eight-point whole blood AUC0-24h and an abbreviated dried blood spot (DBS) AUC0-24h were obtained. The PK data were analyzed using nonlinear mixed effect modeling methods. The PK data were best described using a two-compartment model, including three transit compartments and a mixture model for oral absorption. The best three-sample LSS was T = 0, 2, 6 h. The best four-sample LSSs were T = 0, 2, 6, 8 h and T = 0, 1, 6, 8 h. The developed population PK model adequately described the tacrolimus PK data in a population of elderly kidney transplant recipients. In addition, the developed population PK model and LSS showed an adequate estimation of tacrolimus exposure, and may therefore be used to aid in tacrolimus dose individualization.

Nonlinear modeling of temperature-induced bearing displacement of long-span single-pier rigid frame bridge based on DCNN-LSTM

Long-span single-pier rigid frame bridge may experience excessive bearing displacement under temperature variation, which can result in structural deformation or instability, causing significant engineering accidents and immeasurable losses on society. This paper proposes a spatial-temporal nonlinear modeling method for temperature and temperature-induced bearing displacement (TIBD) of long-span single-pier rigid frame bridge based on DCNN-LSTM network with elastic modulus fusion, relying on the monitoring data of the Second Yangtze River Bridge in Wuhan. This framework introduces a dynamic autoregressive method for model inference to improve the prediction accuracy of the model. This study addresses three major issues: Insufficient rational selection of temperature characteristic values; Inadequate research on correlation between structural temperature and TIBD for the bridges; Poor prediction accuracy of temperature-TIBD regression models. The method proposed in this study achieves prediction accuracy of up to 99.8 % for TIBD, and this model is not affected by seasonal variation.

Deconvoluting Nonlinear Catalyst-Substrate Effects in the Intramolecular Dirhodium-Catalyzed C-H Insertion of Donor/Donor Carbenes Using Data Science Tools.

Interactions between catalysts and substrates can be highly complex and dynamic, often complicating the development of models to either predict or understand such processes. A dirhodium(II)-catalyzed C-H insertion of donor/donor carbenes into 2-alkoxybenzophenone substrates to form benzodihydrofurans was selected as a model system to explore nonlinear methods to achieve a mechanistic understanding. We found that the application of traditional methods of multivariate linear regression (MLR) correlating DFT-derived descriptors of catalysts and substrates leads to poorly performing models. This inspired the introduction of nonlinear descriptor relationships into modeling by applying the sure independence screening and sparsifying operator (SISSO) algorithm. Based on SISSO-generated descriptors, a high-performing MLR model was identified that predicts external validation points well. Mechanistic interpretation was aided by the deconstruction of feature relationships using chemical space maps, decision trees, and linear descriptors. Substrates were found to have a strong dependence on steric effects for determining their innate cyclization selectivity preferences. Catalyst reactive site features can then be matched to product features to tune or override the resultant diastereoselectivity within the substrate-dictated ranges. This case study presents a method for understanding complex interactions often encountered in catalysis by using nonlinear modeling methods and linear deconvolution by pattern recognition.

Dynamic nonlinear soft sensor modelling method using linear slow feature analysis and least squares support vector regression for batch processes

AbstractData‐driven soft sensor models have been extensively utilized in industrial processes. Batch processes are usually employed to manufacture low‐volume and high value‐added products in chemical, materials, and pharmaceutical industries. The most distinctive features of batch process lie in nonlinear, repetition, and slow time varying characteristics. In this paper, a data‐driven soft sensor modelling method based on linear slow feature analysis (LSFA) and least squares support vector regression (LSSVR) is proposed. In this method, LSFA was used to effectively capture the driving force behind the data features that change as slowly as possible. Then, a LSSVR model was constructed between the extracted slow feature variables and quality variables. Finally, a numerical example, industrial penicillin fermentation processes, and cobalt oxalate synthesis process were utilized to confirm the prediction accuracy and model reliability of the proposed approach.

