Technical paper 技术文件

Towards discrete manufacturing workshop-oriented digital twin model: Modeling, verification and evolution
面向离散制造车间的数字孪生模型：建模、验证和演化

Weiwei Qian, Yu Guo *, Litong Zhang, Shengbo Wang, Shaohua Huang *, Sai Geng
钱伟伟，郭宇*，张立彤，王胜波，黄少华*，耿赛The College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, China
中国南京航空航天大学机电工程学院

A R TICLE IN F O
文章

Keywords: 关键词：

Discrete manufacturing workshop (DMW)
离散制造车间（DMW）
Digital twin (DT) model
数字孪生（DT）模型
Migration modeling 迁移建模
Consistency verification
一致性验证
Synchronous evolution 同步进化

Abstract 摘要

Digital twin (DT) is one of key enabling technologies of intelligent transformation in discrete manufacturing workshop (DMW). The construction of digital twin system in DMW has greatly expanded the connotation and extension of smart manufacturing. Although previous studies have shown the success in DT, there is still a lack of clear and systematic methods of DT in DMW. To bridge this gap, this article focuses on how to systematically and effectively construct DT model methods for DMW. Taking the modeling, verification, and evolution of DT model as the main line, three key technologies related to digital twin in DMW are proposed, including model migration based matching modeling technology, graph convolutional network and temporal convolutional network based DT model verification and Adaboost based synchronous evolution of DT model, which will provide a systematic theory and method for the construction of DT model for DMW. The experiment demonstrates that the proposed methods have good performance for physical workshop in industrial environment and provide a holistic understanding of DT model modeling in DMW.
数字孪生（DT）是离散制造车间（DMW）智能化改造的关键使能技术之一。数字孪生系统在离散制造车间的构建，极大地拓展了智能制造的内涵和外延。尽管已有研究表明数字孪生技术取得了成功，但目前仍缺乏清晰、系统的数字孪生技术在离散制造车间中的应用方法。为了弥补这一不足，本文重点探讨了如何系统、有效地构建DMW的DT模型方法。以DT模型的建模、验证和演化为主线，提出了与DMW中数字孪生相关的三项关键技术，包括基于模型迁移的匹配建模技术、基于图卷积网络和时序卷积网络的DT模型验证技术和基于Adaboost的DT模型同步演化技术，为DMW中DT模型的构建提供系统的理论和方法。实验证明，所提出的方法在工业环境下的物理车间具有良好的性能，并对 DMW 中的 DT 模型建模提供了整体的理解。

1. Introduction 1.导言

Facing with fierce market competition and diverse customer needs, the core task of the manufacturing industry has evolved from traditional mass production to matching the trend of mass user customization, which increase the pressure for modern enterprises to improve production efficiency, save costs, and quickly adapt to market changes [1]. In this context, the discrete manufacturing industry also needs to transform and upgrade from “manufacturing” to “smart manufacturing”.
面对激烈的市场竞争和多样化的客户需求，制造业的核心任务已从传统的大规模生产演变为与用户大规模定制趋势相匹配，这为现代企业提高生产效率、节约成本、快速适应市场变化增加了压力[1]。在此背景下，离散制造业也需要从 "制造 "向 "智造 "转型升级。

Discrete manufacturing workshop (DMW) is a large, dynamic and complex system. The accurate analysis, predication and evaluation of the manufacturing performance and manufacturing behavior of DMW under uncertain environment are needed to achieve high quality and mass efficiency [2]. Intelligent technologies, such as the Internet of Things, Big Data Analysis, and artificial intelligence technology have collectively made this requirement possible. Digital twin (DT) is an emerging and vital technology for digital transformation and intelligent upgrade, driven by data and model; it enables high-fidelity monitoring, simulation, prediction, optimization and real-time control of physical manufacturing processes via developing the virtual counterpart of the physical entities and processes in cyberspace [1,3]. Under this background, DT workshop, as a new mode of workshop operation, has been
离散制造车间（DMW）是一个庞大、动态和复杂的系统。要实现高质量、高效率的生产，就需要对不确定环境下 DMW 的制造性能和制造行为进行准确的分析、预测和评估[2]。物联网、大数据分析和人工智能技术等智能技术共同使这一要求成为可能。数字孪生（DT）是以数据和模型为驱动，实现数字化转型和智能化升级的重要新兴技术；它通过在网络空间开发物理实体和过程的虚拟对应物，实现对物理制造过程的高保真监控、模拟、预测、优化和实时控制[1,3]。在此背景下，DT 车间作为一种新的车间运作模式，已被
gradually explored and applied in the manufacturing industry [4,5]. There has been research on construction and verification of DT model issues for DT workshop, such as standard and framework for DT modeling [4], production process DT modeling [6], data-driven featur-e-based verification [7] and mechanism-based verification [8], etc. Besides, some adaptive evolutionary frameworks of DT [9] and the method of online training DT model [10], etc. are also studied. The above-mentioned studies have made great efforts to tackle the construction and verification of DT model issues. However, there still exist such problems as low modeling efficiency, high verification difficulty and status synchronism between physical workshop and virtual workshop, which limits the development of genuine DT applications for smart manufacturing in DMW.
逐渐在制造业中得到探索和应用[4,5]。针对 DT 车间的 DT 模型构建与验证问题，已有一些研究，如 DT 建模的标准与框架[4]、生产过程 DT 建模[6]、基于数据驱动的特征验证[7]和基于机制的验证[8]等。此外，还研究了一些 DT 的自适应演化框架[9]和在线训练 DT 模型的方法[10]等。上述研究为解决 DT 模型的构建与验证问题做出了巨大努力。然而，目前仍存在建模效率低、验证难度大、物理车间与虚拟车间状态同步等问题，限制了DMW智能制造中真正的DT应用的发展。

The DT has progressed from theoretical research to pragmatic implementation, whereas the DT model is a paramount constituent of DT and a prerequisite for successful DT applications [5]. The real-time data and status of physical workshop could be updated to its virtual model, and the simulation and analysis results could act on physical workshop in turn to form closed-loop control, making DT workshop hold the potential for simulation, monitoring and scheduling applications. DT workshop enables the optimizations and decision-making in virtual workshop, which depends on the real-time data updated from physical
DT已从理论研究发展到实际应用，而DT模型是DT的重要组成部分，也是DT成功应用的前提[5]。物理车间的实时数据和状态可以更新到其虚拟模型中，仿真和分析结果可以反过来作用于物理车间，形成闭环控制，这使得 DT 车间在仿真、监控和调度应用方面大有可为。DT 车间使虚拟车间的优化和决策成为可能，而虚拟车间的优化和决策取决于物理车间的实时数据更新。

workshop, through synchronization enabled by sensors and controls. All these are based on a very important prerequisite that the DT model is accurate [11]. So, how to efficiently construct the DT model and how to maintain the accuracy of DT model in application is a key issue. In this study, we discussed three aspects: the modeling, verification, evolution of DT model, and proposed a migration modeling method of DT model, GCN-TCN based verification of DT model and synchronous evolution of DT model respectively. The main contribution of this article can be concluded as follows.
通过传感器和控制装置实现同步。所有这些都基于一个非常重要的前提条件，即 DT 模型的准确性[11]。因此，如何有效地构建 DT 模型以及如何在应用中保持 DT 模型的准确性是一个关键问题。在本研究中，我们从 DT 模型的建模、验证和演进三个方面进行了讨论，并分别提出了 DT 模型的迁移建模方法、基于 GCN-TCN 的 DT 模型验证方法和 DT 模型的同步演进方法。本文的主要贡献可总结如下。

A DT model migration modeling method (DT4M) of DMW is first proposed, which can describe modeling requirements and the transferability between DT models.
首先提出了 DMW 的 DT 模型迁移建模方法（DT4M），它可以描述建模要求和 DT 模型之间的可迁移性。
A GCN-TCN based DT model verification method is first proposed, which can verify the consistency of spatial characteristics and temporal characteristics between the DT model and physical workshop.
首先提出了一种基于 GCN-TCN 的 DT 模型验证方法，该方法可验证 DT 模型与物理车间之间的空间特征和时间特征的一致性。
An Adaboost based synchronous evolution method is proposed, containing the establishment of base learner, accuracy trend detection and synchronous update of DT model, which is essential to enhance the accuracy of DT model applications and to ensure the synchronization of virtual and physical workshop.
提出了一种基于 Adaboost 的同步演化方法，包含基础学习器的建立、精度趋势检测和 DT 模型的同步更新，这对提高 DT 模型应用的精度和确保虚拟车间与物理车间的同步至关重要。

The rest of this article is organized as follows. Section 2 presents an overview of related works on modeling, model verification and evolution of DT workshop. This is followed by an in-depth discussion on the connotation and characteristics of DT in DMW in Section 3. The key technologies of DT modeling of DMW are discussed in Section 4, highlighting how to solve the issue of efficiently constructing DT model and maintaining the accuracy of DT model in application. Section 5 demonstrates the results on performance evaluation for the proposed method through a case study. Section 6 concludes this study.
本文接下来的内容安排如下。第 2 节概述了有关建模、模型验证和 DT 工作坊演变的相关工作。随后，第 3 节深入讨论了 DMW 中 DT 的内涵和特点。第 4 节讨论了 DMW DT 建模的关键技术，重点介绍了如何解决高效构建 DT 模型和在应用中保持 DT 模型准确性的问题。第 5 节通过案例研究展示了所提方法的性能评估结果。第 6 节为本研究的结论。

In this section, several issues relating to this article in complex industrial manufacturing systems, including modeling, verification and evolution techniques for DT model, and the research gaps in industrial applications, are discussed respectively.
本节将分别讨论本文在复杂工业制造系统中的几个相关问题，包括 DT 模型的建模、验证和演化技术，以及在工业应用中的研究空白。

2.1. Digital twin model modeling
2.1.数字孪生模型建模

The DT model modeling of DMW is grounded on modeling theory. In current years, researchers have spared no effort to tackle the construction and verification of DT model issues for DT workshop, which is an important prerequisite for the promotion of the implementation of industrial DT technology and the realiziation of smart manufacturing. Grieves [12] proposed a general and standard DT modeling reference framework, known as three-dimensional DT model, including three dimensions namely physical entities, virtual models, and their connections. But it is still abstract and lack of pertinence in the implementation process, for example, the establishment of DT model in industrial environment. To explore vulnerabilities in industrial DT, Tao et al. [4] put forward the concept of DT workshop, defined the DT model from five dimensions namely physical entity, virtual entity, service, digital twin data and connection, and presented the theoretical support and key technologies of information fusion from four aspects of physical integration, data integration, model integration and service integration. In terms of modeling framework, Zhang et al. [13] proposed a framework of digital workshop, including physical model, ontology-based digital model and virtual model. Zhuang et al. [14] introduced a framework of DT model modeling in workshop, including four dimensions such as modeling object, modeling dimension, real-time condition monitoring and state prediction. However, the proposed dimensional modeling approach for describing the physical shop-floor lacks the consideration for model reuse and efficient modeling, and it may be limited by enforceability in practical applications. To explore virtual-real mapping,
DMW的DT模型建模以建模理论为基础。近年来，研究人员不遗余力地解决 DT 车间 DT 模型的构建和验证问题，这是推动工业 DT 技术实施和实现智能制造的重要前提。Grieves [12]提出了一个通用的标准 DT 建模参考框架，即三维 DT 模型，包括物理实体、虚拟模型及其连接三个维度。但在实施过程中，例如在工业环境中建立 DT 模型时，它仍然比较抽象，缺乏针对性。为了探索工业 DT 中的漏洞，Tao 等人[4]提出了 DT 工作坊的概念，从物理实体、虚拟实体、服务、数字孪生数据和连接五个维度定义了 DT 模型，并从物理集成、数据集成、模型集成和服务集成四个方面提出了信息融合的理论支撑和关键技术。在建模框架方面，张文等[13]提出了数字车间框架，包括物理模型、基于本体的数字模型和虚拟模型。Zhuang 等[14]提出了车间 DT 建模框架，包括建模对象、建模维度、实时状态监测和状态预测等四个维度。然而，所提出的描述物理车间的维度建模方法缺乏对模型重用和高效建模的考虑，在实际应用中可能会受到可执行性的限制。探索虚实映射、

Ding et al. [15] established the virtual-real mapping approach of DT model in smart manufacturing space and the data modeling method in multidimensional spatio-temporal. Uhlemann et al. [16] presented a method of establishing DT production system by real-time data acquisition and processing technology. In terms of production process modeling, Bao et al. [6] introduced a method of modeling DT model from the perspectives of product, technological process and operation. However, this method just focused on generating process models in virtual space, which are hardly to reuse models when new modeling requirements arise in the manufacturing systems. Schleich et al. [17] established an interaction between virtual model design and manufacturing method of skin model shapes. Yi et al. [18] set up a DT model for intelligent assembly process design and built a three-layer application framework based on DT for complex product process design. Wang et al. [19] presented a unified modeling approach to a knowledge-based DT system design, which enables designers to create and document a system model, but it costs too much time in learning such a graphical modeling language and finding suitable models to reuse and lacks DT model reusable cases and patterns. Leng et al. [20] proposed a remote semi-physical commissioning method of flow-type smart manufacturing systems based on DT. Besides, Luo et al. [21] put forward a hybrid predictive maintenance approach driven by DT to promote the development from modeling to application.
Ding等人[15]建立了智能制造空间中DT模型的虚实映射方法和多维时空的数据建模方法。Uhlemann 等[16]提出了一种通过实时数据采集和处理技术建立 DT 生产系统的方法。在生产过程建模方面，Bao 等人[6]介绍了一种从产品、工艺流程和操作角度对 DT 模型进行建模的方法。然而，这种方法只是侧重于在虚拟空间中生成过程模型，当制造系统出现新的建模需求时，很难重复使用模型。Schleich 等人[17]建立了虚拟模型设计与皮肤模型形状制造方法之间的交互。Yi 等[18]建立了智能装配工艺设计的 DT 模型，并构建了基于 DT 的复杂产品工艺设计三层应用框架。Wang 等[19]提出了一种基于知识的 DT 系统设计的统一建模方法，使设计人员能够创建和记录系统模型，但在学习这种图形建模语言和寻找合适的模型重用方面花费了太多的时间，缺乏 DT 模型可重用的案例和模式。Leng 等人[20]提出了一种基于 DT 的流程型智能制造系统远程半实物调试方法。此外，Luo 等人[21]提出了一种以 DT 为驱动的混合预测性维护方法，以促进从建模到应用的发展。

Obviously, previous studies have shown the success in DT framework and manufacturing elements modeling of workshop. However, there is a lack of effective DT migration modeling theory and methods for DMW, which may lead to low efficiency of DT modeling.
显然，以往的研究已经在车间 DT 框架和制造要素建模方面取得了成功。然而，目前还缺乏针对 DMW 的有效 DT 迁移建模理论和方法，这可能会导致 DT 建模效率低下。

2.2. Digital twin model verification
2.2.数字孪生模型验证

The DT model verification of DMW tests the consistency between physical and virtual workshop. Two DT model validity verification methods, data-driven feature-based verification and mechanism-based verification, are widely studied. The former is suitable for complex systems with mechanism unknown and difficult to characterize, but requires massive data and neural network model, while the latter is suitable for equipment or products with relatively clear mechanism. Saratha et al. [7] presented a DT model verification approach for industrial applications, including data modeling, data connection, attribute definition, semantic modeling, information connection and parameter verification. The sensor data from the real-time operation is used to compare with the corresponding data in DT model, which may be easily affected by noisy data. Semenkov et al. [8] put forward a hybrid digital model including virtual machine, simulator and actual hardware to verify large scale control systems. Besides, Qian et al. [22] proposed a model verification approach for DMW based on data characteristics to verify the modeling accuracy of DT model. However, the proposed layout similarity and model similarity index use expert scoring method, which has the characteristics of simple use and strong intuition, but its theoretical and systematic nature is ignored and it is sometimes difficult to ensure the objectivity and accuracy of the evaluation results. Jiang et al. [23] exploited blockchain to propose a new digital twin edge networks framework for enabling flexible and secure digital twin construction, and presented a DT model update and verification method based on blockchain and cooperative federated learning, which is efficient in the case of small data volumes, but may be inappropriate in the case of big data environment due to its properties (e.g., tracing historical data, the delay of the transactions, etc.) of the blockchain. Zhang et al. [24] proposed a consistency evaluation framework towards DT shop-floor models, which mainly contains two phases: before and after model assembly and model fusion. Specifically, the geometry models, physics models, behavior models and rule models are mainly considered before model assembly and model fusion. After model assembly and fusion, the overall performance of assembled models and fusion models is paid more attention. But the analytic hierarchy process is used to form a comprehensive consistency evaluation result for DT shop-floor models,
DMW 的 DT 模型验证测试物理车间与虚拟车间之间的一致性。基于数据驱动的特征验证和基于机理的验证这两种 DT 模型有效性验证方法被广泛研究。前者适用于机理未知、难以表征的复杂系统，但需要海量数据和神经网络模型；后者适用于机理相对清晰的设备或产品。Saratha 等人[7]提出了一种面向工业应用的 DT 模型验证方法，包括数据建模、数据连接、属性定义、语义建模、信息连接和参数验证。实时运行中的传感器数据被用来与 DT 模型中的相应数据进行比较，而 DT 模型中的数据很容易受到噪声数据的影响。Semenkov 等人[8]提出了一种混合数字模型，包括虚拟机、仿真器和实际硬件，用于验证大型控制系统。此外，Qian 等人[22]提出了一种基于数据特征的 DMW 模型验证方法，以验证 DT 模型的建模精度。但提出的布局相似度和模型相似度指标采用专家打分法，具有使用简单、直观性强的特点，但忽略了其理论性和系统性，有时难以保证评价结果的客观性和准确性。Jiang 等人 [23]利用区块链提出了一种新的数字孪生边缘网络框架，以实现灵活、安全的数字孪生构建，并提出了一种基于区块链和合作联合学习的DT模型更新与验证方法，该方法在数据量较小的情况下效率较高，但由于区块链的特性（如历史数据的追溯、交易的延迟等），在大数据环境下可能并不适用。Zhang 等人[24]提出了一种针对 DT 车间模型的一致性评价框架，主要包含两个阶段：模型组装前后和模型融合。具体来说，在模型组装和模型融合之前，主要考虑几何模型、物理模型、行为模型和规则模型。在模型组装和融合之后，则更加关注组装模型和融合模型的整体性能。但分析层次过程用于形成 DT 车间模型的综合一致性评价结果、
which may be easily influenced by subjective factors, resulting in inaccurate evaluation.
这很容易受到主观因素的影响，导致评估不准确。

