During the metal tube bending (MTB) process, high-fidelity reconstruction of full cross-section (FCS) deformation is critical to the robustness of closed-loop control in tube-bending manufacturing systems. However, the distributed nature of industrial data, the spatiotemporal discontinuity of physical sensing, and the heterogeneity of multimodal physical-virtual data hinder effective integration of distributed sources and precise reconstruction of the transient deformation of tube surfaces. To address these challenges, we propose a Federated Split-Learning-Driven Multimodal Physical-Virtual Integration (FSLD-MPVI) framework. Leveraging a hybrid distributed-centralized architecture with cross-level collaborative fusion, FSLD-MPVI enables efficient integration and knowledge sharing of local high-fidelity visual data, global low-fidelity finite-element (FE) simulation data, and static process parameters that are dispersed across manufacturing nodes. Within the split learning (SL) distributed architecture, three cascaded, heterogeneous subnetworks are deployed, each dedicated to fusing a specific class of hybrid modality inputs, thereby providing the infrastructure needed to integrate modalities originating from different workshops. In the federated learning (FL) layer, a centralized server aggregates the parameters of each subnetwork respectively, mitigating cross-node data isolation while preserving data locality. Experiments demonstrate that FSLD-MPVI achieves high-accuracy global reconstruction (R2 = 0.9973); in the 90 degrees bending case, the shape deviation remains within 0.2 mm. These results verify that multimodal physical-virtual integration strongly supports precise global reconstruction of FCS deformation fields and establishes a new paradigm for intelligent process monitoring in advanced manufacturing systems.

Free-bending technology represents a novel manufacturing process for forming spatial tubes, enabling the fabrication of complex-axis tubes through continuous adjustment of die positions. As critical geometric parameters for describing tube shapes, curvature and torsion require precise prediction to ensure forming accuracy. This paper presents novel insights into the formation mechanism of free-bending spatial tubes by considering the dual factors of bending plane rotation and additional torque, and establishes a theoretical model for curvature-torsion prediction. To further elaborate the generation mechanisms of additional torque and twist angle—key forming parameters in the theoretical model—during bending die rotation, and their influence on the torsion of the tube axis, a physics-based fully connected neural network (FCNN) prediction model was proposed. By integrating tube material/geometric parameters with equipment process parameters, this model achieves high-accuracy predictions of curvature and torsion while maintaining strong interpretability, overcoming the limitations of intractable theoretical models. Example verification shows that its predicted curvature and torsion exhibit relative errors within 5% and 8%, respectively. Finally, forming experiments of an ellipsoidal-shaped part validated the model's effectiveness in practical manufacturing, demonstrating its capacity to guide spatial tube forming. Additionally, the model provides an alternative approach to determine key forming parameters (e.g., bending moment, torque, twist angle) with significantly higher accuracy and efficiency than traditional theoretical calculations.

Metal tube bending is widely used in high-tech fields such as aerospace. High-precision tube bending processes typically rely on pre-bending simulations to determine parameters that improve bending quality. Traditional machine learning methods focus only on predicting individual defects. In contrast, Finite Element Analysis (FEA) can simulate the entire deformation region (full-field) throughout the whole bending process (full-process), but it requires high computational resources. To address these challenges, this paper proposes a spatio-temporal deep learning method, SimTubeFormer, for full-field and full-process prediction of tube bending. Unlike existing methods that focus only on the final forming state or predicting individual defects, SimTubeFormer enables multi-frame spatio-temporal prediction across the entire deformation region throughout the bending process. SimTubeFormer also incorporates time-varying process parameters, including the bending speed and pressure die feed rate. This allows the model to accurately represent variable-speed bending operations. SimTubeFormer employs a temporal convolutional network to embed both static and dynamic process parameters. The embedded features are then fed into a spatio-temporal simulator based on a U-Net architecture. To reduce the computational complexity of 3D U-Net architectures, SimTubeFormer uses 2D convolutional modules for spatial feature extraction. Additionally, it integrates multi-head attention along the temporal dimension to capture dynamic evolution and efficiently model spatio-temporal features. The model is trained with a spatio-temporal smoothing loss. This loss function enhances prediction accuracy by enforcing both spatial consistency and temporal continuity. Experimental results show that SimTubeFormer outperforms traditional data-driven methods in prediction accuracy. In real physical experiments, it achieves a nodal error of 0.13 mm, which is comparable to the 0.12 mm error obtained by FEA. Crucially, SimTubeFormer reduces computation time from several hours required by FEA to mere tens of milliseconds, enabling rapid and accurate tube bending simulations. These results show that the proposed SimTubeFormer method can achieve accurate and real-time full-field and fullprocess virtual simulation. They also highlight its potential to support the development of digital twins and optimize tube bending quality.