3D modelling of the interaction between bending and corrosion-induced cracks in reinforced concrete beams

Bending and corrosion-induced cracks interact and impact the durability of Reinforced Concrete (RC) beams in a complex manner. Still, research on the mechanisms behind this interaction is limited. In this study, a 3D nonlinear finite element modelling method of RC beams was developed to explore this interactive mechanism. The models were assessed against experimental benchmarks, showing good agreement in terms of crack pattern and growth. Moreover, the modelling method offered the advantage of visualising the internal condition of beams, including the propagation of internal cracks and the expansion of corrosion products represented by interface elements. A parametric study was conducted to examine the influence of three crucial factors: the transport of the corrosion products, the distance between bending cracks, and bending crack width. The modelling results revealed that the transport of corrosion products appeared to play a decisive role in the location of initiation and evolution of corrosion-induced cracks. Additionally, as the distance between the bending cracks decreased, the onset of corrosion-induced cracks transitioned from the midpoint between the bending cracks to the vicinity of them. Furthermore, an increase in bending crack width had no impact on the initiation and evolution of corrosion-induced cracks, while their widths, and variation of width along the bar, decreased.

Nonlinear modeling and analysis of spatial pipeline with arbitrary shapes supported by clamps considering displacement-dependent characteristic

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes.

A Review of the Evolution Characteristics and Argillization of Clay Interbeds in Rockslides

Weak interlayers in rockslides often become the controlling factor for slope deformation due to their poor physical properties and tendency to undergo argillization. Achieving a comprehensive understanding of the weak interlayers will help us comprehend the mechanism of rock slope failure and protect people’s property. Some weak interlayers will develop into clay interbeds after tectonic activity and long-term groundwater action. A comprehensive review of clay interbeds includes a discussion on the formation conditions, the governing factors of the argillization process, and the investigation methods. The clay minerals in clay interbeds, illite, montmorillonite, and Kaolinite, play a major role in the acceleration of the argillization process. The argillization process can be observed through scanning technology and investigated using nonlinear dynamics, statistics, and numerical modeling methods, which may result in limitations for extracting parameters. It is necessary to conclude a unified evaluation standard to find the basic commonness during the argillization process of clay interbeds. It would be a future trend to establish the quantitative relationships among the mechanical strength, micro-structure and content of clay minerals of the clay interbeds during the argillization process.

Natural Disasters, Economic Growth, and Carbon Emissions: Empirical Analysis of Chinese Data Based on a Nonlinear Auto-Regressive Distributed Lag Model

China has a high frequency of natural disasters and it has become the economy with the largest carbon emissions in recent years. In this study, we mainly investigated the relationships between carbon emissions and natural disaster losses in China, as well as considering important factors such as economic growth and new energy consumption. Time series data for China from 2000 to 2020 were selected and based on the nonlinear auto-regressive distributed lag model method, a short-term error correction model and long-term co-integration relationship model were obtained between carbon emissions and their related factors. The results showed that in the long run, there is a significant nonlinear relationship between carbon emissions, new energy consumption and direct economic losses from natural disasters. There is a significant U-shaped relationship between natural disasters and carbon emissions, that is, natural disaster losses will significantly inhibit carbon emissions before they are below a certain threshold, but fewer natural disaster losses will increase carbon emissions. On the contrary, there is an inverted U-shaped relationship between new energy consumption and carbon emissions. When new energy consumption exceeds a certain threshold, it will help carbon peak early. In the short term, the impact of natural disasters on carbon emissions in the current period is significantly positive and higher than that in the lagged period, but the impact of its square term is negative. The short-term error correction model coefficient is −0.6467, and the error will be corrected when the short-term volatility deviates from the long-term equilibrium. These results suggest that attention should be paid to reducing disaster losses and the low-carbon reconstruction path for natural disasters, as well as continuously improving the level of new energy utilization, accelerating the pace of energy substitution, and promoting economic transformation for achieving “carbon peaking” in China.