Obviously, previous studies have shown it is feasible to evaluate the effectiveness of DT model in terms of its behavior, process and result, etc. However, there are some indicators (e.g., consistency index of geometry model, behavior model and logic model, etc.) that are difficult to quantify, moreover, the evaluation result are easily affected by subjective factors and thus result in poor enforceability.
显然，以往的研究表明，从行为、过程和结果等方面评价 DT 模型的有效性是可行的。但有些指标（如几何模型、行为模型和逻辑模型的一致性指标等）难以量化，而且评价结果容易受主观因素影响，可执行性差。

2.3. Digital twin model evolution
2.3.数字孪生模型的演变

The DT model synchronous evolution of DMW requires DT model to be qualified for tracking the performance degradation state of workshop and realizing the synchronization of the performance of physical workshop in industrial environment. Qian et al. [6] presented a representation model of performance degradation and a synchronous update model with competitive election mechanism to enhance the accuracy of production progress prediction with time in industrial environment. Zheng et al. [9] explored an application framework of DT, and described the implementation process of full parametric virtual modeling and the construction idea for DT application subsystems. In this work, the model of DT for product lifecycle management is analyzed, and the implementation process of full parametric virtual modeling is described. However, there are drawbacks in practical application mode of DT. Wang et al. [25] put forward a big data driven hierarchical DT predictive remanufacturing paradigm, and developed a big data driven layered architecture and the hierarchical digital-twin reconfiguration control mechanism respectively, which makes it possible to predict rapid reconfiguration optimization of sustainable products and remanufacturing processes. But it still lacks detailed reconstruction and evolution methods. Liu et al. [26] proposed an adaptive evolutionary framework for the decision-making models of DT machining system, focusing on insufficient robustness of the DT system in specific scenarios. This framework endows different decision-making models with evolutionary characteristics and can adaptively improve system’s machining quality requirements with product development. However, this research needs to be further in-depth application validated in complex industrial environments. Besides, Liu et al. [27] presented a multi-scale evolution mechanism of DT mimic model and explained the evolution mechanism from data to knowledge. The method can generate product quality knowledge models from product data, explore the relationships between quality indicators and reveal evolution mechanism during machining processes. But it may not be able to adapt to the dynamic evolution process in industrial environments, as it is considered from a multi-scale perspective and not from an adaptive dynamic perspective. Zhang et al. [10] explored an online training DT model method, established a DT model including state attributes, static performance attributes and fluctuating performance attributes, and realized the DT model training online by using real-time data. It facilitates data sharing and reuse to meet the dynamic update requirements of DT model and mainly focuses on the theoretical framework. However, the specific evolution and application technique are not clear to reflect production process, entities status dynamically.
DMW 的 DT 模型同步演化要求 DT 模型具备跟踪车间性能退化状态的能力，实现工业环境下物理车间的性能同步。Qian等[6]提出了一种性能退化表示模型和具有竞争选举机制的同步更新模型，以提高工业环境下生产进度随时间变化预测的准确性。Zheng 等[9]探讨了 DT 的应用框架，阐述了全参数虚拟建模的实现过程和 DT 应用子系统的构建思路。本文分析了 DT 在产品生命周期管理中的模型，阐述了全参数化虚拟建模的实现过程。然而，DT 在实际应用模式中存在缺陷。Wang等[25]提出了大数据驱动的分层DT预测再制造范式，并分别开发了大数据驱动的分层体系结构和分层数字-双子重构控制机制，使得预测可持续产品和再制造过程的快速重构优化成为可能。但它仍然缺乏详细的重构和演化方法。Liu 等人[26]针对 DT 系统在特定场景下鲁棒性不足的问题，提出了针对 DT 加工系统决策模型的自适应进化框架。该框架赋予不同的决策模型以进化特性，并能随着产品的发展自适应地提高系统的加工质量要求。不过，这项研究还需要在复杂的工业环境中进一步深入应用验证。此外，Liu 等人[27] 提出了 DT 模拟模型的多尺度演化机制，并解释了从数据到知识的演化机制。该方法可以从产品数据中生成产品质量知识模型，探索质量指标之间的关系，揭示加工过程中的演化机理。但由于该方法是从多尺度角度考虑问题，而不是从自适应动态角度考虑问题，因此可能无法适应工业环境中的动态演化过程。Zhang 等人[10]探索了一种在线训练 DT 模型的方法，建立了包括状态属性、静态性能属性和波动性能属性的 DT 模型，并利用实时数据实现了 DT 模型的在线训练。它促进了数据共享和重用，满足了 DT 模型的动态更新要求，主要集中在理论框架上。但在动态反映生产过程、实体状态等方面的具体演化和应用技术还不明确。

Obviously, previous studies have shown the success in DT model evolution framework, parameter evolution, etc. However, the dynamic updating mechanism of DT model in application of a manufacturing system has not yet been established, which may result in the decline of DT model accuracy and even failure.
显然，以往的研究已经在 DT 模型演化框架、参数演化等方面取得了成功。然而，DT 模型在制造系统应用中的动态更新机制尚未建立，这可能会导致 DT 模型精度下降，甚至失效。

To sum up, some challenges including modeling, verification and evolution of DT model still exist in DMW. Thus, taking “model modelingmodel verification-model evolution” as the main line, we begin with the concept and characteristics of DT model in DMW, then propose three key technologies of DMW, and verify the actual scenarios application effect, providing guide for the development of DT in discrete manufacturing workshop.
综上所述，在DMW中，DT模型的建模、验证和演进仍然存在一些挑战。因此，我们以 "建模-验证-演进 "为主线，从离散制造车间 DT 模型的概念和特点入手，提出离散制造车间 DT 的三大关键技术，并验证实际场景应用效果，为离散制造车间 DT 的发展提供指导。

3. Connotation and characteristics of digital twin in discrete manufacturing workshop
3.离散制造车间数字孪生的内涵和特点

3.1. Concept of digital twin in discrete manufacturing workshop
3.1.离散制造车间数字孪生的概念

Definition 1. The discrete manufacturing workshop (DMW) is a type of workshop (system) that produces according to discrete tasks and work units. For each order or product, the workshop needs to dispatch and arrange processes, operations and resources, etc. to meet different production needs. In the current multi-variety, variable-batch, and highly complex production environment, it has characteristics such as diversity, multi-disturbance, uncertainty and dynamic variability [2,11], which are described in more detail as follows.
定义 1.离散制造车间（DMW）是一种按照离散任务和工作单元进行生产的车间（系统）。针对每个订单或产品，车间需要调度和安排工序、作业和资源等，以满足不同的生产需求。在当前多品种、变批量、高复杂度的生产环境下，它具有多样性、多干扰性、不确定性和动态多变性等特点[2,11]，下面对这些特点进行详细介绍。

Diversity. The DMW has a wide range of manufacturing equipment, product types, production processes and technical states, and generally requires simultaneous manufacturing activities for multiple products and subassemblies.
多样性。DMW 的生产设备、产品类型、生产流程和技术状态多种多样，通常需要同时进行多种产品和组件的生产活动。
Multi-disturbance. There are various disturbance factors in the internal and external environment of DMW, including customer demand change, emergency order insertion, process adjustment, equipment failure, material shortage and product rework, etc.
多重干扰。DMW 的内外部环境存在多种干扰因素，包括客户需求变化、紧急订单插入、工艺调整、设备故障、材料短缺和产品返工等。
Uncertainty. The uncertainty of DMW runs through the whole production process of products and subassemblies, which is manifested in the dimensions of spacetime and state, such as the uncertainty of processing time and material transfer status.
不确定性。DMW 的不确定性贯穿于产品和组件的整个生产过程，表现在时空和状态两个维度上，如加工时间和材料转移状态的不确定性。
Dynamic variability. The manufacturing elements such as equipment, materials, tooling and personnel, etc. involved in the production process of products and subassemblies are constantly under dynamic changes.
动态变化。产品和组件生产过程中所涉及的设备、材料、工具和人员等制造要素一直处于动态变化之中。

With the in-depth research and application of DT in various industries, the understanding of DT tends to be diversified. It is widely acknowledged that the terminology was first introduced as “digital equivalent to a physical product” by Michael Grieves at University of Michigan in 2003 [28]. After then, the terminologies such as digital replication [29], dynamic virtual representation [30] and information mirroring model [31] emerged. Although the terminology has changed, the core concept remains, namely, establishing the mirroring model of physical object in digital space, and then mirroring model-based to enhance management ability and application effect.
随着 DT 在各行各业的深入研究和应用，人们对 DT 的理解也趋于多元化。众所周知，2003 年密歇根大学的 Michael Grieves 首次提出了 "数字等同于实体产品 "的术语[28]。此后，又出现了数字复制[29]、动态虚拟表示[30]和信息镜像模型[31]等术语。虽然术语发生了变化，但核心理念没有变，即在数字空间建立实物的镜像模型，然后基于镜像模型提高管理能力和应用效果。

DMW is a large, dynamic and complex system, and its virtual workshop should be completely consistent and synchronized with physical workshop in the explicit and implicit characteristics such as geometry, behavior, logic and performance of DT model. The concept of DT in DMW is defined as follows.
DMW 是一个庞大、动态和复杂的系统，其虚拟车间在 DT 模型的几何、行为、逻辑和性能等显性和隐性特征方面应与物理车间完全一致和同步。DMW 中 DT 的概念定义如下。

Definition 2. The DT of DMW is a technology system that is supported by manufacturing technologies such as Internet of things, Big data, artificial intelligence etc. It makes virtual replication of the characteristics, behavior, logic, and performance of manufacturing elements of DMW, performs the operations of description, monitoring, simulation, prediction, optimization and control of DMW, and enables visible status, measurable performance, optimized scheme, and controllable process [32,33].
定义 2.DMW 的 DT 是以物联网、大数据、人工智能等制造技术为支撑的技术系统。它对 DMW 制造要素的特征、行为、逻辑和性能进行虚拟复制，对 DMW 进行描述、监测、仿真、预测、优化和控制等操作，实现状态可视、性能可测、方案可优、过程可控[32,33]。

3.2. Characteristics of digital twin in discrete manufacturing workshop
3.2.离散制造车间数字孪生的特点

The word “twin” is interpreted as ‘the same or very similar metaphor’ in the dictionary, so it means that physical system and DT model have the same gene, and should have very similar or the same physical laws and operating mechanism. Moreover, DT model should have ability to simulate, track physical system performance changes, and optimize the physical system constantly. Thus, the DT in discrete manufacturing workshop also needs to have the following characteristics.
在字典中，"孪生 "一词被解释为 "相同或非常相似的比喻"，因此它意味着物理系统和 DT 模型具有相同的基因，应该具有非常相似或相同的物理规律和运行机制。此外，DT 模型还应具备模拟、跟踪物理系统性能变化并不断优化物理系统的能力。因此，离散制造车间的 DT 还需要具备以下特征。

The virtual workshop mirroring the physical workshop. DT model should be the same as physical entity contained in physical workshop with similar or the same explicit and implicit characteristics. Explicit characteristics refer to geometrical shapes, material quality, illumination, size, layout etc. Implicit characteristics refer to physical laws and operating mechanisms, as well as the requirements of high accuracy in production progress, capacity, energy consumption and other performance index of physical workshop.
虚拟车间是实体车间的镜像。DT 模型应与实体车间中的实体相同，具有相似或相同的显性和隐性特征。显性特征指几何形状、材料质量、照明、尺寸、布局等。隐含特征指物理规律和运行机制，以及对物理车间的生产进度、产能、能耗等性能指标的高精度要求。
The virtual-physical fusion and virtual controlling physical. Physical and virtual workshop are two-way real mapping and real-time interaction, existing in parallel, one-to-one correspondence and coevolution. The performance of manufacturing elements such as equipment and tools may deteriorate over time during the life cycle of DMW. The accuracy of DT model varies greatly with time, resulting in poor consistency of physical laws between physical entity evolution and that contained in DT model. Virtual workshop tracks the performance changes of physical entity by continuously accumulating real-time data of physical workshop, and truly records the evolution process of physical workshop; then DT model can accurately represent the dynamic evolution law of physical entity by constantly modifying its structure and parameters. Virtual workshop simulates and optimizes the operating state of physical workshop according to real-time data, regulating physical workshop in real time. Physical workshop reproduces the production process defined by virtual workshop, and produces in strict accordance with the analysis results by virtual workshop.
虚实融合、虚实相控。物理车间和虚拟车间是双向真实映射和实时交互，并行存在，一一对应，共同演进。在 DMW 的生命周期中，设备和工具等制造要素的性能可能会随着时间的推移而退化。DT 模型的精度随时间变化很大，导致物理实体演化与 DT 模型所包含的物理规律之间的一致性很差。虚拟车间通过不断积累物理车间的实时数据，跟踪物理实体的性能变化，真实记录物理车间的演化过程，然后通过不断修改物理实体的结构和参数，使 DT 模型能够准确表达物理实体的动态演化规律。虚拟车间根据实时数据模拟和优化物理车间的运行状态，实时调节物理车间。物理车间再现虚拟车间定义的生产过程，并严格按照虚拟车间的分析结果进行生产。

4. Key technologies of digital twin modeling in discrete manufacturing workshop
4.离散制造车间数字孪生建模的关键技术

DT, as an effective medium to realize the fusion of physics and information, is applied in DMW for production monitoring and management. The framework of DT system for DMW is shown in Fig. 1.
DT 作为实现物理与信息融合的有效媒介，被应用于 DMW 的生产监控和管理。用于 DMW 的 DT 系统框架如图 1 所示。

The framework is composed of physical layer, virtual layer and application layer [22,32,33]. Physical layer is the source of the actual data of the production process, including objective entities of manufacturing elements such as materials, tooling, equipment and environment, etc., as well as various production activities in the production process.
该框架由物理层、虚拟层和应用层组成[22,32,33]。物理层是生产过程实际数据的来源，包括材料、工具、设备和环境等制造要素的客观实体，以及生产过程中的各种生产活动。

Virtual layer is the core of DT system, which realizes the digital reconstruction of workshop production activities in virtual space, supports data analysis, mechanism model and control model construction, etc., and completes the law discovery of the global data uploaded from the physical layer. And it interacts with the application layer to realize the global optimization of the workshop production process. The virtual layer is composed of model block, data middle platform and technical support block, etc. The model block is the geometric mirror of physical workshop, completing the visualization of physical entities. The data middle platform completes data processing and feature fusion, etc. to realize the law discovery of global data, model warehouse construction, rule/controlling model management and application service visualization, which mainly completes the definition of rule model, control model and incorporates tacit knowledge, etc., making the digital twin model smarter.
虚拟层是 DT 系统的核心，实现车间生产活动在虚拟空间的数字化重构，支持数据分析、机构模型和控制模型构建等，完成对物理层上传的全局数据的规律发现。并与应用层交互，实现车间生产过程的全局优化。虚拟层由模型块、数据中间平台和技术支持块等组成。模型块是物理车间的几何镜像，完成物理实体的可视化。数据中间平台完成数据处理、特征融合等，实现全局数据的规律发现、模型仓库构建、规则/控制模型管理和应用服务可视化，主要完成规则模型、控制模型的定义，并融入隐性知识等，使数字孪生模型更加智能。

In addition, the behavior model describes the real-time response and behavior of physical entities under the influence of external environment and internal operating mechanisms. It can be functions possessed by physical entities such as turning, milling, drilling, grinding, etc. of machine tools. The physical entities are exposed to a variety of uncertainties in practical operation, and behavioral models tend to be discrepant as a result [5]. Therefore, it is necessary to construct based on constraints. For example, if the turning function of machine tools cannot be used under current conditions, the turning function of the behavior model will be shielded. The logic model maps the actual production process of the physical workshop and the internal logic of the workshop operation, such as technological process and production scheduling
此外，行为模型描述了物理实体在外部环境和内部运行机制影响下的实时响应和行为。它可以是机床的车、铣、钻、磨等物理实体所具有的功能。物理实体在实际操作过程中会面临各种不确定性，行为模型往往因此而出现偏差[5]。因此，有必要基于约束条件进行构建。例如，如果机床的车削功能在当前条件下无法使用，行为模型的车削功能将被屏蔽。逻辑模型映射物理车间的实际生产流程和车间运行的内部逻辑，如工艺流程和生产调度等。
rules, etc. 规则等。
Application layer is based on the law discovery of global data in virtual layer, combined with real-time information to support intelligent planning, distribution and manufacturing. The simulation block plays an important role in this layer, which provides multi-level service functions such as monitoring analysis, prediction and cause analysis of production process for the workshop, so as to improve the overall production efficiency of the workshop.
应用层基于虚拟层对全局数据的规律发现，结合实时信息，支持智能计划、分配和制造。仿真块在这一层中扮演着重要角色，为车间提供生产过程的监控分析、预测、原因分析等多层次服务功能，从而提高车间的整体生产效率。