The multi-roller bending (MRB) process, characterized by its high stability and flexible mold adaptability, is widely employed in the bending forming of spatial metal tubes (such as spatial spiral tubes (SSTs)). However, due to the fewer mold constraints of the already bent-formed section, the bent tube exhibits irregular axial spring-back, resulting in uncontrollable axial deviations. To improve forming accuracy, this study proposes a novel AWPSO-FECAM-LSTM framework that predicts the axis coordinates of the SSTs formed with MRB. The framework incorporates two prediction modes: the Angle-Regulation-Based (ARB) model, which predicts points based on the same angle, and the Segment-Regulation-Based (SRB) model, which predicts points based on the same segment. The FECAM module extracts frequency-domain features, thereby enhancing the model's ability to capture both temporal and frequency characteristics. Meanwhile, AWPSO optimizes hyperparameters using time-decay inertia weights and adaptive acceleration coefficients. Validated through bending experiments and finite element (FE) simulations, the model achieves a mean absolute percentage error (MAPE) of 0.98% and a mean squared error (MSE) of 0.000042, outperforming baseline models such as PSO-LSTM and vanilla LSTM. The ARB and SRB models collectively enable precise prediction of tube axis coordinates, with progressive prediction modes effectively reducing error accumulation. This framework demonstrates significant potential for real-time compensation in digital twin applications, advancing high-precision manufacturing of spatial metal tubes.

Free-bending (FB) technology enables the efficient processing of spatially complex-shaped tubes. Springback causes variations in curvature and torsion of the tube axis during the FB process. The mapping relationship of bent tube curvature and torsion from ideal to actual values can be abstracted as nonlinear physical operators. This paper first proposes a novel six-axis FB processing method that can control geometric features of tube transition segments. Then, an operator learning-based springback behavior prediction (OL-SBP) framework is presented, which includes an OL module and an SBP module. A feature-information-enhanced deep operator network (FIE-DeepONet) is integrated into the first module to learn tube springback operators. The curvature and torsion predicted by the OL module are then fed into the SBP module to calculate the overall shape of the springback axis. This paper also introduces a set of similarity evaluation indicators that are independent of the curve's spatial attitude. Planar and spatial bent tubes are selected as case studies. Results show that the framework yields more accurate predictions compared to the analytical model. The framework also exhibits excellent generalization performance. Once FIE-DeepONet has learned the springback operators, it can accurately predict the springback curvature and torsion, even for tube shapes not present during training.

Spatial multi-bend tubes are widely used in high-end equipment because they provide flexible spatial routing and lightweight structural integration. However, their forming quality is difficult to predict because the deformation of each bend influences the subsequent bends, leading to cumulative axial and cross-sectional errors. This study proposes a graph-based dual-attention framework for predicting the forming quality of spatial multi-bend tubes under given process parameters. A closed basis spline representation in a polar-coordinate cross-sectional frame is introduced to describe continuous cross-sectional deformation, while a kinematics-based key-point representation is adopted to characterize axial forming accuracy. A hierarchical graph attention module is used to capture intra-section and inter-segment geometric dependencies, and a segment-to-tube decoder integrates crosssectional features with process parameters for axial prediction. The framework is trained on finite-element data generated for rotary draw bending of 316L stainless steel tubes and is further examined using physical bending experiments. The results show that the proposed method provides accurate prediction of both crosssectional deformation and axial forming accuracy, demonstrating its potential for data-driven quality evaluation in multi-bend tube manufacturing.