Structural Behavior of Water Flow Glazing: Stress and Elastic Deformation Considering Hydrostatic Pressure

The fluid inside a Water Flow Glazing (WFG) panel creates a linear pressure distribution along the vertical dimension. Tension stress can cause problems with the sealant; compression stress can cause deflections in the glass panel. Increasing the glass thickness until the deflection is below the required limit implies more weight and cost of the glazing. Another solution is to limit glass deflection by inserting pillars or stripes into the cavity between two glass panes. The novelty of this article was to test a High-Order Finite Difference Method for linear and non-linear models to evaluate the effect of hydrostatic pressure produced by the fluid chamber on WFG panels. The methodology was tested on two case studies to assess the tension and deflection of the glass panes to guarantee the structural stability of WFG. The main conclusion drawn was that a linear plate model was sufficient to dimension the width, length, and thickness of the WFG panel. Furthermore, the mathematical model provided criteria to keep the glass tension below 45 MPa for tempered glass and the maximum deflection as the minimum between 1‰ of the glass height and 10% of the water chamber thickness. Introducing pillars or stripes solved hydrostatic pressure problems when the panel’s height was above 1.5 m.

Finite Element Analyses of Cast-in-situ RC Flat Slab with an Equivalent Frame for Horizontal Loads

In structural construction practice, the use of reinforced concrete slabs is extremely common due to their structural and economic advantages. An important issue is the study of structural behavior given to horizontal effects. The significance of this is centered not only on the scales caused by wind and construction loads, which are considered to be “normal” i.e., the quasi-static, which is essentially one-way monotonically increasing but also the most researched seismic, i.e., cyclically varying, direction and magnitude effects. The most widespread methods for sizing flat plate slabs to unidirectional, quasi-static horizontal loads are the so-called Equivalent Frame Method and the Effective Beam Width Method. In addition to analytical and basically linear numerical theoretical methods, several laboratory experiments have been performed and published. In this paper, we investigate the behavior of column-supported flat plate slabs against unidirectional, monotonically increasing horizontal loads using an advanced nonlinear numerical modeling method. Numerical models constructed with different geometric dimensions were created with ATENA 3D three-dimensional nonlinear finite element software. In the numerical studies, in addition to the vertical loads in the global sense, the horizontal loads were also taken into account. In our studies, we analyzed the global behavior of the structure, crack propagation, and internal stresses. The results were illustrated on force - displacement diagrams and compared with the results of the laboratory experiments used, thus showing the accuracy and limitations of the numerical modeling procedure. The numerical results were also compared with the results determined on the basis of the equivalent framework models.

Nonlinear modelling of shared mooring concepts for floating offshore wind turbines

Shared mooring has been proposed as one of the innovative technologies to reduce mooring and anchoring expenses for floating offshore wind turbines (FOWTs). Despite cost saving benefits, shared moorings introduce more complexity for the connected FOWTs than the conventional moorings. In this paper, we propose a nonlinear modelling approach for mooring configurations in Simpack and investigate the impact of a shared line mooring configuration on floater motions and mooring tensions of two connected FOWTs. First, we validate the nonlinear mooring modelling method through a comparison between OpenFAST and Simpack simulations of a 15-MW semi-submersible FOWT. Second, two 15-MW FOWTs with a shared mooring line configuration are further analyzed in Simpack to investigate the shared mooring impact. Different performances of the wind-wave loaded and unloaded FOWTs in two cases reflects the importance of considering nonlinear mooring stiffness in the modelling approach. By comparing the simulation results of two FOWTs using shared moorings and the single FOWT using conventional moorings, we identify the challenges of applying a shared mooring configuration that the surge displacements of the connected FOWTs increase significantly.

Implementation of Iwan-Type Nonlinear Rheology in a 3D High-Order Staggered-Grid Finite-Difference Method

ABSTRACT We have implemented and verified a parallel-series Iwan-type nonlinear model in a 3D fourth-order staggered-grid velocity–stress finite-difference method. The Masing unloading and reloading behavior is simulated by tracking an overlay of concentric von Mises yield surfaces. Lamé parameters and failure stresses pertaining to each surface are calibrated to reproduce the stress–strain backbone curve, which is controlled by the reference strain assigned to a given depth level. The implementation is successfully verified against established codes for 1D and 2D SH-wave benchmarks. The capabilities of the method for large-scale nonlinear earthquake modeling are demonstrated for an Mw 7.8 dynamic rupture ShakeOut scenario on the southern San Andreas fault. Although ShakeOut simulations with a single yield surface reduces long-period ground-motion amplitudes by about 25% inside a waveguide in greater Los Angeles, Iwan nonlinearity further reduces the values by a factor of 2. For example, inside the Whittier Narrows corridor spectral accelerations at a period of 3 s are reduced from 1g in the linear case to about 0.8 in the bilinear case and to 0.3–0.4g in the multisurface Iwan nonlinear case, depending on the choice of reference strain. Normalized shear modulus reductions reach values of up to 50% in the waveguide and up to 75% in the San Bernardino basin at the San Andreas fault. We expect the implementation to be a valuable tool for future nonlinear 3D dynamic rupture and ground-motion simulations in models with coupled source, path, and site effects.