As shown in Fig. 1, DT model includes all aspects of the whole life cycle of DMW and provides overall technical and method support for the intelligent management of the production process. However, the effective DT model needs to be established from multiple aspects, such as model modeling, model verification and model evolution. As there are many key technologies to be broken through in the construction of DT workshop, and DT modeling is the basis for building DT workshop, it is necessary to first establish DT model of DMW, verify DT model according to the construction index and track the synchronous evolution of physical entity performance degradation. This study explores the technical methods of DT modeling in DMW by taking “model modelingmodel verification-model evolution” as the main line
如图 1 所示，DT 模型包含了 DMW 全生命周期的各个环节，为生产过程的智能化管理提供了全面的技术和方法支持。然而，有效的 DT 模型需要从建模、模型验证、模型演化等多个方面建立。由于 DT 车间建设需要突破的关键技术较多，而 DT 建模是建设 DT 车间的基础，因此首先需要建立 DMW 的 DT 模型，根据建设指标对 DT 模型进行验证，并跟踪物理实体性能退化的同步演化。本研究以 "建模-模型验证-模型演化 "为主线，探索DMW中DT建模的技术方法。

The meanings for acronyms and models’ notation are listed in Table 1 for easy reading.
表 1 列出了缩略语和模型符号的含义，以便于阅读。

4.1. Model migration based matching modeling technology
4.1.基于匹配建模技术的模型迁移

DT model is a physical workshop-based integrated model of operation modes and control methods with different granularities such as units, production lines and workshops, which has various components and complex-coupled logic. Physical workshop is one set containing the collection of physical entities, processes, functions etc., which involves in different aspects such as people, machines, materials, methods and environments, and has functions of data acquisition, state perception and real-time transmission. In detail, it includes:(1) sensing resources such as sensors and radio frequency identification systems (radio frequency identification (RFID), Ultra wide band (UWB), etc.), (2) network resources such as wireless networks and intelligent gateways, (3) control and execution resources such as machine tools, automated guided vehicles (AGVs) etc., (4) software resources such as enterprise resource planning, manufacturing execution systems etc., (5) technological process (process chain, technological parameter, etc.) and products (semimanufactured goods, finished goods, etc.).
DT 模型是基于物理车间的运行模式和控制方法的集成模型，具有单元、生产线和车间等不同粒度，包含各种组件和复杂的耦合逻辑。物理车间是包含物理实体、过程、功能等的集合，涉及人、机、料、法、环等不同方面，具有数据采集、状态感知、实时传输等功能。具体包括：(1) 传感器和射频识别系统（射频识别（RFID）、超宽带（UWB）等）等传感资源；(2) 无线网络和智能网关等网络资源；(3) 机床、自动导引车（AGV）等控制和执行资源；(4) 企业资源计划、制造执行系统等软件资源；(5) 工艺流程（工艺链、工艺参数等）和产品（半成品、成品等）。

With the development of DT technology, we are in the era of big data and gradually entering the era of “big model”. Similar to big data, the model warehouse formed by massive DT model also contains huge potential value. Establishing a new DT model requires a lot of time and expensive computing resources. Therefore, it is necessary to consider the technical and economic issues of model reuse, focusing on whether these established models can be reasonably reused from the DT model warehouse.
随着 DT 技术的发展，我们正处于大数据时代，并逐渐进入 "大模型 "时代。与大数据类似，由海量 DT 模型形成的模型仓库也蕴含着巨大的潜在价值。建立一个新的 DT 模型需要大量的时间和昂贵的计算资源。因此，有必要考虑模型复用的技术和经济问题，重点关注这些已建立的模型能否从 DT 模型仓库中合理复用。

Previous studies have shown the success in the transfer learning of models or features [34]. Inspired by them, we transform the DT modeling problem into the transferability between DT models.
以往的研究表明，模型或特征的迁移学习是成功的[34]。受其启发，我们将 DT 建模问题转化为 DT 模型之间的可迁移性问题。

Problem Description. The DT modeling requirement set (domain) of physical workshop is recorded as $P W_{need} = {p w_{need}^{1}, p w_{need}^{2}, \dots, p w_{need}^{I}}$ , the DT model set (domain) is recorded as $V W_{p r} = {v w_{p r}^{1}, v w_{p r}^{2}, \dots, v w_{p r}^{J}}$ , and the interaction matrix between $P W_{need}$ and $V W_{p r}$ is recorded as. Then, given the ${I N}_{p w \leftrightarrow ww}$ and knowledge graph $G$ , the goal of migration modeling is to predict whether the model in DT model warehouse will be selected according to the modeling requirements. The prediction formula can be expressed as follows:
问题描述。物理车间的 DT 建模需求集（域）记为 $P W_{need} = {p w_{need}^{1}, p w_{need}^{2}, \dots, p w_{need}^{I}}$ ，DT 模型集（域）记为 $V W_{p r} = {v w_{p r}^{1}, v w_{p r}^{2}, \dots, v w_{p r}^{J}}$ ， $P W_{need}$ 和 $V W_{p r}$ 之间的交互矩阵记为。然后，给定 ${I N}_{p w \leftrightarrow ww}$ 和知识图谱 $G$ ，迁移建模的目标就是预测 DT 模型仓库中的模型是否会根据建模要求被选中。预测公式可表示如下：
$i n_{p w \leftrightarrow vw}^{predict} = f_{predict} (p w_{need}, ν w_{p r}, ω, G)$
Where, $i n_{p w \leftrightarrow v w}^{predict}$ represents the probability of selecting $v w_{p r}$ for migration modeling based on the modeling requirement $p w_{need}$ , and $ω$ represents
其中， $i n_{p w \leftrightarrow v w}^{predict}$ 代表根据建模要求 $p w_{need}$ 选择 $v w_{p r}$ 进行迁移建模的概率， $ω$ 代表

Fig. 1. The framework of DT system for DMW.
图 1.用于 DMW 的 DT 系统框架。

Table 1 表 1
The Meanings for acronyms and models’ notation.
缩略语和模型符号的含义。

Acronyms or Symbols 缩略语或符号	Description 说明
DT	Digital twin 数字孪生
DMW	Discrete manufacturing workshop 离散制造车间
DT4M	Digital twin model migration modeling method 数字孪生模型迁移建模方法
GCN	Graph convolutional network 图卷积网络
TCN	Temporal convolutional network 时序卷积网络
RFID	Radio frequency identification 无线电频率识别
UWB	Ultra wide band 超宽带
AGVs AGV	Automated guided vehicles 自动制导车辆
GM	Geometry model 几何模型
BM	Behavior model 行为模式
LM	Logic model 逻辑模型
PM	Performance model 性能模式
WIP	Work in process 正在进行的工作
$P W_{need}$	Represents the DT modeling requirement set (domain) of physical workshop. 代表物理车间的 DT 建模需求集（域）。
$V_{P} W_{r}$	Represents the DT model set (domain). 代表 DT 模型集（域）。
${I N}_{pw↔vw}$	Represents the interaction matrix between $P W_{need}$ and $V W_{p r}$ . 代表 $P W_{need}$ 和 $V W_{p r}$ 之间的交互矩阵。
G	Represents knowledge graph. 代表知识图谱。
in $i_{p w \leftrightarrow pred}^{prict}$ 在 $i_{p w \leftrightarrow pred}^{prict}$ 中	Represents the probability of selecting $v w_{p r}$ for migration modeling based on the modeling requirement $p w_{need}$ . 表示根据建模要求 $p w_{need}$ 选择 $v w_{p r}$ 进行迁移建模的概率。
$f_{predict}$ (.)	Represents the recommendation model. 代表推荐模型。
$ω$	Represents parameters to be trained for the prediction model. 代表预测模型需要训练的参数。
$p w_{need}^{i}$	Represents the $i$ -th modeling requirement. 代表 $i$ -th 建模需求。
$v w_{p r}^{j}$	Represents the $j$ -th DT model. 代表 $j$ -th DT 模型。
AM	Represents the adjacency matrix. 代表邻接矩阵。
DM	Represents the degree matrix. 代表度矩阵。
${P W}_{M}$	Represents the graph model of physical workshop 代表物理工作室的图模型
V	Represents the node set. 代表节点集。
E	Represents the edge set. 代表边缘集。
L	Represents the correlation function. 代表相关函数。
M	Represents the workstations. 代表工作站。
O	Represents the manufacturing orders, and represents processes in the workshop. 代表生产订单，并代表车间的流程。
W	Represents the work in process. 代表正在进行的工作。
$M_{P}$	Represents the processes in the workshop. 代表研讨会的进程。
$d (v_{n^{'}})$	Represents the attribute information stored by the nodev $_{n}$ . 代表 nodev $_{n}$ 存储的属性信息。
£	Represents the spatial characteristic of physical workshop. 代表实体车间的空间特征。
£ $^{}$ $^{}$ 英镑	Represents the spatial characteristic of virtual workshop. 代表虚拟车间的空间特征。
$Δ_{P W \leftrightarrow V W}^{S F}$	Represents the proportion of spatial characteristic consistency number of physical workshop and virtual workshop in the total. 表示实体车间和虚拟车间的空间特征一致性数量占总数的比例。
da $_{s m} (k)$	Represents manufacturing information of the $m$ -th station at time $k$ . 代表时间 $k$ 时第 $m$ -th 台的制造信息。
$l_{s m}^{h} (k)$	Represents the position of the $h$ -th manufacturing element at time $k$ . 表示时间 $k$ 时， $h$ -th 制造元素的位置。
$^{sm}$ h $(k)$	Represents the serial number of the $h$ -th manufacturing element. 代表第 $h$ 个制造元素的序列号。
${con}_{s m}^{h} (k)$	Represents the production state of the $h$ -th manufacturing element. 代表第 $h$ 个制造元素的生产状态。
$λ$	Represents the temporal characteristic of physical workshop. 代表物理车间的时间特征。
$λ^{*}$	Represents the temporal characteristic of virtual workshop. 代表虚拟车间的时间特性。
$Δ_{P W ↩ V W}^{T F}$	Represents the proportion of temporal characteristic consistency number of physical workshop and virtual workshop in the total. 代表实体车间和虚拟车间数量在总数中所占的时间特征一致性比例。
TA	Represents the test accuracy of performance model at time $t$ . 表示时间 $t$ 时性能模型的测试精度。
${\hat{y}}_{i *, t}$	Represents the actual value of the $i ψ$ -th performance index at time $t$ . 表示时间 $t$ 时 $i ψ$ -th 性能指标的实际值。
$f (x^{#}; i^{\#; t)}$	Represents the predicted value of the $i ♯$ -th performance index at time $t$ . 表示时间 $t$ 时 $i ♯$ -th 性能指标的预测值。
$f_{TD}$ (.)	Represents the trend detection function. 代表趋势检测功能。
$Δ^{Δ^{*}}$	Represents the value of trend detection function $f_{T D}$ (.). 代表趋势检测函数 $f_{T D}$ (.) 的值。
$α_{u}$	Represents the weight coefficient of the $u$ -th base learner. 代表第 $u$ 个基础学习器的权重系数。
$ε_{u}$	Represents the accuracy error of the $u$ -th base learner. 代表第 $u$ 个基础学习器的精度误差。
F * ( $()$ F * ( $()$ )	Represents the evolved performance model. 代表演变后的性能模型。

Acronyms or Symbols Description DT Digital twin DMW Discrete manufacturing workshop DT4M Digital twin model migration modeling method GCN Graph convolutional network TCN Temporal convolutional network RFID Radio frequency identification UWB Ultra wide band AGVs Automated guided vehicles GM Geometry model BM Behavior model LM Logic model PM Performance model WIP Work in process PW_("need ") Represents the DT modeling requirement set (domain) of physical workshop. V_("P ")W_("r ") Represents the DT model set (domain). IN_("pw↔vw ") Represents the interaction matrix between PW_("need ") and VW_(pr). G Represents knowledge graph. in i_(pw harr" pred ")^("prict ") Represents the probability of selecting vw_(pr) for migration modeling based on the modeling requirement pw_("need "). f_("predict ") (.) Represents the recommendation model. omega Represents parameters to be trained for the prediction model. pw_("need ")^(i) Represents the i-th modeling requirement. vw_(pr)^(j) Represents the j-th DT model. AM Represents the adjacency matrix. DM Represents the degree matrix. PW_("M ") Represents the graph model of physical workshop V Represents the node set. E Represents the edge set. L Represents the correlation function. M Represents the workstations. O Represents the manufacturing orders, and represents processes in the workshop. W Represents the work in process. M_(P) Represents the processes in the workshop. d(v_(n^('))) Represents the attribute information stored by the nodev _(n). £ Represents the spatial characteristic of physical workshop. £ ^(**) Represents the spatial characteristic of virtual workshop. Delta_(PW harr VW)^(SF) Represents the proportion of spatial characteristic consistency number of physical workshop and virtual workshop in the total. da _(sm)(k) Represents manufacturing information of the m-th station at time k. l_(sm)^(h)(k) Represents the position of the h-th manufacturing element at time k. ^("sm ") h (k) Represents the serial number of the h-th manufacturing element. con_(sm)^(h)(k) Represents the production state of the h-th manufacturing element. lambda Represents the temporal characteristic of physical workshop. lambda^(**) Represents the temporal characteristic of virtual workshop. Delta_(PW↩VW)^(TF) Represents the proportion of temporal characteristic consistency number of physical workshop and virtual workshop in the total. TA Represents the test accuracy of performance model at time t. widehat(y)_(i**," t ") Represents the actual value of the i psi-th performance index at time t. f(x^(#);i^("\# ";" t ")):} Represents the predicted value of the i♯-th performance index at time t. f_("TD ") (.) Represents the trend detection function. Delta^(Delta^("* ")) Represents the value of trend detection function f_(TD) (.). alpha_("u ") Represents the weight coefficient of the u-th base learner. epsi_(u) Represents the accuracy error of the u-th base learner. F * ( () Represents the evolved performance model.

| Acronyms or Symbols | Description | | :---: | :---: | | DT | Digital twin | | DMW | Discrete manufacturing workshop | | DT4M | Digital twin model migration modeling method | | GCN | Graph convolutional network | | TCN | Temporal convolutional network | | RFID | Radio frequency identification | | UWB | Ultra wide band | | AGVs | Automated guided vehicles | | GM | Geometry model | | BM | Behavior model | | LM | Logic model | | PM | Performance model | | WIP | Work in process | | $P W_{\text {need }}$ | Represents the DT modeling requirement set (domain) of physical workshop. | | $V_{\text {P }} \boldsymbol{W}_{\text {r }}$ | Represents the DT model set (domain). | | $\mathbf{I N}_{\text {pw↔vw }}$ | Represents the interaction matrix between $\boldsymbol{P} \boldsymbol{W}_{\text {need }}$ and $\boldsymbol{V} \boldsymbol{W}_{p r}$. | | G | Represents knowledge graph. | | in $i_{p w \leftrightarrow \text { pred }}^{\text {prict }}$ | Represents the probability of selecting $v w_{p r}$ for migration modeling based on the modeling requirement $p w_{\text {need }}$. | | $f_{\text {predict }}$ (.) | Represents the recommendation model. | | $\omega$ | Represents parameters to be trained for the prediction model. | | $p w_{\text {need }}^{i}$ | Represents the $i$-th modeling requirement. | | $v w_{p r}^{j}$ | Represents the $j$-th DT model. | | AM | Represents the adjacency matrix. | | DM | Represents the degree matrix. | | $\mathbf{P W}_{\text {M }}$ | Represents the graph model of physical workshop | | V | Represents the node set. | | E | Represents the edge set. | | L | Represents the correlation function. | | M | Represents the workstations. | | O | Represents the manufacturing orders, and represents processes in the workshop. | | W | Represents the work in process. | | $M_{P}$ | Represents the processes in the workshop. | | $d\left(v_{n^{\prime}}\right)$ | Represents the attribute information stored by the nodev ${ }_{\mathrm{n}}$. | | £ | Represents the spatial characteristic of physical workshop. | | £ $^{*}$ | Represents the spatial characteristic of virtual workshop. | | $\Delta_{P W \leftrightarrow V W}^{S F}$ | Represents the proportion of spatial characteristic consistency number of physical workshop and virtual workshop in the total. | | da $_{s m}(k)$ | Represents manufacturing information of the $m$-th station at time $k$. | | $l_{s m}^{h}(k)$ | Represents the position of the $h$-th manufacturing element at time $k$. | | ${ }^{\text {sm }}$ h $(k)$ | Represents the serial number of the $h$-th manufacturing element. | | $\operatorname{con}_{s m}^{h}(k)$ | Represents the production state of the $h$-th manufacturing element. | | $\lambda$ | Represents the temporal characteristic of physical workshop. | | $\lambda^{*}$ | Represents the temporal characteristic of virtual workshop. | | $\Delta_{P W \hookleftarrow V W}^{T F}$ | Represents the proportion of temporal characteristic consistency number of physical workshop and virtual workshop in the total. | | TA | Represents the test accuracy of performance model at time $t$. | | $\widehat{y}_{\mathrm{i} *, \text { t }}$ | Represents the actual value of the $i \psi$-th performance index at time $t$. | | $\mathrm{f}\left(\mathbf{x}^{\#} ; \mathrm{i}^{\text {\# } ; \text { t })}\right.$ | Represents the predicted value of the $i \sharp$-th performance index at time $t$. | | $f_{\text {TD }}$ (.) | Represents the trend detection function. | | $\Delta^{\Delta^{\text {* }}}$ | Represents the value of trend detection function $f_{T D}$ (.). | | $\alpha_{\text {u }}$ | Represents the weight coefficient of the $u$-th base learner. | | $\varepsilon_{u}$ | Represents the accuracy error of the $u$-th base learner. | | F * ( $\mathbf{(})$ | Represents the evolved performance model. |

parameters to be trained for the prediction model

f_{predict} (.) . i_{p w \leftrightarrow vw}^{predict}

can be expressed as the dot product

(d p = sigmoid (p w_{need} \otimes v w_{p r}))

p w_{need}

and

v w_{p r}

, while

\otimes

represents the dot product of the vector. The interaction between

p w_{n e e d}

and label of

v w_{p r}

is the actual migration, that is, when

i n_{p w, v w} = 1

, it indicates

v w_{p r}

is selected for migration based on

p w_{need,}

, and while

i n_{p w, v w} = 0

, indicating

v w_{p r}

is not selected. The
预测模型

f_{predict} (.) . i_{p w \leftrightarrow vw}^{predict}

的待训练参数可以表示为

p w_{need}

和

v w_{p r}

的点积

(d p = sigmoid (p w_{need} \otimes v w_{p r}))

，而

\otimes

表示向量的点积。

p w_{n e e d}

与

v w_{p r}

的标签之间的相互作用是实际迁移，即当

i n_{p w, v w} = 1

时，表示根据

p w_{need,}

选择

v w_{p r}

进行迁移，而

i n_{p w, v w} = 0

时，表示不选择

v w_{p r}

。在
framework of DT4M is shown in Fig. 2.
DT4M 的框架如图 2 所示。
The DT4M includes two aspects. Firstly, the modeling requirements of geometry model (GM), behavior model (BM), logic model (LM) and performance model (PM) of physical workshop need to be represented, and the DT model existing in warehouse also needs to be represented. Secondly, a recommendation model needs to be established, which is used to calculate and recommend best model from warehouse for migration. Accordingly, there emerge two difficulties: (1) the representation of modeling requirements and existing DT model, and (2) the establishment of a recommendation model

f_{predict} (

) .