Spatially bent tubes are essential components for high-end manufacturing equipment, enabling efficient routing and integration in confined spaces. Free-bending (FB) technology provides a flexible forming method for manufacturing complex-shaped tubes. However, dynamic springback during the FB process leads to irregular deviations between the tube's actual and intended geometries. This study first proposes a process-parameter-adjustable tube FB (PPA-TFB) method, which controls the post-springback curvature and torsion through two global and two local coefficients. Then, a multi-step springback prediction (MSP) strategy and its implementation framework based on a global-local multi-channel Fourier neural operator (GLMC-FNO) are developed to facilitate the realization of PPA-TFB. The MSP strategy discretizes the entire FB process using a series of time nodes. At each time node, the framework predicts the curvature and torsion of the forming segment at the next two time nodes based on the current springback state. Since the dominant factors influencing springback are different between processed and newly processed segments at each time node, a segmented prediction method is adopted to make GLMC-FNO learn distinct springback operators separately. The effectiveness of the MSP framework is validated through numerical simulations and experiments. The results show that the framework achieves accurate, efficient, and generalizable predictions for dynamic springback in FB-processed tubes. The spatiotemporal evolution patterns of curvature and torsion of bent tubes during the FB process are also investigated. Curvature exhibits local self-equilibrium and slight irregular attenuation, while torsion shows pronounced fluctuations in both time and space.

During the operation of a free-bending die, its fillets inevitably wear out, thereby reducing their “sharpness” and decreasing the forming accuracy of tubes. To clarify the influence law of fillet radius on the wall thickness of freely bent pipes, this study employs a combined approach of finite element simulation and experimental testing to systematically analyze the strain characteristics and wall thickness distribution of the pipes. The results indicate that the outer surface of the pipe exhibits the maximum thinning rate when the mold ceases upward movement. During the mold movement stage, increasing the fillet radius can mitigate tube wall thinning by expanding the contact area between the mold and the pipe. The thinnest region of the bent pipe is located at the end of the radius formation section, with a maximum thinning rate of 5.4%. Furthermore, an increase in mold offset distance, a rise in friction coefficient, and the adoption of thin-walled pipe structures all exacerbate the thinning phenomenon in this vulnerable region, which warrants special attention in engineering practice. With the increase of fillet radius, the maximum thinning rate presents a variation pattern of “first decreasing rapidly and then decreasing extremely slowly”. Within the range where the fillet radius-to-tube radius ratio (RB/r) is less than 0.33, the rate of decrease is significantly faster than that in the range where RB/r > 0.33. When RB/r = 0.33, a sudden enhancement of the thinning-inhibiting effect is observed. Subsequently, further increasing the fillet radius exerts no significant inhibitory effect on the maximum thinning rate. The findings of this study provide a theoretical basis and engineering guidance for the optimization of fillet parameters of free bending dies and the control of pipe forming accuracy.

Bending and torsion processes are commonly used for the forming of spatial tubes. However, buckling and wrinkling pose a significant challenge to the high-quality and stable forming of spatial tubes. To reveal the interaction mechanism of bending-torsion buckling (BTB) during the forming process, a quadratic model of BTB under plastic buckling instability was proposed. According to the buckling phenomena of simulation and experiment, the BTB state was divided into three zones, namely Zone I (torsion-dominated zone), Zone II (bending-torsion transition zone), and Zone III (bending-dominated zone). The characteristics of bending-torsion response changes, buckling wrinkle features, and critical load variation in the three zones were studied. Considering the complex interaction of bending and torsional buckling, an analytical method for the critical load of tubes under combined bending-torsion action has been provided. This method is based on the energy approach, incorporating pure bending and pure torsion buckling, along with the quadratic model of BTB. The effectiveness of the proposed theoretical model was verified by finite element (FE) simulation, and the influence of tube geometric characteristics, material parameters, and initial imperfection amplitude on the BTB interaction was discussed.

Accurate axial forming of bent tubes is critical for securing the reliability of aeroengine systems. The growing demand for advanced, customized, and smoothly operating aeroengines has driven the adoption of metal bent tubes with spatially free axes, formed by free bending (FB). However, the complexity of the FB process limits data accessibility and diminishes the axial prediction accuracy of deep-learning approaches. To address these challenges, a novel cross-forming-process transfer (CFPT) learning framework is proposed herein. The CFPT realizes precise axis prediction by transferring knowledge from a data-rich rotary-draw bending (RDB) process to a data-limited FB process. The CFPT employs graph neural networks (GNNs) as the base learner. Process reconstruction based on sparse coding realizes effective knowledge transfer by alleviating the effects of heterogeneity among the forming processes. This step is followed by adversarial domain adaption to extract the domain-invariant features between the source and target domains. Finally, a unified learning framework simplifies the model-training steps and prevents overfitting. Evaluation results revealed that CFPT significantly enhances the predictive accuracy, achieving a coefficient of determination (R2) of 0.985, a root mean squared error (RMSE) of 6.645, and a mean absolute error (MAE) of 4.778 on the validation set, along with an R2 of 0.970, an RMSE of 8.423, and an MAE of 5.766 on the test set-reflecting 30 % improvement on the validation set and 20 % improvement on the test set compared to the conventional methods. Further experiments also indicated that each of the transfer-learning strategies in CFPT is essential for successful knowledge transfer. These results demonstrated that the proposed CFPT precisely predicted spirally bent tube axes with limited data. This capability of the CFPT is expected to enable the smart manufacturing system for tube bending to respond flexibly to complex and variable bending demands.