A Multipath and Multiscale Siamese Network Based on Spatial-Spectral Features for Few-Shot Hyperspectral Image Classification

Deep learning has been demonstrated to be a powerful nonlinear modeling method with end-to-end optimization capabilities for hyperspectral Images (HSIs). However, in real classification cases, obtaining labeled samples is often time-consuming and labor-intensive, resulting in few-shot training samples. Based on this issue, a multipath and multiscale Siamese network based on spatial-spectral features for few-shot hyperspectral image classification (MMSN) is proposed. To conduct classification with few-shot training samples, a Siamese network framework with low dependence on sample information is adopted. In one subnetwork, a spatial attention module (DCAM), which combines dilated convolution and cosine similarity to comprehensively consider spatial-spectral weights, is designed first. Then, we propose a residual-dense hybrid module (RDHM), which merges three-path features, including grouped convolution-based local residual features, global residual features and global dense features. The RDHM can effectively propagate and utilize different layers of features and enhance the expression ability of these features. Finally, we construct a multikernel depth feature extraction module (MDFE) that performs multiscale convolutions with multikernel and hierarchical skip connections on the feature scales to improve the ability of the network to capture details. Extensive experimental evidence shows that the proposed MMSN method exhibits a superior performance on few-shot training samples, and its classification results are better than those of other state-of-the-art classification methods.

A low-cost discrete Vis-NIR optical sensing method for the determination of pear internal blackheart

In this study, a moldy crown pear core detection system based on a micro-optical sensor was developed. The micro-optical sensor has seven specific wavelengths, 425, 455, 515, 615, 660, 700, and 850 nm, and a cost-effective advantage. For the discrete spectrum, 7 kinds of preprocessing methods were compared. Traditional preprocessing methods, such as the standard normal transform (SNV) and multiple scattering correction (MSC) methods, cannot improve the efficiency of the spectrum. It was verified that the Savitzky − Golay (SG) convolution smoothing preprocessing method could be applied to preprocess discrete spectral data. The correlation of the spectrum after SG preprocessing in the partial least squares regression (PLSR) prediction model was 0.86, and the root mean square error (RMSE) was 0.19. Furthermore, the difference between the nonlinear modeling method without preprocessing and the PLS prediction model after preprocessing was compared. The results showed that the accuracy of the nonlinear modeling method for the discrete spectrum was much higher than that of the PLS linear modeling. The average model accuracy was above 0.9, and the k nearest neighbor (KNN) algorithm had the best effect, reaching an accuracy of 0.96. Finally, a prediction model accuracy of 0.98 was obtained by combining SG + KNN. In summary, the micro-optical sensor system had the advantages of low-cost performance and high precision, which are convenient for popularization and application in practice.

Regression-based identification and order reduction method for nonlinear dynamic structural models

In this paper, we propose a regression-based nonlinear reduced-order model for nonlinear structural dynamics problems, called the Nonlinear Identification and Dimension-Order Reduction (NLIDOR) algorithm. We evaluate the algorithm using a simple toy model, a chain of coupled oscillators and an actual three-dimensional flat plate. The results show that NLIDOR can accurately identify the natural frequencies and modes of the system and capture the nonlinear dynamical features, while the linear Dynamic Mode Decomposition (DMD) method can only capture linear features and is influenced by nonlinear terms. Compared with the full-order model (FOM), NLIDOR can effectively reduce computational cost, while compared with DMD, NLIDOR significantly improves computational accuracy. The results demonstrate the effectiveness and potential of NLIDOR for solving nonlinear dynamic problems in various applications.