DT4M 包括两个方面。首先，需要表示物理车间的几何模型（GM）、行为模型（BM）、逻辑模型（LM）和性能模型（PM）的建模要求，还需要表示仓库中现有的 DT 模型。其次，需要建立一个推荐模型，用于计算和推荐仓库中的最佳迁移模型。因此，出现了两个难点：(1) 建模需求和现有 DT 模型的表示；(2)

f_{predict} (

推荐模型的建立。

) .

(1) The modeling requirements of physical workshop cover four aspects GM, BM, LM and PM. Let the

i

-th modeling requirement be represented as

p w_{need}^{i} = {g m_{need}^{i}, b m_{need}^{i}, l m_{need}^{i}, p m_{need}^{i}}

and let the

j

-th DT model be represented as

v w_{p r}^{j} = {g m_{p r}^{j}, b m_{p r}^{j}, l m_{p r}^{j}, p m_{p r}^{j}}

. There is a coupling correlation between the GM, BM, LM and PM, which is manifested by the fact that the accuracy of the PM is affected by the accuracy of BM and LM. For example, when selecting in

v w_{p r}^{1} = {g m_{p r}^{1} (0.2)

b m_{p r}^{1} (0.3), l m_{p r}^{1} (0.3), p m_{p r}^{1} (0.2)}

and

v w_{p r}^{2} = {g m_{p r}^{2} (0.2), b m_{p r}^{2} (0.2)

{lm}_{p r}^{2} (0.2), p m_{p r}^{2} (0.4)}

, if focusing on selecting

b m_{p r}^{2} (0.2), 0.2

represents the importance at the same group of model, then the associated

g m_{p r}^{2} (0.2), l m_{p r}^{2} (0.2), p m_{p r}^{2} (0.4)

is selected together, that is,

v w_{p r}^{2}

, and when focusing on selecting

b m_{p r}^{2} (0.3)

, then the associated

g m_{p r}^{2} (0.2)

l m_{p r}^{2} (0.3), p m_{p r}^{2} (0.2)

is also selected together, that is,

v w_{p r}^{1}

. Therefore, we use the attention mechanism to capture the interaction/preference relation between the modeling requirements and the selected DT model, so, the

j

-th DT model can be expressed as

v w_{p r}^{j} =

Attention

(g m_{p r}^{j} \oplus b m_{p r}^{j} \oplus l m_{p r}^{j} \oplus p m_{p r}^{j})

.
(1) 物理车间的建模要求包括 GM、BM、LM 和 PM 四个方面。假设第

i

个建模需求用

p w_{need}^{i} = {g m_{need}^{i}, b m_{need}^{i}, l m_{need}^{i}, p m_{need}^{i}}

表示，假设第

j

个 DT 模型用

v w_{p r}^{j} = {g m_{p r}^{j}, b m_{p r}^{j}, l m_{p r}^{j}, p m_{p r}^{j}}

表示。GM、BM、LM 和 PM 之间存在耦合关联，表现为 PM 的精度受 BM 和 LM 的精度影响。例如，在

v w_{p r}^{1} = {g m_{p r}^{1} (0.2)

、

b m_{p r}^{1} (0.3), l m_{p r}^{1} (0.3), p m_{p r}^{1} (0.2)}

和

v w_{p r}^{2} = {g m_{p r}^{2} (0.2), b m_{p r}^{2} (0.2)

、

{lm}_{p r}^{2} (0.2), p m_{p r}^{2} (0.4)}

中选择时，如果重点选择

b m_{p r}^{2} (0.2), 0.2

代表同一组模型的重要性、则相关的

g m_{p r}^{2} (0.2), l m_{p r}^{2} (0.2), p m_{p r}^{2} (0.4)

也会一起被选中，即

v w_{p r}^{2}

，而当重点选择

b m_{p r}^{2} (0.3)

时，相关的

g m_{p r}^{2} (0.2)

l m_{p r}^{2} (0.3), p m_{p r}^{2} (0.2)

也会一起被选中，即

v w_{p r}^{1}

。因此，我们使用关注机制来捕捉建模需求与所选 DT 模型之间的交互/偏好关系，所以第

j

个 DT 模型可以表示为

v w_{p r}^{j} =

关注

(g m_{p r}^{j} \oplus b m_{p r}^{j} \oplus l m_{p r}^{j} \oplus p m_{p r}^{j})

。
(2) The establishment of recommendation model

f_{predict} (

) . T h e p r o -

cess of establishing

f_{predict} (

) i s t o t r a i n t h e p a r a m e t e r s ω

according to the known parameters of

p w_{need}, v w_{p r}

and

I N_{p w \leftrightarrow v w}

, which is shown in Fig. 3.
(2) 建立推荐模式

f_{predict} (

.根据已知的

p w_{need}, v w_{p r}

和

I N_{p w \leftrightarrow v w}

参数，建立

f_{predict} (

) i s t o t r a i n t h e p a r a m e t e r s ω

，如图 3 所示。

Transformer as a special case of Graph Neural Network is used to learn the preference information between modeling requirements and DT model migration, and to establish a recommendation model

f_{predict} (p w_{need}, v w_{p r}, ω, G)

. When new modeling requirements appear in physical workshop, DT models are pushed from model warehouse through model

f_{predict (.)}

for migration. The selection criterion of DT model from the existing warehouse is to comprehensively calculate the transferability of each candidate model from four aspects (GM, BM, LM and PM ) according to the modeling requirements.
变换器作为图神经网络的特例，用于学习建模需求与 DT 模型迁移之间的偏好信息，并建立推荐模型

f_{predict} (p w_{need}, v w_{p r}, ω, G)

。当物理车间出现新的建模需求时，通过模型

f_{predict (.)}

从模型仓库推送 DT 模型进行迁移。从现有仓库中选择 DT 模型的标准是根据建模需求，从 GM、BM、LM 和 PM 四个方面综合计算每个候选模型的可迁移性。

4.2. GCN-TCN based digital twin model verification
4.2.基于 GCN-TCN 的数字孪生模型验证

The DT model verification is to investigate whether the virtual workshop model can accurately represent the physical workshop system in the sense of modeling purpose. DT emphasizes the consistency with virtual workshop of physical workshop in spatio-temporal characteristics. The production process of physical workshop exhibits spatiotemporal correlations, as manufacturing elements change with time and spatial position. Moreover, the coupling states between BM, LM and PM in the production process are closely related. For example, when equipment fails, its real-time response and behavioral characteristics will change, and when the distribution path is congested, the congestion effect will spread to neighboring paths, resulting in changes in the distribution logic of the production process. Therefore, on the basis of the modeling DT model of DMW, DT model is verified by the spatial consistency and temporal consistency, which can be described from manufacturing elements in the production process of physical and virtual workshop. The framework of verification of DT model is shown in Fig. 4.
DT 模型验证是研究虚拟车间模型在建模目的意义上是否能准确代表物理车间系统。DT 强调虚拟车间与物理车间在时空特征上的一致性。物理车间的生产过程具有时空相关性，生产要素随着时间和空间位置的变化而变化。此外，生产过程中 BM、LM 和 PM 之间的耦合状态也密切相关。例如，当设备发生故障时，其实时响应和行为特征会发生变化；当配送路径拥塞时，拥塞效应会扩散到相邻路径，导致生产流程的配送逻辑发生变化。因此，在对 DMW 的 DT 模型进行建模的基础上，通过空间一致性和时间一致性对 DT 模型进行验证，空间一致性和时间一致性可以从物理车间和虚拟车间生产过程中的制造要素进行描述。DT 模型验证框架如图 4 所示。

Spatial characteristics consistency verification. By establishing the production process graph models of physical and virtual workshop, the spatial characteristics consistency verification is converted into
空间特征一致性验证。通过建立实体车间和虚拟车间的生产流程图模型，将空间特征一致性验证转化为

Fig. 2. The framework of DT model migration modeling method.
图 2.DT 模型迁移建模方法框架。

Fig. 3. Process of establishing recommendation model

f_{predict}

.
图 3.建立推荐模型

f_{predict}

的过程。

Fig. 4. The framework of verification of DT model.
图 4.DT 模型的验证框架。
the spatial characteristic calculation between the graph models. So, the spatial characteristic verification process of DT model is shown in Fig. 5.
图形模型之间的空间特征计算。因此，DT 模型的空间特征验证过程如图 5 所示。
“(1)” represents the graph model is established by using the production process data of physical and virtual workshop. The “(2)” represents the adjacency matrix (AM) and the degree matrix (DM) are established by the graph model. The “(3)” represents the Graph Convolutional Network (GCN) model is trained and migrated based on the actual production process data of physical workshop. The “(4)” represents performing spatial characteristic mining by GCN on physical and virtual workshop. The “(5)” represents evaluating the spatial characteristics consistency of physical and virtual workshop. The five processes are described in detail as below.
"(1) "表示利用物理车间和虚拟车间的生产过程数据建立的图模型。(2) "表示通过图模型建立邻接矩阵（AM）和度矩阵（DM）。(3) "表示根据物理车间的实际生产流程数据训练和迁移图卷积网络（GCN）模型。(4) "表示通过 GCN 对物理车间和虚拟车间进行空间特征挖掘。(5) "表示对物理车间和虚拟车间的空间特征一致性进行评估。这五个过程的详细描述如下。
(1)Let

{P W}_{M} = {V, E, L}

is the graph model of physical workshop,

V (P W) = {v_{1}, v_{2}, \dots, v_{n^{'}}}

is the node set,

E (P W)

is the edge set.

L (P W)

is the correlation function, which represents the mapping between

E (P W)

and

V (P W)

. The graph model establishes the connection relations
(1）设

{P W}_{M} = {V, E, L}

为物理车间图模型，

V (P W) = {v_{1}, v_{2}, \dots, v_{n^{'}}}

为节点集，

E (P W)

为边集。

L (P W)

是相关函数，表示

E (P W)

和

V (P W)

之间的映射关系。图模型建立了连接关系
between the manufacturing elements in the workshop, containing two core elements, nodes and edges. The nodes represent manufacturing elements, and the edges represent the connection and circulation relation between nodes in production process. Therefore, when establishing the production process graph model of physical workshop, the production process is decomposed into the combination of “order-product-process-equipment”, that is, the order completes the processing of specific processes on equipment (e.g. machine tool) according to the technological process, and thus the manufacturing circulation process is formed in the space. In order to facilitate the description, the definition is presented below.
车间制造元素之间的关系，包含节点和边两个核心元素。节点代表生产要素，边代表生产过程中节点之间的连接和循环关系。因此，在建立物理车间生产过程图模型时，将生产过程分解为 "订单-产品-工序-设备 "的组合，即订单按照工艺流程在设备（如机床）上完成具体工序的加工，从而在空间上形成制造循环过程。为便于说明，现将定义介绍如下。

Definition 3. The production process graph model of physical workshop is a model using equipment, orders, products and processes as nodes, and describing the production process by using real-time correlation state between equipment, orders, products, and processes as edges.
定义 3.物理车间的生产流程图模型是以设备、订单、产品和工序为节点，以设备、订单、产品和工序之间的实时关联状态为边来描述生产流程的模型。

The graph model of physical workshop is established as follows:
物理工作室的图模型建立如下：

Spatial characteristic of virtual workshop
虚拟研讨会的空间特征
Fig. 5. GCN based spatial characteristic verification process of DT model.
图 5.基于 GCN 的 DT 模型空间特征验证过程。

V = {M, O, W, M_{P}}

E = {M ⊙ O, M ⊙ W, \dots, W ⊙ M_{P}}

Where,

M = {m_{1}, m_{2}, \dots, m_{n^{'}}}

represents workstations,

O = {o_{1}, o_{2}

\dots, o_{k^{'}}}

represents manufacturing orders,

W = {w_{1}, w_{2}, \dots, w_{r^{'}}}

represents work in process (WIP), and

M_{P} = {m_{p 1}, m_{p 2}, \dots, m_{p s^{'}}}

represents processes in the workshop. Similarly, the production process graph model of virtual workshop can be established by using the same method.
其中，

M = {m_{1}, m_{2}, \dots, m_{n^{'}}}

表示工作站，

O = {o_{1}, o_{2}

，

\dots, o_{k^{'}}}

表示生产订单，

W = {w_{1}, w_{2}, \dots, w_{r^{'}}}

表示在制品（WIP），

M_{P} = {m_{p 1}, m_{p 2}, \dots, m_{p s^{'}}}

表示车间内的工序。同样，也可以用同样的方法建立虚拟车间的生产流程图模型。
(2) The

A M

and

D M

can be established by the production process graph model of physical workshop, which can be expressed as follows:
(2）

A M

和

D M

可以通过物理车间的生产流程图模型建立，其表达式如下：

A M (P W) = [\begin{array}{cccc} x_{11} & x_{12} & \dots & x_{1 n^{'}} \\ x_{21} & x_{22} & \dots & x_{2 n^{'}} \\ ⋮ & ⋮ & x_{i i^{'}} & ⋮ \\ x_{n^{'} 1} & x_{n^{'} 2} & \dots & x_{n^{'} n^{'}} \end{array}]

D M (P W) = [\begin{array}{cccc} d (v_{1}) & 0 & \dots & 0 \\ 0 & d (v_{2}) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & x_{n^{'} 2} & \dots & d (v_{n^{'}}) \end{array}]

Where,

x_{{ij}^{'}}

represents whether the node

v_{i^{'}}

of physical workshop has a directed edge pointing to

v_{j^{'}}

. If there is a directed edge of

v_{i^{'}}

pointing to

v_{j^{'}}

, then

x_{i^{'} j^{'}} = 1

, otherwise

x_{i^{'} j^{'}} = 0

. Similarly,

x_{{ji}^{'}}

represents whether the node

v_{j^{'}}

of physical workshop has a directed edge pointing to

v_{i^{'}} \cdot d (v_{n^{'}})

represents the attribute information stored by the node

v_{n}

.
其中，

x_{{ij}^{'}}

表示物理车间的节点

v_{i^{'}}

是否有指向

v_{j^{'}}

的有向边。如果

v_{i^{'}}

有指向

v_{j^{'}}

的有向边，则表示

x_{i^{'} j^{'}} = 1

，否则表示

x_{i^{'} j^{'}} = 0

。同样，

x_{{ji}^{'}}

表示物理车间的节点

v_{j^{'}}

是否有指向

v_{i^{'}} \cdot d (v_{n^{'}})

的有向边，

v_{i^{'}} \cdot d (v_{n^{'}})

表示节点

v_{n}

存储的属性信息。
(3) GCN [35] is considered as an effective method for graph data feature extraction, which can fully mine the spatial characteristics existing in the production process. The GCN is trained by using the actual data of physical workshop, and then the parameter matrix

W (P W), A M (P W)

and

D M (P W)

are migrated, so that it makes the spatial characteristic consistency evaluation model of virtual and physical workshop identical.
(3) GCN[35]被认为是图形数据特征提取的有效方法，可以充分挖掘生产过程中存在的空间特征。利用物理车间的实际数据对 GCN 进行训练，然后对参数矩阵

W (P W), A M (P W)

和

D M (P W)

进行迁移，使虚拟车间和物理车间的空间特征一致性评价模型完全一致。
(4)The spatial characteristic mining of production process data from physical and virtual workshop is carried out respectively based on GCN, and the formulas can be expressed as follows:
(4)基于 GCN 分别对物理车间和虚拟车间的生产过程数据进行空间特征挖掘，其公式可表示如下：

£ = f_{G C N} (D a (P W))

£^{*} = f_{G C} (D a (V W))

(5) The consistency of

£

and

£^{*}

is evaluated, which can be expressed as follows:
(5) 评估

£

和

£^{*}

的一致性，可表示如下：

Δ_{P W \leftrightarrow V W}^{S F} = \frac{\sum (£^{Δ} £^{*})}{n^{'}}

Δ_{P W \leftrightarrow V W}^{S F}

represents the proportion of spatial characteristic consistency number of physical and virtual workshop in the total.