Point cloud registration for evaluating the shape of 3D bent tubes is a preferred method for improving the forming quality and reducing fabrication costs. In this process, large nonlinear deformations, smooth regions, and low overlap result in massive outliers, making accurate registration for forming iterative optimization a challenging yet indispensable technique. We propose a new registration method based on implicit structural feature compatibility to predict the global-local rigid transform for multi-unit 3D bent tubes, called ISFC. In the two-stage tactic of ISFC for the alignment of the cross-source point cloud, the rigid compatibility in overlap regions and non-rigid compatibility in deformation regions are discriminated by the soft-distance consistency metric for global correspondence initialization. A new implicit axial structure constraint is established by evolution from the surface point to the interior based on the grassfire analogy, which joins faithful anchor points to generate a robust global correspondence hypothesis. Based on the global pose transformation, an innovative multipliers method named PC-ADMM is proposed for sequential local registration, which introduces a processing constraint into the optimization objective of the Lie group to refine tube unit transformation. The robustness and accuracy of the proposed method are confirmed by extensive registration experiments on synthetic and realworld tube datasets.

In response to the growing demand for small-batch bending tube production, traditional bending dies require separate customization for each tube size, resulting in extended design cycles and high costs. To meet bending requirements for tubes of different diameters using a single mandrel, a novel adjustable diameter mechanism (DAM) and its optimization design method are proposed. Initially, the DAM based on a planetary bevel gearscrew transmission set is developed for bending tubes of varying diameters. Subsequently, a domain knowledge-integrated optimization design framework is introduced. To reduce the cost of acquiring training samples for training surrogate models, a monotonicity-constrained neural network based on cascade boosting architecture (CB-MCNN) is introduced that enhances prediction accuracy while maintaining monotonicity. To improve the optimization speed and quality of Evolutionary Algorithms (EAs), a domain knowledge-guided EA (DK-EA) method is proposed, incorporating domain knowledge into the population initialization phase. The results indicate that: (1) CB-MCNN outperforms traditional methods and shows excellent performance on smallsample datasets. (2) DK-EA accelerates optimization processes and produces better outcomes. As a result, the domain knowledge-integrated optimization design framework enables the DAM to achieve a wider diameter variation range and enhanced reliability. The optimized DAM demonstrates the capability to bend tubes with diameters of 46-60 mm.

The thermal-mechanical bending process is a promising forming method for high-strength hollow tubular structures, enabling the forming limit's extension, particularly for difficult-to-bend metal tubes such as TA18 tubes. However, this forming process complicates the springback characteristics due to the thermal-mechanical coupling effect under multi-die heating and constraint strategies. To further reveal the elastic release mechanism of TA18 thin-walled tubes postunloading, we propose an analytical springback modeling method for the warm rotary draw bending (WRDB) process. A continuous steady-state heat transfer theoretical model across the tube cross-section is developed to model the thermal field and exactly capture the material flow behavior. Based on the experimental TA18 temperature softening behavior, the distribution of asymmetric strain and the nonlinear thermal field is introduced into the springback modeling, revealing the behavior of neutral layer shifting (NLS) and the evolution of yield surface across the cross-section under non-isothermal loading. Compared to finite element (FE) modeling, quantitative results show that the theoretical model can accurately predict tube springback behavior in experiments, achieving an average absolute relative error (AARE) below 1.4 % with low computational cost. The temperature effect on springback under different heating strategies is attributed to NLS and stress-strain redistribution induced by material temperature softening. The temperature difference of the local thermal field aggravates the NLS and contributes to the nonuniform tensile and compressive deformation, which is significant in the bending case with large diameter tubes and induces pronounced springback behavior.