£ ≜ £^{*}

represents the absolute value of

£

minus

£

. For example, when the three spatial characteristics of physical and virtual workshop are [0.052,

0.038, 0.022

] and [0.005, 0.018, 0.102], respectively. If set

£ ≜ £^{*} \leq

0.05 as threshold, then the third characteristics cannot be considered to meet the requirements, because of

| 0.022 - 0.102 | = 0.08 > 0.05

, so

Δ_{P W \leftrightarrow V W}^{S F} = (3 - 1) / 3 \approx 0.667

Δ_{P W \leftrightarrow V W}^{S F}

表示实体车间和虚拟车间的空间特性一致性数量在总数中所占的比例。

£ ≜ £^{*}

表示

£

减去

£

的绝对值。例如，当物理车间和虚拟车间的三个空间特性分别为 [0.052,

0.038, 0.022

] 和 [0.005, 0.018, 0.102]时。如果设置

£ ≜ £^{*} \leq

0.05 为阈值，则不能认为第三个特征满足要求，因为

| 0.022 - 0.102 | = 0.08 > 0.05

，所以

Δ_{P W \leftrightarrow V W}^{S F} = (3 - 1) / 3 \approx 0.667

。
2) Temporal characteristics consistency verification. As the production process is endowed with time sequence and strong continuity, in addition to the spatial characteristics, it is necessary to evaluate the temporal characteristics consistency of physical and virtual workshop. The temporal characteristics verification process of DT model is shown in Fig. 6.
2) 时间特征一致性验证。由于生产过程具有时序性和较强的连续性，除了空间特征外，还需要对物理车间和虚拟车间的时间特征一致性进行评估。DT 模型的时间特性验证过程如图 6 所示。
“(1)” represents the organization of the production process data of physical and virtual workshop, “(2)” represents Temporal Convolutional Network (TCN) [36] model is trained and migrated based on the actual production process data of physical workshop, “(3)” represents performing temporal characteristic mining by TCN on physical and virtual workshop, and “(4)” represents evaluating the temporal characteristics consistency of physical and virtual workshop. The four processes are described in detail as below.
"(1) "表示对物理车间和虚拟车间的生产过程数据进行整理，"(2) "表示根据物理车间的实际生产过程数据对时态卷积网络（TCN）[36] 模型进行训练和迁移，"(3) "表示通过 TCN 对物理车间和虚拟车间进行时态特征挖掘，"(4) "表示对物理车间和虚拟车间的时态特征一致性进行评估。这四个过程的详细描述如下。
(1) According to the time series combination, the production process data of physical workshop can be described as follows:
(1）根据时间序列组合，物理车间的生产过程数据可描述如下：

D a (k) = {{d a}_{s 1} (k), {d a}_{s 2} (k), \dots, {d a}_{s m} (k), \dots, {d a}_{s M} (k)}

{d a}_{s m} (k) = {{d a}_{s m}^{1} (k), {d a}_{s m}^{2} (k), \dots, {d a}_{s m}^{h} (k), \dots, {d a}_{s m}^{H} (k)}

Where,

M

represents the number of workstations,

{d a}_{s m} (k)

represents
其中，

M

表示工作站数量，

{d a}_{s m} (k)

表示

Fig. 6. TCN based temporal characteristics verification process of DT model.
图 6.基于 TCN 的 DT 模型时间特性验证过程。
manufacturing information of the

m

-th station at time

k, H

represents serial number of manufacturing elements (e.g. equipment, tooling, materials, WIP, etc.) of the

m

-th station, and

{d a}_{s m}^{h} (k)

represents the information of the

h

-th manufacturing element at time

k

, which can be expressed as follows:

m

-th工位在

k, H

时间的制造信息代表

m

-th工位的制造要素（如设备、工装、材料、WIP等）序列号，

{d a}_{s m}^{h} (k)

代表

h

-th工位在

k

时间的制造要素信息，可表示如下：

{d a}_{s m}^{h} (k) = {k, l_{s m}^{h} (k), I D_{s m}^{h} (k), {con}_{s m}^{h} (k)}

Where,

l_{s m}^{h} (k)

represents the position of the

h

-th manufacturing element at time

k

, and

I_{s m}^{h} (k)

represents the serial number of the

h

-th manufacturing element;

{con}_{s m}^{h} (k)

represents the production state of the

h

-th manufacturing element (e.g. “waiting”, “processing”, “completed” of WIP, etc.). According to the time series combination, the data link can be expressed as follows [37]:
其中，

l_{s m}^{h} (k)

表示第

h

个制造元件在第

k

个时间点的位置，

I_{s m}^{h} (k)

表示第

h

个制造元件的序列号；

{con}_{s m}^{h} (k)

表示第

h

个制造元件的生产状态（如 "等待"、"加工"、WIP的 "完成 "等）。根据时间序列组合，数据链路可表示如下[37]：

{D a}_{s m}^{h} = [\begin{array}{cccc} k_{1} & l_{s m}^{h} (k_{1}) & I D_{s m}^{h} (k_{1}) & c o n s m_{h}^{(k + 1)} \\ k_{2} & l_{s m}^{h} (k_{2}) & I D_{s m}^{h} (k_{2}) & c o n_{s m}^{h} (k_{2}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ k_{i} & l_{s m}^{h} (k_{i}) & I D_{s m}^{h} (k_{i}) & c o n_{s m}^{h} (k_{i}) \\ ⋮ & ⋮ & ⋮ & ⋮ \end{array}]

Where,

W_{s m}^{h}

represents the data link of the

h

-th manufacturing element of the

m

-th workstation.
其中，

W_{s m}^{h}

表示

h

-th 工作站的

m

-th 制造元件的数据链路。
(2) TCN is considered as an effective method for time series prediction, which can fully mine the temporal characteristics existing in the production process data. The TCN is trained by using

D a (k) = {{d a}_{s 1} (k)

{d a}_{s 2} (k), \dots, {d a}_{s m} (k), \dots, {d a}_{s M} (k)}

, which is the actual data of physical workshop, and the network structure and parameters are migrated, so that it makes the temporal characteristic consistency evaluation model of virtual and physical workshop identical.
(2) TCN 被认为是一种有效的时间序列预测方法，可以充分挖掘生产过程数据中存在的时间特征。利用实体车间的实际数据

D a (k) = {{d a}_{s 1} (k)

、

{d a}_{s 2} (k), \dots, {d a}_{s m} (k), \dots, {d a}_{s M} (k)}

对 TCN 进行训练，并对网络结构和参数进行迁移，使虚拟车间和实体车间的时间特性一致性评价模型完全一致。
(3) The temporal characteristic mining of production process data from physical and virtual workshop is carried out respectively based on TCN , and the formulas can be expressed as follows:
(3) 基于 TCN，分别对物理车间和虚拟车间的生产过程数据进行时间特征挖掘，其公式可表示如下：

λ = f_{T}_{N} (D a (P W))

λ^{*} = f_{T}_{N} (D a (V W))

(4) The consistency of

λ

and

λ^{*}

is evaluated, which can be expressed as follows:
(4) 评估

λ

和

λ^{*}

的一致性，可表示如下：

Δ_{P W ↛ V W}^{T F} = \frac{\sum (λ ≜ λ^{*})}{n^{'}}

Δ_{P W \leftrightarrow V W}^{T F}

represents the proportion of temporal characteristic consistency number of physical and virtual workshop in the total. For example, when the three temporal characteristics of physical and virtual workshop are

[0.788, 0.350, 0.784]

and

[0.732, 0.347, 0.737]

, respectively. If set

λ ≜ λ^{*} \leq 0.1

as threshold, then all of this three characteristics can be considered to meet the requirements, because of

| 0.788 - 0.732 | = 0.056 < 0.1, | 0.350 - 0.347 | = 0.003 < 0.1

and

| 0.784 - 0.737 | = 0.047 < 0.1

, so

Δ_{P W \leftrightarrow V W}^{T F} = 3 / 3 = 1

Δ_{P W \leftrightarrow V W}^{T F}

表示物理车间和虚拟车间的时间特性一致性数量在总数中所占的比例。例如，当物理车间和虚拟车间的三个时间特征分别为

[0.788, 0.350, 0.784]

和

[0.732, 0.347, 0.737]

时。如果设置

λ ≜ λ^{*} \leq 0.1

为阈值，那么这三个特征都可以认为是满足要求的，因为有

| 0.788 - 0.732 | = 0.056 < 0.1, | 0.350 - 0.347 | = 0.003 < 0.1

和

| 0.784 - 0.737 | = 0.047 < 0.1

，所以

Δ_{P W \leftrightarrow V W}^{T F} = 3 / 3 = 1

。

Since the virtual workshop should faithfully map the physical workshop, the consistency of spatio-temporal characteristics need to be verified simultaneously when performing model verification. That is,

Δ_{P W ↩ V W}^{S F}

and

Δ_{P W ↩ V W}^{T F}

need to be satisfied at the same time to pass the verification.
由于虚拟车间应忠实映射物理车间，因此在进行模型验证时，需要同时验证时空特征的一致性。也就是说，

Δ_{P W ↩ V W}^{S F}

和

Δ_{P W ↩ V W}^{T F}

需要同时满足才能通过验证。

4.3. Digital twin model synchronous evolution
4.3.数字孪生模型同步演化

As the performance of manufacturing elements (e.g. equipment) in physical workshop declines in its life cycle, the production process performance of physical workshop will change. If DT model does not evolve synchronously in time, it will lead to a large deviation of DT model prediction results. Therefore, DT model needs to evolve synchronously according to the state of physical workshop. Synchronous
随着物理车间内制造元件（如设备）在其生命周期内性能的下降，物理车间的生产过程性能也会发生变化。如果 DT 模型不能及时同步演化，就会导致 DT 模型预测结果出现较大偏差。因此，DT 模型需要根据物理车间的状态同步演化。同步
evolution needs to solve two problems: when to evolve and how to evolve. To solve this problem, a DT model synchronous evolution method based on Adaboost is proposed to ensure the synchronization of virtual and physical workshop. The framework is shown in Fig. 7.
进化需要解决两个问题：何时进化和如何进化。为了解决这个问题，我们提出了一种基于 Adaboost 的 DT 模型同步进化方法，以确保虚拟车间和物理车间的同步。框架如图 7 所示。

The synchronous evolution of DT model includes two main parts: “(1)” the accuracy trend detection of DT model and “(2)” the synchronous update of PM based on Adaboost, which can be described in detail as below.
DT 模型的同步演化包括两个主要部分："(1) "DT 模型的精确度趋势检测和"(2) "基于 Adaboost 的 PM 同步更新，具体描述如下。

The evolution in this paper refers to the synchronous updating of PM in DT model, denoted as

{DTM}_{P M}

. The

{DTM}_{P M}

is an intelligent analysis model based on operation rules of workshop, including production progress prediction model, energy consumption model and production capacity model etc., which is the core of DT model. The

{DTM}_{P M}

is defined as follows:
本文中的演化是指 DT 模型中 PM 的同步更新，用

{DTM}_{P M}

表示。

{DTM}_{P M}

是基于车间运行规律的智能分析模型，包括生产进度预测模型、能耗模型、产能模型等，是 DT 模型的核心。

{DTM}_{P M}

的定义如下：

D {TM}_{P M} = (D_{T M}_{M M}, D T M_{D D M})

{DTM}_{M M}

represents mechanism model and

{DTM}_{D D M}

represents datadriven model. The two processes of

{DTM}_{P M}

synchronous updating are described in detail as below.

{DTM}_{M M}

表示机制模型，

{DTM}_{D D M}

表示数据驱动模型。下面将详细介绍

{DTM}_{P M}

同步更新的两个过程。

The accuracy trend detection of $D T M_{P M}$ . The inaccuracy of the model and parameters is considered to be the main factor for the obvious deviation between the simulation results and the actual system dynamic behavior. There is a coupling correlation between BM, LM and PM , which is manifested by the fact that the accuracy of PM is affected by the accuracy of BM, and LM. For example, if BM or LM is not accurate, then PM established on this basis may not accurately track the operation laws of physical workshop. Thus, the consistency of change trend of performance index is used to detect the consistency of physical laws between physical entity evolution and that contained in ${DTM}_{P M}$ , which is used as a generator to trigger the evolution of ${DTM}_{P M}$ .
$D T M_{P M}$ 的精度趋势检测。模型和参数的不准确性被认为是仿真结果与实际系统动态行为之间存在明显偏差的主要因素。BM、LM 和 PM 之间存在耦合相关性，表现为 PM 的精度受 BM 和 LM 精度的影响。例如，如果 BM 或 LM 不准确，那么在此基础上建立的 PM 可能无法准确跟踪物理车间的运行规律。因此，利用性能指标变化趋势的一致性来检测物理实体演化与 ${DTM}_{P M}$ 中包含的物理规律的一致性，并以此作为触发 ${DTM}_{P M}$ 演化的发生器。
Given $T A_{t}$ is the test accuracy of ${DTM}_{P M}$ at time $t$ , the $T A_{t}$ calculation can be expressed as follows:
假设 $T A_{t}$ 是 ${DTM}_{P M}$ 在 $t$ 时间的测试精度，则 $T A_{t}$ 的计算可表示如下：
$T A_{t}^{i^{*}} = 1 - \frac{| {\hat{y}}_{i \neq t} - f (x^{#}; i^{#}; t) |}{{\hat{y}}_{i^{#}, t}}$
${\hat{y}}_{i}^{*}, andf (x^{#}; i^{#}; t)$ indicate the actual value of the $i^{i^{#} -th}$ performance index at time $t$ and the predicted value respectively. When the test accuracy at time $t + 1$ is $T A_{t + 1}^{i}$ , the change of the test accuracy of ${DTM}_{P M}$ can be expressed as follows:
${\hat{y}}_{i}^{*}, andf (x^{#}; i^{#}; t)$ 分别表示 $i^{i^{#} -th}$ 性能指标在时间 $t$ 时的实际值和预测值。当时间 $t + 1$ 的测试精度为 $T A_{t + 1}^{i}$ 时， ${DTM}_{P M}$ 的测试精度变化可表示如下：
$Δ T A_{t, t + 1}^{i #} = T A_{t + 1}^{i #} - T A_{t}^{i #}$

Then, the change of test accuracy of the

i^{#}

-th performance index of

{DTM}_{P M}

complete can be expressed as follows:
那么，

{DTM}_{P M}

完整的

i^{#}

-th 性能指标的测试精度变化可表示如下：

Δ {T A}^{i #} = {Δ T A_{t}^{i #}}_{t = 1}^{T}

By fitting the change in

Δ {T A}^{i^{*}}

, the trend changes of

{DTM}_{P M}

that is obtained can be expressed as follows:
通过拟合

Δ {T A}^{i^{*}}

的变化，可以得到

{DTM}_{P M}

的趋势变化如下：

Δ^{i^{*}} = f_{T D} {{T A}^{i^{#}}}

Where,

f_{T D}

(.) represents trend detection function, such as linear fitting, Non-linear fitting. And it is mainly used to reveal the trend change of

{DTM}_{P M}

. When the accuracy of

{DTM}_{P M}

is decreasing with time, the update mechanism of

{DTM}_{P M}

is triggered.
其中，

f_{T D}

（.）表示趋势检测函数，如线性拟合、非线性拟合。它主要用于揭示

{DTM}_{P M}

的趋势变化。当

{DTM}_{P M}

的精度随时间下降时，就会触发

{DTM}_{P M}

的更新机制。
2) The synchronous evolution of

D T M_{P M}

. The performance index with performance degradation is calculated from the

{DMW}_{P M}

(e.g. production progress prediction model

F (X)

, production capacity prediction model

H (X)

and energy consumption prediction model

G (X)

)
2)

D T M_{P M}

的同步演变。由

{DMW}_{P M}

计算出性能下降的性能指标（如生产进度预测模型

F (X)

、生产能力预测模型

H (X)

和能耗预测模型

G (X)

）。

Fig. 7. Synchronous evolution of DT model in DMW.
图 7.DT 模型在 DMW 中的同步演化。
by using formula (17)-(20), and then selecting model needs to evolve synchronously from

{DMW}_{P M}

.
式 (17)-(20)，然后选择模型需要从

{DMW}_{P M}

同步演化。

The

{DMW}_{P M}

(e.g.

F (X)

) selected is integrated as a base learner. However, an important premise here is that

{DMW}_{P M}

needs to satisfy the integrated condition before integrating by Adaboost algorithm to form a strong learner integration model. The Adaboost based integration can be expressed as follows:
选定的

{DMW}_{P M}

（例如

F (X)

）将作为基础学习器进行整合。但是，这里有一个重要的前提，即

{DMW}_{P M}

需要满足整合条件，然后再通过 Adaboost 算法进行整合，以形成一个强学习器整合模型。基于 Adaboost 的整合可表示如下：

F^{*} (x^{i^{*}}) = \sum_{u = 1}^{U} α_{u} f_{u} (x^{i^{*}})

α_{u}

represents the weight coefficient of the

u

-th base learner, which is determined by the accuracy error

ε_{u}

of base learner (e.g.

F (X)

) in the data set.

α_{u}

can be calculated and expressed as follow:

α_{u}

表示第

u

个基础学习器的权重系数，它由数据集中基础学习器（例如

F (X)

）的精度误差

ε_{u}

决定。

α_{u}

的计算和表达式如下：

α_{u} = \frac{1}{2} \ln [\frac{1 - ε_{u}}{ε_{u}}]

s.t.

ε_{u} = \sum_{i^{*}}^{m *} \frac{| f_{u} (x^{i^{*}}) - y^{i^{*}} |}{sup_{i^{*}} [| f_{u} (x^{i^{*}}) - y^{i^{*}} |]}

F * (X)

represents the evolved

{DMW}_{P M}

. The parameters of

{DMW}_{P M}

are dynamically adjusted based on Adaboost algorithm by incremental data from physical workshop, which realizes the performance of virtual and physical workshop updated synchronously.

F * (X)

表示演化后的

{DMW}_{P M}

。

{DMW}_{P M}

的参数基于 Adaboost 算法，通过物理车间的增量数据进行动态调整，实现虚拟车间和物理车间的性能同步更新。

The synchronous evolution of DT model runs through the entire production process of workshop, aiming at tracking the performance of manufacturing elements (e.g. equipment). So, when the production process performance of physical workshop changes, DT model needs to evolve synchronously in time, so as to ensure the accuracy of prediction and regulation based on DT model.
DT 模型的同步演化贯穿车间的整个生产过程，旨在跟踪生产要素（如设备）的性能。因此，当物理车间的生产过程性能发生变化时，DT 模型需要及时同步演化，以确保基于 DT 模型的预测和调节的准确性。

5. Experiments and discussion
5.实验和讨论

The experiment of the proposed method of a typical machining workshop with 13 stations is provided in this section to validate its effectiveness, which are engaged in small structural parts processing. We analyze the effectiveness of migration modeling, model verification and model evolution in turn according to the main line.
本节将对一个拥有 13 个工位的典型加工车间进行实验，以验证所提方法的有效性，该车间从事小型结构件加工。我们按照主线依次分析了迁移建模、模型验证和模型演化的有效性。

5.1. Analysis of model modeling
5.1.模型建模分析

To demonstrate the advantages and potential of the DT model migration modeling method (DT4M), one application of the proposed modeling approach is briefly introduced in this section, including two parts. (1) The first part is to test the performance of DT4M, which contains (1) the construction of knowledge graph of manufacturing resources, and (2) the testing of recommended model. (2) The seconded part is to describe actual established process of DT model, that is, the actual fine-tuning process after DT model migration, including the detailed modeling process of GM, BM, LM and PM.
为了展示 DT 模型迁移建模方法（DT4M）的优势和潜力，本节简要介绍了所提建模方法的一个应用，包括两个部分。(1）第一部分是测试 DT4M 的性能，包括（1）制造资源知识图谱的构建；（2）推荐模型的测试。(2）第二部分是描述 DT 模型的实际建立过程，即 DT 模型迁移后的实际微调过程，包括 GM、BM、LM 和 PM 的详细建模过程。
(1) To test the performance of DT4M. (1) According to the semantic data of different manufacturing resources in DMW, the knowledge graph of workshop manufacturing resources is established as shown in Fig. 8, which mainly represents the production relationship between manufacturing resources and hides the data relationship of individual instances.
(1）检验DT4M的性能。(1）根据 DMW 中不同制造资源的语义数据，建立如图 8 所示的车间制造资源知识图谱，该图谱主要表示制造资源之间的生产关系，隐藏了单个实例的数据关系。

In this paper, the ontology is mainly applied to the formal representation, organization and management of DT workshop related data and knowledge. By adopting the ontology modeling method, the related data and domain knowledge can be modeled in a structured manner, and the semantic association between concepts, attributes and relations can be defined. The WIPs is connected to the AGV through “beTransportedby” and connected to the stations through “beProcessedin”. Besides, the “hasProcess” is used to connect stations and manufacturing processes. The “beTransportedto” is used to establish a relationship between station and AGV. The “isLocatedin” is used to connect between station and machine. And the “hasStatus” establishes relationships between the manufacturing state and machine, WIP, AGV to display the real-time production state of manufacturing resources. For example, the WIP_04 is connected to the Station_03 through “beprocessedin” and connected to the AGV_01 through “beTransportedby”, etc.
本文主要将本体应用于 DT 车间相关数据和知识的正式表示、组织和管理。通过采用本体建模方法，可以对相关数据和领域知识进行结构化建模，并定义概念、属性和关系之间的语义关联。WIPs 通过 "beTransportedby "与 AGV 连接，通过 "beProcessedin "与工位连接。此外，"hasProcess "用于连接工位和生产流程。beTransportedto "用于在站点和 AGV 之间建立关系。isLocatedin "用于连接站点和机器。而 "hasStatus "则用于建立制造状态与机器、WIP、AGV 之间的关系，以显示制造资源的实时生产状态。例如，WIP_04 通过 "beprocessedin "连接到 Station_03，通过 "beTransportedby "连接到 AGV_01，等等。
(2) To test the recommendation model

(f_{predict} () .)

, based on the above-mentioned knowledge graph of manufacturing resources of DMW, the attribute information of GM, BM, LM and PM are further described respectively, which describes the migration of DT model is shown in Table 2.
(2）为了检验推荐模型

(f_{predict} () .)

，在上述DMW制造资源知识图谱的基础上，分别对GM、BM、LM和PM的属性信息进行了进一步描述，DT模型的迁移描述如表2所示。

The DT model migration mainly involves selecting basic models from the existing warehouse, and then fine-tuning the model selected to meet
DT 模型迁移主要涉及从现有仓库中选择基本模型，然后对所选模型进行微调，以满足以下要求

Fig. 8. The knowledge graph (part) of manufacturing resources in DMW.
图 8.DMW 中制造资源的知识图谱（部分）。

Table 2 表 2
Transferable attribute information of digital twin model.
数字孪生模型的可转移属性信息。

Model type 型号	Unique attribute 独特属性	Common attribute 共同属性
GM	Assembly, Material, Scale, Mass, $\dots$ 装配, 材料, 标尺, 质量, $\dots$	Name,Id, 姓名、身份证号
BM	Part, Rule, Action, Assembly,... 部分, 规则, 动作, 装配, ...	Creator, 创作者
LM	Part, Assembly, Process,... 部件、装配、工艺、...	Position, 职位：
PM	Input, Output, Accuracy,... 输入、输出、精度...	Number, $\dots$ 编号， $\dots$

DT modeling requirements of workshop. The transferability of GM, BM, LM and PM is obtained by calculating the difference of unique attributes. The precise information of the models included in warehouse and DT modeling requirement information are shown in Table 3.
车间的 DT 建模要求。GM、BM、LM 和 PM 的可转移性是通过计算独特属性的差异得到的。仓库所含模型的精确信息和 DT 建模要求信息如表 3 所示。

The model A is required to be established and it need to select models from the DT model set (model number

B \sim F

) to migration, and then finetuning. The effectiveness of the recommendation model (

f_{predict} (

)) i s

verified by self-made data sets. The parametric examples of the experiment and the detailed parameters of

f_{predict} (

) a r e s h o w n i n T a b l e 4

and Table 5 respectively. Table 4 shows that the DT model can be divided into three levels, such as unit, line and workshop, with each level containing four types of models: GM, BM, LM and PM. The "

0.2 - 0.2 - 0.3 - {0.3}^{''}

represents the value of the four types of models in unit, line or workshop level. Similarly, the "

2 - 1 - 2

" represents the identification number of unit, line and workshop respectively.
需要建立模型 A，并从 DT 模型集中选择模型（模型编号

B \sim F

）进行迁移，然后进行微调。推荐模型（

f_{predict} (

)) i s

通过自制数据集进行验证。实验的参数示例以及

f_{predict} (

) a r e s h o w n i n T a b l e 4

和表 5 分别给出。从表 4 可以看出，DT 模型可分为单元、生产线和车间等三个层次，每个层次包含四种类型的模型：GM、BM、LM 和 PM。

0.2 - 0.2 - 0.3 - {0.3}^{''}

表示单位、生产线或车间级四类模型的值。同样，"

2 - 1 - 2

"分别代表单位、生产线和车间的标识号。

The Transformer includes network structure, number of layers, maximum number of epochs, number of heads, learning rate. With MSE as output, the convergence processes of the Transformer are recorded in Fig. 9. The

f_{predict} (

) i s t r a i n e d a c c o r d i n g t o f o r m u l a (1) .

变换器包括网络结构、层数、最大历元数、头数和学习率。以 MSE 作为输出，图 9 记录了变换器的收敛过程。

f_{predict} (

) i s t r a i n e d a c c o r d i n g t o f o r m u l a (1) .

It clearly indicates that Transformer converges to lower MSE rapidly. The testing results of overall accuracy rate from workshop level are listed in Table 6, in which each number modeling requirement is set to
这清楚地表明，Transformer 能迅速收敛到较低的 MSE。表 6 列出了车间级总体准确率的测试结果，其中每个数字建模要求设置为

Table 4 表 4
Parametric examples of the experiment.
实验参数示例

Data set 数据集

GM, BM, LM, PM (values)
GM、BM、LM、PM（数值）

型号

Model

number

Choose 选择

unit 单位

line 线条

workshop 工作坊

0.2, 0.2, 0.3,

...

0.2, 0.2, 0.3

2, 1, 2

Yes 是

0.3 ,

0.3

No 没有

0.2, 0.2, 0.2

...

0.2, 0.2, 0.2

10, 3, 10

No 没有

0.4 ,

0.4

Yes 是

0.2, 0.1, 0.3,

...

0.2, 0.1, 0.3

6, 7, 6

0.4,

0.4

0.1, 0.1, 0.4,

...

0.1, 0.1, 0.4

12, 6, 12

0.4 ,

0.4

...

Table 5 表 5
The Detailed Parameters of Transformer.
变压器的详细参数。

Parameters 参数	Value 价值
No. of layers 层数	2
Maximum No. of epochs 最大纪元数	1000
No. of heads 头数	2
Learning rate in training 培训学习率	0.0001

be different, such as "

0.1 - 0.1 - 0.1 - 0.7

", etc. When modeling requirements of GM, BM, LM and PM are set from "

0.1 - 0.1 - 0.1 - 0.7

" to "

0.1 - 0.1 - 0.3 - 0.5

" or "

0.1 - 0.3 - 0.3 - 0.3

", the DT model number is pushed from model warehouse through proposed model.
不同，如"

0.1 - 0.1 - 0.1 - 0.7

"等。当 GM、BM、LM 和 PM 的建模要求从"

0.1 - 0.1 - 0.1 - 0.7

"设置为"

0.1 - 0.1 - 0.3 - 0.5

"或"

0.1 - 0.3 - 0.3 - 0.3

"时，DT 型号将从型号仓库通过建议型号推送。

For each performance measure, " " represents the priority indicators to be considered. It clearly indicates that DT model number is pushed according to different requirement, which demonstrates

f_{predict} (

对于每个性能指标，" "代表需要考虑的优先指标。这清楚地表明，DT 型号是根据不同的要求推送的，这表明

f_{predict} (

Table 3 表 3
Model details included in the digital twin model warehouse and DT modeling requirement.
数字孪生模型仓库中包含的模型细节和 DT 建模要求。

Model number 型号	Model Type 型号	Workshop size 车间面积	Storage 存储	Composition 组成	Migration 迁移	Completion time 完成时间	Requirement 要求
A	Machining shop 加工车间	13 workstations 13 个工作站	local 当地	GM, BM, LM, PM	Need 需要	?	$\sqrt{}$
B	Machining shop 加工车间	8 workstations 8 个工作站	local 当地	GM, BM, LM, PM	No 没有	about 4 months 约 4 个月	/
C	Machining shop 加工车间	8 workstations 8 个工作站	local 当地	GM, BM, LM, PM	No 没有	about 4 months 约 4 个月	/
D	Assembly shop 装配车间	4 workstations 4 个工作站	local 当地	GM, BM, LM, PM	No 没有	about 1.5 months 约 1.5 个月	/
E	Assembly shop 装配车间	4 workstations 4 个工作站	local 当地	GM, BM, LM, PM	No 没有	about 1.5 months 约 1.5 个月	/
F	Assembly shop 装配车间	8 workstations 8 个工作站	local 本地	GM, BM, LM, PM	No 没有	about 3 months 约 3 个月	/

Fig. 9. Convergence curves of Transformer.
图 9.变压器的收敛曲线。

Table 6 表 6
The Testing Results of Transformer.
变压器的测试结果。

测试装置

Test

set

DT 型号

DT model

number

建议的模式

Proposed

model

(0.1)

(0.1)

(0.1)

(0.7)

(0.1)

(0.1)

(0.3)

(0.5)

) as a recommendation model can calculate and recommend the best model from warehouse for migration. The reason for this result may be that

f_{predict} (

) c a n c a p t u r e i n t e r a c t i o n / p r e f e r e n c e r e l a t i o n b e t w e e n t h e

modeling requirements and the selected DT model.
作为推荐模型，

f_{predict} (

可以从仓库中计算并推荐最佳模型用于迁移。造成这一结果的原因可能是

f_{predict} (

) c a n c a p t u r e i n t e r a c t i o n / p r e f e r e n c e r e l a t i o n b e t w e e n t h e

建模要求和所选的 DT 模型。
(2) To describe actual established process of GM, BM, LM and PM. The model B is selected as basic model, and model A is established on the basis of Model B after fine-tuned. The importance of model migration is

v w_{p r}^{4} = {g m_{p r}^{4} (0.1), b m_{p r}^{4} (0.1), l m_{p r}^{4} (0.3), p m_{p r}^{4} (0.5)}

, that is, the LM and PM are given priority at the same group of models when model migrating.
(2) 描述 GM、BM、LM 和 PM 的实际建立过程。选取模型 B 作为基本模型，在模型 B 的基础上进行微调后建立模型 A。模型迁移的重要性为

v w_{p r}^{4} = {g m_{p r}^{4} (0.1), b m_{p r}^{4} (0.1), l m_{p r}^{4} (0.3), p m_{p r}^{4} (0.5)}

，即模型迁移时，在同一组模型中优先考虑 LM 和 PM。

The modeling process of GM, BM, LM and PM is shown in Fig. 10.
GM、BM、LM 和 PM 的建模过程如图 10 所示。

The GM ( ${DTM}_{G M}$ ) modeling. The model B and PlantSimulation platform based, (1) For the manufacturing elements (machine tools, AGV, etc.) existing geometric model, the GM is established by importing its corresponding 3D model, light-weighting and setting animation. (2) For the manufacturing elements (tooling, part of the equipment, etc.) not existing GM, the GM is established by model migration from historical 3D model base, and then adjusting the dimensions, assembling relations, materials, etc.
GM（ ${DTM}_{G M}$ ）建模。基于模型 B 和 PlantSimulation 平台，（1）对于已有几何模型的制造元素（机床、AGV 等），通过导入其对应的三维模型、轻量化和设置动画来建立 GM。(2）对于不存在 GM 的制造要素（工装、部分设备等），通过从历史三维模型库中迁移模型，然后调整尺寸、装配关系、材料等来建立 GM。
The BM $({DTM}_{B M})$ modeling. The model B and PlantSimulation platform based, the finite state machine is used to establish BM by defining the state machine of each ME, and constructing station state machine model, transport equipment state machine model and storage state machine model, etc., and then they are used as the submodel of BM of the production line to realize their own state transfer. Further, the BM of production lines are assembled into the BM of workshop according to the logical relation between manufacturing elements.
BM $({DTM}_{B M})$ 建模。基于 B 模型和 PlantSimulation 平台，采用有限状态机建立 BM，定义各 ME 的状态机，构建工位状态机模型、运输设备状态机模型和存储状态机模型等，作为生产线 BM 的子模型，实现各自的状态转移。然后，根据制造要素之间的逻辑关系，将生产线的 BM 组装成车间的 BM。
The LM $({DTM}_{L M})$ modeling. Similarly, the model $B$ based, and the establishment of the LM model based on the physical entities meets the following conditions. First, the production information such as
LM $({DTM}_{L M})$ 建模。同样，基于 $B$ 模型，基于物理实体的 LM 模型的建立也满足以下条件。首先，生产信息如
size information, layout and technological process (warehouse picking, WIP transfer, material distribution and finished goods storage, etc.). Second, the constraint rules (picking rules, transfer rules, distribution rules, storage rules, etc.) in the production process. Third, the behavior and operation rules that occur in actual production process PlantSimulation platform based. The LM model is endowed with behavior characteristics, response mechanism and state transition.
尺寸信息、布局和工艺流程（仓库拣选、在制品转移、物料配送和成品储存等）。第二，生产过程中的约束规则（拣选规则、转移规则、配送规则、存储规则等）。第三，基于 PlantSimulation 平台的实际生产过程中的行为和操作规则。LM 模型具有行为特征、响应机制和状态转换。
The PM $({DTM}_{P M})$ modeling. Due to the variety of PMs such as production progress prediction model, energy consumption model and production capacity model etc., the PM of model A is established by migrating the parameters of model B and retraining the production progress prediction model of PM as an example based on Long shortterm memory (LSTM) is established in this paper, and the detailed process in Section 5.3. The algorithm was coded in Python, meanwhile, realizing communication with PlantSimulation platform by using Socket interface. The DT model modeling of workshop containing 13 stations need 1.5 months through DT migration, that is, the completion time “?” in Table 3 is just about 1.5 months. Compared with other non-migration modeling methods (e.g. model number B F), DT4M can save a lot of time and improve modeling efficiency.
PM $({DTM}_{P M})$ 建模。由于生产进度预测模型、能耗模型和产能模型等 PM 种类繁多，本文以基于长短期记忆（LSTM）的 PM 生产进度预测模型为例，通过迁移模型 B 的参数建立模型 A 的 PM，并对其进行再训练，具体过程见 5.3 节。算法采用 Python 编写，同时使用 Socket 接口实现与 PlantSimulation 平台的通信。通过 DT 迁移对包含 13 个站的车间进行 DT 建模需要 1.5 个月，即表 3 中的完成时间"? "仅为 1.5 个月左右。与其他非迁移建模方法（如模型编号 B F）相比，DT4M 可以节省大量时间，提高建模效率。

5.2. Analysis of model verification
5.2.模型验证分析

To test the effectiveness of the proposed approach, two groups of experiments were designed. (1) The first group tests the effectiveness of spatial characteristics consistency verification method of DT model, (2) the second group tests the effectiveness of temporal characteristics consistency verification method of DT model. Some parameters can be calculated according to formula (6) to formula (8) and formula (13) to formula (15).
为了检验所提出方法的有效性，我们设计了两组实验。(1）第一组测试 DT 模型空间特征一致性验证方法的有效性；（2）第二组测试 DT 模型时间特征一致性验证方法的有效性。部分参数可根据式（6）至式（8）和式（13）至式（15）计算。
(1) Analysis of spatial characteristics consistency verification of DT model. The spatial characteristics consistency verification scoring parameters of

£

and

£^{*}

in DT model are presented in Fig. 11.
(1）DT 模型空间特征一致性验证分析。DT 模型中

£

和

£^{*}

的空间特征一致性验证评分参数如图 11 所示。

The spatial characteristics mining manufacturing process data of physical and virtual workshop is carried out based on GCN, and the ordinate represents the values of 11 sets of characteristics.