Accurate forming shape prediction and process optimization are crucial for ensuring the quality of tubular components throughout both the design and iteration phases. However, the nonlinear multi-physics coupling between plastic deformation and process attributes presents significant complexity. Although the two tasks are inherently interdependent, they are often treated as separate paradigms in industrial applications. This lack of a synergistic approach impedes the establishment of an efficient closed-loop manufacturing process. We propose a physical-modulated dual-branch prediction framework, called Forming Process to Three-Dimension (FP-3D). It operates under a unified feature scale, which interactively maps from the process attributes to the three-dimensional (3D) tube shape. It bridges branches by extracting structurally embedded geometric latent features as a reliable intermediate representation. The process optimization branch contrasts shape features to learn latent disparity in pairs. It alleviates limitations posed by sample quantity and encourages the model to learn process attribute adjustments as a historically measured shape deformed to target one. A physical-increment-modulated (PIM) layer is proposed to facilitate the accelerated learning of physical increments that are sensitive to process attributes. In the shape inference branch, we propose radially transferring features toward the implicit skeleton, which enables physical information to intervene in the latent space for controllable shape generation. Under the sole condition of process parameters, FP-3D allows the conditional generation of point-wise features to decode refined 3D shapes. The extensive experiments conducted on diverse tube and benchmark datasets demonstrate that FP-3D exhibits state-of-the-art performance.

As a foundational configuration of spatial tubes, the spiral metal tube has been widely used in the industrial tube line system. Unfortunately, its precise forming remains a challenge up till now. In this paper, an improved analytical model is presented to reveal the forming mechanism of the spiral tube taking various processing parameters into account and verified by the FE simulations and four-axis free-bending (FFB) bending experiments. The method of springback prediction for the spatial configuration is given, and the transformation between four-axis and six-axis free bending process is provided. The conclusions can be drawn that the curvature radius primarily decreases with the increment of the offset, and the pitch mainly diminishes with the growth of the ratio defined as pushing velocity versus angular velocity of the panel. When the ratio is constant, the forming result remains unchanged, which is consistent with the theoretical model. Meanwhile, the evolution mechanism of forming quality is explored to provide a certain reference for the actual forming process. For the same tube configuration, reduction/thickening of wall thickness and cross-section distortion can be improved with the synchronous increase of the ratio of pushing velocity and angular velocity. It was innovatively found that the nonuniform distribution of shear stress under lower loading velocities is the dominant reason for the lower forming quality. This research effectively reveals the spiral tube forming mechanism and the evolution mechanism of forming quality, which establishes the foundation for analyzing the forming issues of complex spatial tubes in this field.

In order to improve the forming quality of largesize nickel-based alloy chemical tubes, the influence of different tube specifications and mandrel/tube clearances on the bending forming quality of largesize Hastelloy C-276 alloy tubes was studied. A finite element (FE) analysis model for NC rotary drawing bending (RDB) was established and verified. The results show that with the increasement of the clearance, the wall-thinning ratio [Formula: see text] of different tube specifications gradually decreases, and the wall-thickening ratio [Formula: see text] and ovality [Formula: see text] gradually increase. As the measuring angle increases, the [Formula: see text] of different specifications initially increases and then decreases. The [Formula: see text] exhibits a trend of initially increasing and then decreasing. It is reasonable to take about 1/8 of the wall thickness t for the mandrel/tube clearance. It is of great significance for guiding the bending forming of largesize nickel-based alloy chemical tubes and development of intelligent forming equipment.

The high-performance virtual sample generation (VSG) method has been extensively introduced to solve the problem of small sample sizes. Data distribution information is a key element of current VSG methods at the data-driven level. Herein, we propose an improved VSG method with a Gaussian distribution and explore the relationship between the Gaussian function and data expansion. To obtain more feasible virtual samples, information expanded based on the Gaussian membership function (GMIE) was established. For further improvement, a hybrid surrogate model based on transfer (THSM) is proposed, which differs from the general hybrid surrogate model (HSM) methods that only mix single models. Using a prevailing evaluation method, our proposed method, which combines a Gaussian membership function with a hybrid surrogate model, outperforms other competing approaches in 12 numerical cases owing to its feasibility and efficacy. Additionally, the proposed approach is applied to a metal tube rotary draw bending (RDB) prediction problem to illustrate its ability to support complex engineering designs.

Wrinkling is one of the most fatal defects of metal tube bending, which may seriously affect the forming quality and even lead to forming failure. Traditional wrinkling prediction methods fail to provide accurate results due to the complexity of multi-die coupling in the bending process and the neglect of time-varying effects. To this end, a novel early wrinkling prediction method is proposed in this paper, distinct from conventional methods, realizing to forecast future wrinkling trends during the bending process and laying the foundation for real-time wrinkling prediction. It leverages the wrinkling factor (WrF), calculated using the energy method, as temporal data during the bending process to indirectly predict future tube wrinkling trends. Since the wrinkling occurs at the beginning of the bending process, a multi-state informer-based early prediction of tube wrinkling is put forward utilizing the limited WrF collected at the start of the bending process. To meet the demand for high accuracy and efficiency of wrinkling early prediction in a dynamic process, the model pre-trained by the multi-state fusion wrinkling data from the fully bent tube is migrated to the target model through the transfer learning approach. A stainless-steel tube bending case is conducted as the verification experiment, which is simultaneously compared with the finite element analysis (FEA) result. The results show the superior prediction accuracy and higher efficiency of the proposed method mainly compared with the traditional Informer model, Transformer model, and Long Short-Term Memory (LSTM).

To satisfy lightweight design requirements, aerospace ducts frequently employ ultra-thin-walled tubes with a diameter-to-thickness ratio (D/t) exceeding 100.However, ultra-thin-walled tubes present significant forming challenges, and the mandrel plays a critical role in their bending. Therefore, investigating the effect of mandrel structure on the quality of ultra-thin-walled tubes formed through NC bending is of considerable importance. In this study, utilizing the Abaqus nonlinear finite element platform, an asymmetric thickness ball design method is proposed. Based on the positioning of the asymmetric balls within the mandrel, seven distinct designs for asymmetric thickness mandrels are developed. This study conducts a finite element analysis of the NC rotary draw bending (RDB) process for ultra-thin-walled 304 stainless steel tubes and validates the corresponding experiments. The results indicate that as the asymmetric thickness mandrel is positioned further from the mandrel, the stress on the outer side of the tube near the bend initiation first increases and then decreases, while the stress on the inner side of the tube, after the midpoint of the bend, initially decreases and then increases. The use of asymmetric thickness mandrels significantly reduces both the thinning and thickening rates of the tubes, though their impact on improving the ellipticity is less pronounced. The core ball nearest to the mandrel is designated as Ball 1, with subsequent balls further from the mandrel labeled as Ball 2 and Ball 3, respectively. The placement of the asymmetric thickness balls improves the thinning rate in the order: Ball 2 > Ball 1 > Ball 3; enhances the thickening rate in the order: Ball 1 > Ball 2 > Ball 3; and optimizes the ellipticity in the order: Ball 3 > Ball 1 > Ball 2.

A metal tube system is known as the industrial blood vessel, among which the bent section is the most vulnerable part. The cross-sectional defects (CSDs) of the bent tube cause the flow fluctuation of the fluid inside the tube. The existing defect characterization methods are roughly presented by describing CSDs in some specific cross-sections, which results in the lack of the tube full-bent section (FBS) characteristic information. To comprehensively describe and predict the tube FBS characteristics, an advanced physics-embedded CSDs prediction framework is proposed. This framework includes an FBS-neutral layer displacement angle (NLDA) prediction module and an FBS-CSDs prediction module, which uses the method that integrates the analytical model and BiLSTM-based deep learning (DL) models to predict the CSDs in the FBS of the tube. A novel analytical model of CSDs that considers both three-directional stresses and strains during tube bending is embedded in the FBS-CSDs prediction module. The analytical model provides the initial predicted values of CSDs through the NLDA sequence obtained from the FBS-NLDA module. The inaccurate CSDs are then treated as physical information to be fed into DL models for further correction and prediction. The prediction performance of this framework is validated through numerical simulations and experiments. The results prove that the framework can accurately predict the CSDs in the tube FBS. The integration of DL models with the analytical model not only overcomes the limitations of the analytical model, but also improves the prediction accuracy and convergence speed of DL models.