£, £^{*}

and

Δ_{P W \leftrightarrow V W}^{S F}

are calculated according to formula (6) to formula (8) and recorded in Fig. 11. By comparing the

£

and

£^{*}

values, if set

£ ≜ £^{*} \leq 0.05

, then the characteristic 1 , characteristic 3 , characteristic 5 and 9 cannot be considered to meet the requirements. According to the formula (8), we can get

Δ_{P W \leftrightarrow V W}^{S F} = \frac{7}{11} \approx 63.6 %

. Similarly, if the

Δ_{P W \leftrightarrow V W}^{S F}

threshold is assumed to be

90 %

, it clearly indicates that the spatial characteristics consistency of physical and virtual workshop cannot meet the requirements.
基于 GCN 对物理车间和虚拟车间的制造过程数据进行空间特征挖掘，序号表示 11 组特征值。根据式（6）至式（8）计算出

£, £^{*}

和

Δ_{P W \leftrightarrow V W}^{S F}

，并记录在图 11 中。通过比较

£

和

£^{*}

值，如果设置

£ ≜ £^{*} \leq 0.05

，则不能认为特性 1、特性 3、特性 5 和 9 符合要求。根据公式 (8)，我们可以得到

Δ_{P W \leftrightarrow V W}^{S F} = \frac{7}{11} \approx 63.6 %

。同样，如果假定

Δ_{P W \leftrightarrow V W}^{S F}

临界值为

90 %

，则清楚地表明物理车间和虚拟车间的空间特性一致性不能满足要求。
(2) Analysis of temporal characteristics consistency verification of DT model. The temporal characteristics consistency verification scoring parameters of

λ

and

λ^{*}

in DT model are presented in Fig. 12.
(2) DT 模型的时间特性一致性验证分析。图 12 列出了 DT 模型中

λ

和

λ^{*}

的时间特性一致性验证评分参数。

The temporal characteristics mining manufacturing process data of physical and virtual workshop is carried out based on TCN, and the ordinate represents the values of 11 sets of characteristics.

λ, λ^{*}

and

Δ_{P W \leftrightarrow V W}^{T F}

are calculated according to formula (13) to formula (15) and recorded in Fig. 12. By comparing the

λ

and

λ^{*}

values, if set

λ ≜ λ^{*} \leq 0.1

, it can be observed that the characteristic 2 cannot be considered to meet the requirements. According to the formula (15), we can get

Δ_{P W \leftrightarrow V W}^{T F} = \frac{10}{11} \approx 90.9 %

. Similarly, if the

Δ_{P W \leftrightarrow V W}^{T F}

threshold is assumed to be 90%, it shows that temporal characteristics consistency of physical and virtual workshop meet the requirements.
基于 TCN 对物理车间和虚拟车间的制造过程数据进行时间特征挖掘，序号表示 11 组特征值。根据式（13）至式（15）计算出

λ, λ^{*}

和

Δ_{P W \leftrightarrow V W}^{T F}

，并记录在图 12 中。通过比较

λ

和

λ^{*}

值，如果设置

λ ≜ λ^{*} \leq 0.1

，可以看出特性 2 不能满足要求。根据公式 (15)，我们可以得到

Δ_{P W \leftrightarrow V W}^{T F} = \frac{10}{11} \approx 90.9 %

。同样，如果假定

Δ_{P W \leftrightarrow V W}^{T F}

临界值为 90%，则表明物理车间和虚拟车间的时间特性一致性符合要求。

In summary, compared with spatial and temporal characteristics of
总之，与《全球海洋观测系统》的空间和时间特征相比

Fig. 10. The modeling process of GM, BM, LM and PM of DMW.
图 10.DMW 的 GM、BM、LM 和 PM 的建模过程。

Fig. 11. Spatial characteristics consistency verification scoring parameters in DT model.
图 11.DT 模型中验证评分参数的空间特征一致性。
physical and virtual workshop, it is obvious that DT model still has shortcomings in terms of spatial characteristics consistency and needs further improvement.
通过对物理车间和虚拟车间的分析，可以明显看出 DT 模型在空间特征一致性方面仍有不足，需要进一步改进。

5.3. Analysis of model evolution
5.3.模型演变分析

To test the effectiveness of the proposed approach, two groups of experiments were designed. (1) The first group tests the effectiveness of accuracy trend detection of

{DTM}_{P M}

, (2) the second group tests the
为了检验所提出方法的有效性，我们设计了两组实验。(1) 第一组测试

{DTM}_{P M}

的准确性趋势检测的有效性， (2) 第二组测试

{DTM}_{P M}

的准确性趋势检测的有效性， (3) 第三组测试

{DTM}_{P M}

的准确性趋势检测的有效性。

Fig. 12. Temporal characteristics consistency verification scoring parameters in DT model.
图 12.DT 模型中一致性验证评分参数的时间特征。
effectiveness of synchronous evolution of

{DTM}_{P M}

. Some parameters can be calculated according to formula (17) to formula (22).

{DTM}_{P M}

同步演化的有效性。一些参数可根据式 (17) 至式 (22) 计算。
(1) Analysis of the accuracy trend detection of

D T M_{P M}

. The production progress prediction model as one of the PM is established by LSTM using historical and real-time data in DMW. That is, the performance model

G (X)_{none} =

LSTM and its prediction results are shown in Fig. 13, and the detailed real and predicted values are listed in Table 7.
(1)

D T M_{P M}

的精度趋势检测分析。作为 PM 之一的生产进度预测模型是利用 DMW 中的历史数据和实时数据，通过 LSTM 建立的。即性能模型

G (X)_{none} =

LSTM 及其预测结果如图 13 所示，详细的实际值和预测值如表 7 所示。

A total of 40 groups were conducted to test and compare, which from test set 1 , test set 2 and test set 3 , accounting for 12, 12, 16 groups, respectively. For each group measure, the numbers in bold are the real values and the others are predicted values by

G (X)_{none.}

. The coincidence of blue and red dots in Fig. 13 shows that

G (X)_{none}

has high prediction accuracy at test set 1 and test set 2 . The prediction accuracy is lower at test set 3 than at test set 1 and test set 2 . The reason for this result may be that the consistency of performance between the virtual and physical workshop decreases with time [11]. Therefore, it is necessary to update

G (X)_{none}

synchronously to track the degradation process of the performance of the manufacturing elements in physical workshop.
共进行了 40 组测试和比较，分别来自测试集 1、测试集 2 和测试集 3，占 12、12、16 组。对于每组测量值，粗体数字为真实值，其他数字为

G (X)_{none.}

的预测值。图 13 中蓝点和红点的重合表明，

G (X)_{none}

在测试集 1 和测试集 2 中具有较高的预测准确率。而测试集 3 的预测准确率则低于测试集 1 和测试集 2。造成这一结果的原因可能是虚拟车间和物理车间的性能一致性随着时间的推移而降低 [11]。因此，有必要同步更新

G (X)_{none}

以跟踪物理车间中制造元件性能的下降过程。

To further verify the degradation process of

{DTM}_{P M} (G (X)_{nопе})

, a comparison between some predicted values of physical and virtual workshop on eleven test sets is intuitively illustrated with curve plot, and the experimental results of

G (X)_{none}

and the fitting process of model
为了进一步验证

{DTM}_{P M} (G (X)_{nопе})

的退化过程，我们用曲线图直观地说明了物理车间和虚拟车间在 11 个测试集上的一些预测值的比较，以及

G (X)_{none}

的实验结果和模型的拟合过程。
degradation are intuitively illustrated in Fig. 14.
图 14 直观地说明了降解过程。
The coincidence of blue and red lines in Fig. 14 shows that

G (X)_{none}

has high prediction accuracy at test set 3 and test set 4 . Where, the ordinate represents the completion percentage value for products that are included in a production task. The prediction accuracy is lower from test set 5 to test set 11 . And the accuracy of physical and virtual workshop decrease linearly (linear fitting is a selected method in this paper, but not limited to this) according to formula (20), which indicates that the accuracy of

G (X)_{none}

is decreasing, tracking physical workshop performance changes is weakening, and the

G (X)_{none}

needs to be updated synchronously. The reason for this result may be that the data distribution from test set 5 to test set 11 is different from that of test set 3 and test set 4.
图 14 中蓝线和红线的重合表明，

G (X)_{none}

在测试集 3 和测试集 4 中具有较高的预测精度。其中，纵坐标表示生产任务中包含的产品的完成百分比值。测试集 5 到测试集 11 的预测准确率较低。而根据公式（20），物理车间和虚拟车间的准确度呈线性下降（本文选择线性拟合的方法，但不限于此），这表明

G (X)_{none}

的准确度在下降，跟踪物理车间性能变化的能力在减弱，而

G (X)_{none}

需要同步更新。造成这一结果的原因可能是测试集 5 到测试集 11 的数据分布与测试集 3 和测试集 4 的数据分布不同。
(2) Analysis of the synchronous evolution of

D T M_{P M}

. It is necessary to update

G (X)_{none}

synchronously to track the degradation process of the performance of the manufacturing elements in physical workshop. And this study also adopts some other commonly used evaluation measures, including mean absolute error (MAE), root mean square error (RMSE), symmetric mean absolute percentage error (SMAPE) and R-square (R2). The prediction results of

G

(X)_{none}

synchronous evolution is shown in Fig. 15.
(2)

D T M_{P M}

的同步演化分析。有必要同步更新

G (X)_{none}

以跟踪物理车间制造元件的性能退化过程。本研究还采用了其他一些常用的评估指标，包括平均绝对误差（MAE）、均方根误差（RMSE）、对称平均绝对百分比误差（SMAPE）和 R 平方（R2）。

G

(X)_{none}

同步演化的预测结果如图 15 所示。

In terms of the MAE and RMSE results, the

G (X)_{update} (G (X)

synchronous evolution based on Adaboost according to formula (21) and
从 MAE 和 RMSE 结果来看，基于 Adaboost 的

G (X)_{update} (G (X)

同步演化根据公式 (21) 和

Fig. 13. Comparison of the actual and predicted values on 200 random test samples using

G (X)_{none}

.
图 13.使用

G (X)_{none}

的 200 个随机测试样本的实际值和预测值对比。
Table 7 表 7
The Testing Results of

G (X)_{none.}

G (X)_{none.}

的测试结果。

Index 索引	Sets 设置	Comparison group number (total 40 groups, Test Dataset 1 and 2 account for 12 groups respectively, and the remaining 16 groups belong to Test Dataset 3 ) 对比组数（共 40 组，测试数据集 1 和 2 分别占 12 组，其余 16 组属于测试数据集 3）
Index 索引	Sets 设置	1		2		3		4		5		6		...
Production Progress (/h) 生产进度（小时）	Test Dataset 1 测试数据集 1	403.552	401.676	181.103	177.676	98.199	97.621	243.078	241.621	334.073	335.836	392.638	391.676	...
		316.977	318.836	391.406	393.836	157.998	153.676	159.106	163.836	396.266	398.836	119.677	123.836	...
		52.689	43.989	244.289	242.621	36.809	29.836	255.124	253.676	359.124	357.989	200.629	204.836	...
		47.827	43.836	131.279	125.676	33.314	19.676	22.213	9.836	174.347	170.676	236.530	234.676	...
		179.171	175.676	42.242	36.836	51.334	39.676	129.391	133.836	256.941	259.836	65.006	54.676	...
	Test Dataset 2 测试数据集 2	18.396	4.621	348.048	348.621	88.252	92.031	255.025	257.031	237.393	240.031	127.345	126.825	...
		524.469	529.079	118.834	124.031	248.078	246.903	407.820	408.031	299.628	300.621	327.378	334.079	...
		394.609	400.079	35.822	21.079	468.466	473.079	396.642	402.079	265.960	267.621	399.979	400.031	...
		52.641	47.079	74.625	77.031	114.030	119.079	326.969	327.031	341.652	348.079	411.875	417.079	.
		632.501	639.079	488.543	493.079	120.394	124.621	216.689	219.621	442.253	447.079	44.485	41.621	...
	Test Dataset 3 测试数据集 3	53.717	51.132	480.439	484.399	370.060	374.389	272.317	278.132	493.916	517.466	154.850	161.389	...
		157.439	156.466	197.764	205.132	41.380	38.399	125.008	131.132	614.080	648.466	48.941	45.132	...
		507.295	506.132	415.081	417.399	107.532	112.399	51.159	28.466	293.627	299.389	211.252	215.466	.
		404.214	406.399	95.184	99.132	513.042	517.389	453.212	453.132	21.576	0.132	450.383	453.399	...
		174.603	175.466	209.068	213.399	234.614	238.399	51.723	51.399	127.269	122.466	408.170	410.399	...

| Index | Sets | Comparison group number (total 40 groups, Test Dataset 1 and 2 account for 12 groups respectively, and the remaining 16 groups belong to Test Dataset 3 ) | | | | | | | | | | | | | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | | | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | ... | | Production Progress (/h) | Test Dataset 1 | 403.552 | 401.676 | 181.103 | 177.676 | 98.199 | 97.621 | 243.078 | 241.621 | 334.073 | 335.836 | 392.638 | 391.676 | ... | | | | 316.977 | 318.836 | 391.406 | 393.836 | 157.998 | 153.676 | 159.106 | 163.836 | 396.266 | 398.836 | 119.677 | 123.836 | ... | | | | 52.689 | 43.989 | 244.289 | 242.621 | 36.809 | 29.836 | 255.124 | 253.676 | 359.124 | 357.989 | 200.629 | 204.836 | ... | | | | 47.827 | 43.836 | 131.279 | 125.676 | 33.314 | 19.676 | 22.213 | 9.836 | 174.347 | 170.676 | 236.530 | 234.676 | ... | | | | 179.171 | 175.676 | 42.242 | 36.836 | 51.334 | 39.676 | 129.391 | 133.836 | 256.941 | 259.836 | 65.006 | 54.676 | ... | | | Test Dataset 2 | 18.396 | 4.621 | 348.048 | 348.621 | 88.252 | 92.031 | 255.025 | 257.031 | 237.393 | 240.031 | 127.345 | 126.825 | ... | | | | 524.469 | 529.079 | 118.834 | 124.031 | 248.078 | 246.903 | 407.820 | 408.031 | 299.628 | 300.621 | 327.378 | 334.079 | ... | | | | 394.609 | 400.079 | 35.822 | 21.079 | 468.466 | 473.079 | 396.642 | 402.079 | 265.960 | 267.621 | 399.979 | 400.031 | ... | | | | 52.641 | 47.079 | 74.625 | 77.031 | 114.030 | 119.079 | 326.969 | 327.031 | 341.652 | 348.079 | 411.875 | 417.079 | . | | | | 632.501 | 639.079 | 488.543 | 493.079 | 120.394 | 124.621 | 216.689 | 219.621 | 442.253 | 447.079 | 44.485 | 41.621 | ... | | | Test Dataset 3 | 53.717 | 51.132 | 480.439 | 484.399 | 370.060 | 374.389 | 272.317 | 278.132 | 493.916 | 517.466 | 154.850 | 161.389 | ... | | | | 157.439 | 156.466 | 197.764 | 205.132 | 41.380 | 38.399 | 125.008 | 131.132 | 614.080 | 648.466 | 48.941 | 45.132 | ... | | | | 507.295 | 506.132 | 415.081 | 417.399 | 107.532 | 112.399 | 51.159 | 28.466 | 293.627 | 299.389 | 211.252 | 215.466 | . | | | | 404.214 | 406.399 | 95.184 | 99.132 | 513.042 | 517.389 | 453.212 | 453.132 | 21.576 | 0.132 | 450.383 | 453.399 | ... | | | | 174.603 | 175.466 | 209.068 | 213.399 | 234.614 | 238.399 | 51.723 | 51.399 | 127.269 | 122.466 | 408.170 | 410.399 | ... |

(22)) has lower MAE and RMSE than

G (X)_{none} (G (X)

no synchronous evolution). And it also has the highest overall accuracies for three test data sets, accounting for 100%, respectively. Moreover, the overall error bar of

G (X)_{update}

is shorter than that of

G (X)_{none}

no synchronous evolution, which reveals the less error between the predicted value and the real value, and thus the proposed model has very good robustness performance.
(22)) 相比，

G (X)_{none} (G (X)

（无同步演化）的 MAE 和 RMSE 更低。在三个测试数据集中，它的总体准确率也是最高的，分别占 100%。此外，

G (X)_{update}

的整体误差条比

G (X)_{none}

无同步演化的误差条更短，这表明预测值与实际值之间的误差更小，因此所提出的模型具有很好的鲁棒性能。

Similarly, according to the SMAPE and R2 results, the

G (X)_{update}

has better SMAPE and R2 index values than

G (X)_{none.}

. In detail, the

G (X)_{update}

has better performance in 2 of 3 data sets, accounting for

66.66 %

and likewise in 3 of 3 data sets, accounting for

100 %

, respectively. The reason for

G (X)_{update}

has better performance than

G (X)_{none}

is that the synchronous evolution mechanism plays a significant role in

G (X)

.
同样，根据 SMAPE 和 R2 结果，

G (X)_{update}

的 SMAPE 和 R2 指标值优于

G (X)_{none.}

。具体来说，

G (X)_{update}

在 3 个数据集中的 2 个数据集（即

66.66 %

）中的表现更好，同样，在 3 个数据集中的 3 个数据集（即

100 %

）中的表现也更好。之所以

G (X)_{update}

的性能优于

G (X)_{none}

，是因为同步进化机制在

G (X)

中发挥了重要作用。

In summary, comparing

G (X)_{update}

and

G (X)_{none}, G (X)_{update}

obviously has optimal comprehensive performance, strong robustness and generalization performance and can effectively track the evolution process of production progress in DMW.
综上所述，比较

G (X)_{update}

和

G (X)_{none}, G (X)_{update}

显然具有最优的综合性能、较强的鲁棒性和泛化性能，可以有效跟踪 DMW 生产进度的演变过程。

5.4. Discussion 5.4.讨论情况

From the case study, it can be found that the proposed models and methods are suitable for model modeling, verification and evolution for DMW. Moreover, the work can make it easily known, which part of DT model does not meet the requirement, so as to timely modify and synchronous evolution. The first two methods correspond to the process before the DT model is used, that is, DT modeling process, while the latter method corresponds to the process when DT model is using, that is, DT model evolution process, which is a minor extension of the DT modeling theory. However, there are still some issues to discuss.
从案例研究中可以发现，所提出的模型和方法适用于 DMW 的建模、验证和演化。此外，这项工作还可以使人们很容易地知道 DT 模型的哪些部分不符合要求，从而及时修改和同步演化。前两种方法对应的是 DT 模型使用前的过程，即 DT 建模过程，而后一种方法对应的是 DT 模型使用时的过程，即 DT 模型演化过程，是 DT 建模理论的一个小扩展。不过，还有一些问题需要讨论。
(1) As for digital twin model modeling, the proposed migration modeling method can improve the efficiency of DT modeling. The PlantSimulation is used to establish the geometry model, behavior model and logic model of DMW. Due to the limited capacity of the model warehouse, the application of model migration based matching modeling technology is only adopt to select from five similar models in the historical DT model warehouse. Experiments with massive models need to be further confirmed.
(1) 在数字孪生建模方面，所提出的迁移建模方法可以提高 DT 建模的效率。利用 PlantSimulation 建立 DMW 的几何模型、行为模型和逻辑模型。由于模型仓库容量有限，应用基于模型迁移的匹配建模技术只能从历史 DT 模型仓库中的五个相似模型中进行选择。大规模模型的实验还需要进一步证实。
(2) As for digital twin model verification, the proposed verification method is based on the microscopic perspective of objective data and is not affected by subjective factors. The verification of characteristics is just using the production process data. The more data selected, the more accurate the evaluation. The GCN-TCN based DT model verification method is implicit characteristic verification. Besides, the explicit characteristic verification, including the authenticity of virtual workshop 3D model and workshop layout of physical workshop system has been explored in [22].
(2) 在数字孪生模型验证方面，所提出的验证方法基于客观数据的微观视角，不受主观因素的影响。特性验证只需使用生产过程数据。选择的数据越多，评估就越准确。基于 GCN-TCN 的 DT 模型验证方法属于隐式特性验证。此外，[22]还探讨了显式特性验证，包括虚拟车间三维模型和物理车间系统的车间布局的真实性。
(3) As for digital twin model evolution, the DT model synchronous evolution method proposed can be used to track the performance degradation state of production progress and realize the prediction of production progress, but the limitation of this approach is requiring

{DMW}_{P M}

to satisfy the integrated condition before integrating by Adaboost algorithm to form a strong learner integration model. And the traceability analysis method is not accessible when the DT model is inaccurate, which needs to be addressed in future research.
(3）在数字孪生模型演化方面，提出的DT模型同步演化方法可用于跟踪生产进度的性能退化状态，实现生产进度的预测，但该方法的局限性在于要求

{DMW}_{P M}

满足集成条件后再通过Adaboost算法进行集成，形成强学习者集成模型。而且当 DT 模型不准确时，无法使用溯源分析方法，这需要在今后的研究中加以解决。

6. Conclusion 6.结论

DT technology offers an enabling tool for intelligent transformation of DMW. This article focuses on a novel method system for the modeling-verification-evolution of DT model in DMW, which contains DT model modeling, DT model verification and DT model synchronous evolution method. The modeling and verification are aimed at the pre-actual use
DT 技术为 DMW 的智能化改造提供了有利工具。本文主要介绍一种新颖的 DT 模型在 DMW 中的建模-验证-演化方法体系，包括 DT 模型建模、DT 模型验证和 DT 模型同步演化方法。建模和验证的目的是在实际使用前

Fig. 14. Accuracy trend detect verification values of

G (X)_{none}

.
图 14.检测

G (X)_{none}

验证值的精度趋势。

Fig. 15. Column graphs of MAE, RMSE, SMAPE and R2 value in three new test datasets of experiments.
图 15.三个新测试数据集的 MAE、RMSE、SMAPE 和 R2 值柱状图。
stage of DT model, and the synchronous evolution is aimed at the actual use stage of DT model. The main contributions are summarized as follows.
同步演化针对的是 DT 模型的实际使用阶段。主要贡献概述如下。

This article first proposed a DT model migration modeling method (DT4M) of DMW to describe the modeling requirements and the transferability between DT models, and realize the selection of DT models from warehouse for migration, which can obtain high efficiency of DT modeling.
本文首先提出了 DMW 的 DT 模型迁移建模方法（DT4M），以描述建模要求和 DT 模型之间的可迁移性，并实现从仓库中选择 DT 模型进行迁移，从而获得高效的 DT 建模。
A DT model verification method based on graph convolutional network (GCN) and temporal convolutional network (TCN) is also first proposed to verify the consistency of spatial characteristics and temporal characteristics between the DT model and physical workshop, which can provide some new indicators to quantify, and enhance the enforceability and accuracy for DT model verification.
同时，首次提出了基于图卷积网络（GCN）和时序卷积网络（TCN）的 DT 模型验证方法，验证 DT 模型与物理车间的空间特征和时间特征的一致性，为 DT 模型验证提供了一些新的量化指标，增强了 DT 模型验证的可执行性和准确性。
This article also proposed an Adaboost based synchronous evolution method to enhance the accuracy of DT model applications and to ensure the synchronization of virtual and physical workshop.
本文还提出了一种基于 Adaboost 的同步演化方法，以提高 DT 模型应用的准确性，并确保虚拟车间和物理车间的同步性。

Finally, an experiment is carried out on an aerospace machining workshop. Case analysis shows that the proposed methods are effective and have good performance.
最后，在一个航空航天加工车间进行了实验。案例分析表明，所提出的方法是有效的，而且性能良好。

The limitation of this approach is requiring one model management system (including model warehouse) to organize DT model in industrial environment. For example, when a new DT model is established by migration, it needs to be stored and organized, etc. for subsequent modification and reuse. Moreover, it requires accurate sensor data to support model verification and model evolution in industrial environment. So, limited by the accuracy of sensor data collection, these data errors may affect the accuracy of model verification and model evolution.
这种方法的局限性在于需要一个模型管理系统（包括模型仓库）来组织工业环境中的 DT 模型。例如，当通过迁移建立新的 DT 模型时，需要对其进行存储和组织等，以便后续修改和重用。此外，还需要精确的传感器数据来支持工业环境中的模型验证和模型演化。因此，受限于传感器数据采集的准确性，这些数据误差可能会影响模型验证和模型演化的准确性。

Considering the research content and field development of this paper, future work will focus on the method system of DT model evolution-error traceability and the establishment of the inversion model of manufacturing elements (e.g., equipment) performance, which can guide the optimization control and improve the performance of manufacturing system.
考虑到本文的研究内容和领域拓展，今后的工作重点将放在 DT 模型演化-误差溯源的方法体系和制造要素（如设备）性能反演模型的建立上，从而指导优化控制，提高制造系统的性能。

Declaration of Competing Interest
竞争利益声明

No conflict of interest exits in the submission of this manuscript, and the manuscript is approved for publication by all authors. I would like to declare on behalf of my co-authors that the work described is original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.
提交本手稿不存在任何利益冲突，所有作者均同意发表本手稿。我谨代表我的合著者声明，所描述的工作是原创性研究，此前未曾发表，也未考虑在其他地方发表全部或部分内容。所有作者均已批准随附的手稿。

Acknowledgment 鸣谢

This work was supported in part by the Natural Science Foundation of Jiangsu Province under Grant BK20202007, in part by the National Natural Science Foundation of China under Grant 52105522, in part by China Postdoctoral Science Foundation under Grant 2022M721597.
本研究部分得到江苏省自然科学基金（BK20202007）、国家自然科学基金（52105522）和中国博士后科学基金（2022M721597）的资助。

References 参考资料

[1] Lu Y, Liu C, Wang K, Huang H, Xu X. Digital Twin-driven smart manufacturing: connotation, reference model, applications and research issues. Robot ComputIntegr Manuf 2020;61:101837.
[1] Lu Y, Liu C, Wang K, Huang H, Xu X. Digital Twin-driven smart manufacturing: connotation, reference model, applications and research issues.Robot ComputIntegr Manuf 2020;61:101837.
[2] Lu Y, Xu X, Wang L. Smart manufacturing process and system automation-a critical review of the standards. J Manuf Syst 2020;56:312-25.
[2] Lu Y，Xu X，Wang L. 《智能制造过程与系统自动化--标准评述》。J Manuf Syst 2020;56:312-25.
[3] Liu S, Lu Y, Zheng P, Shen H, Bao J. Adaptive reconstruction of digital twins for machining systems: a transfer learning approach. Robot Comput-Integr Manuf 2022;78:102390.
[3] Liu S, Lu Y, Zheng P, Shen H, Bao J. Adaptive reconstruction of digital twins for machining systems: a transfer learning approach.Robot Comput-Integr Manuf 2022;78:102390.
[4] Tao F, Qi Q. Make more digital twins. Nature 2019;573:490-1.
[4] Tao F，Qi Q.制造更多数字双胞胎。自然》2019；573：490-1.
[5] Tao F, Xiao B, Qi Q, Cheng J, Ji P. Digital twin modeling. J Manuf Syst 2022;64: 372-89.
[5] Tao F, Xiao B, Qi Q, Cheng J, Ji P. 数字孪生建模.J Manuf Syst 2022;64: 372-89.
[6] Bao J, Guo D, Li J, Zhang J. The modelling and operations for the digital twin in the context of manufacturing. Enterp Inf Syst 2019;13(4):534-56.
[6] Bao J，Guo D，Li J，Zhang J.制造业背景下的数字孪生建模与操作.Enterp Inf Syst 2019；13（4）：534-56.
[7] Saratha S, Grimm C, Wawrzik F. A digital twin with runtime-verification for industrial development-operation integration. IEEE Int Conf Eng, Technol Innov 2021.
[7] Saratha S, Grimm C, Wawrzik F. A digital twin with runtime-verification for industrial development-operation integration.IEEE Int Conf Eng, Technol Innov 2021.
[8] Semenkov K, Promyslov V, Poletykin A. Verification of large-scale control systems with hybrid digital models and digital twins. Int Russ Autom Conf 2020.
[8] Semenkov K, Promyslov V, Poletykin A. 使用混合数字模型和数字双胞胎验证大规模控制系统。Int Russ Autom Conf 2020.
[9] Zheng Y, Yang S, Cheng H. An application framework of digital twin and its case study. J Ambient Intell Humaniz Comput 2019;10(3):1141-53.
[9] Zheng Y，Yang S，Cheng H. 数字孪生的应用框架及其案例研究.J Ambient Intell Humaniz Comput 2019;10(3):1141-53.
[10] Zhang L, Guo Y, Qian W, Wang W, Liu D, Liu S. Modelling and online training method for digital twin workshop. Int J Prod Res 2022.
[10] Zhang L，Guo Y，Qian W，Wang W，Liu D，Liu S. 数字孪生工作室的建模与在线培训方法。Int J Prod Res 2022.
[11] Qian W, Guo Y, Zhang H, Huang S, Zhang L, Zhou H, et al. Digital twin driven production progress prediction for discrete manufacturing workshop. Robot Comput-Integr Manuf 2023;80:102456.
[11] Qian W、Guo Y、Zhang H、Huang S、Zhang L、Zhou H 等. 离散制造车间的数字孪生驱动生产进度预测.Robot Comput-Integr Manuf 2023;80:102456.
[12] Grieves M., Digital Twin: Manufacturing Excellence through Virtual Factory Replication, 〈https://www.researchgate.net/publication/275211047_Digital_ Twin_Manufacturing_Excellence_through_Virtual_Factory_Replication

⟩, 2015

.
[13] Zhang Q, Zhang X, Xu W, Liu A, Zhou Z, Pham D. Modeling of digital twin workshop based on perception data. 10th International Conference on Intelligent Robotics and Applications. Cham: Springer; 2017.
[13] Zhang Q，Zhang X，Xu W，Liu A，Zhou Z，Pham D. 基于感知数据的数字孪生车间建模。第十届国际智能机器人及应用大会。Cham：Springer; 2017.
[14] Zhuang C, Miao T, Liu J, Xiong H. The connotation of digital twin, and the construction and application method of shop-floor digital twin. Robot ComputIntegr Manuf 2021;68:102075.
[14] 庄C，苗T，刘J，熊H.数字孪生的内涵及车间数字孪生的构建与应用方法.Robot ComputIntegr Manuf 2021;68:102075.
[15] Ding K, Chan F, Zhang X, Zhou G, Zhang F. Defining a Digital Twin-based CyberPhysical Production System for autonomous manufacturing in smart shop floors. Int J Prod Res 2019;57(20):6315-34.
[15] Ding K, Chan F, Zhang X, Zhou G, Zhang F. Defining a Digital Twin-based CyberPhysical Production System for autonomous manufacturing in smart shop floors.Int J Prod Res 2019;57(20):6315-34.
[16] Uhlemann HJ, Lehmann C, Steinhilper R. The digital twin: realizing the cyberphysical production system for industry 4.0. Procedia Cirp 2017;61:335-40.
[16] Uhlemann HJ, Lehmann C, Steinhilper R. The digital twin: realizing the cyberphysical production system for industry 4.0.Procedia Cirp 2017;61:335-40.
[17] Schleich B, Anwer N, Mathieu L, Wartzack S. Shaping the digital twin for design and production engineering. CIRP Ann-Manuf Technol 2017;66(1):141-4.
[17] Schleich B, Anwer N, Mathieu L, Wartzack S. Shaping the digital twin for design and production engineering.CIRP Ann-Manuf Technol 2017;66(1):141-4.
[18] Yi Y, Yan Y, Liu X, Ni Z, Feng J, Liu J. Digital twin-based smart assembly process design and application framework for complex products and its case study. J Manuf Syst 2021;58:94-107.
[18] Yi Y，Yan Y，Liu X，Ni Z，Feng J，Liu J. 基于数字孪生的复杂产品智能装配流程设计与应用框架及其案例研究.J Manuf Syst 2021;58:94-107.
[19] Wang H, Li H, Wen X, Luo G. Unified modeling for digital twin of a knowledgebased system design. Robot Comput-Integr Manuf 2021;68:102074.
[19] Wang H, Li H, Wen X, Luo G. 基于知识的系统设计的数字孪生统一建模.Robot Comput-Integr Manuf 2021;68:102074.
[20] Leng J, Zhou M, Xiao Y, Zhang H, Liu Q, Shen W, et al. Digital twins-based remote semi-physical commissioning of flow-type smart manufacturing systems. J Clean Prod 2021;320:128836.
[20] Leng J, Zhou M, Xiao Y, Zhang H, Liu Q, Shen W, et al.J Clean Prod 2021;320:128836.
[21] Luo W, Hu T, Ye Y, Zhang C, Wei Y. A hybrid predictive maintenance approach for CNC machine tool driven by Digital Twin. Robot Comput-Integr Manuf 2020;65: 101974.
[21] Luo W, Hu T, Ye Y, Zhang C, Wei Y.数字双胞胎驱动的数控机床混合预测性维护方法。Robot Comput-Integr Manuf 2020;65: 101974.
[22] Qian W, Guo Y, Cui K, Wu P, Fang W, Liu D. Multidimensional data modeling and model validation for digital twin workshop. J Comput Inf Sci Eng 2021;21(3):1-13.
[22] Qian W, Guo Y, Cui K, Wu P, Fang W, Liu D. 数字孪生工作坊的多维数据建模与模型验证.J Comput Inf Sci Eng 2021;21(3):1-13.
[23] Jiang L, Zheng H, Tian H, Xie S, Zhang Y. Cooperative federated learning and model update verification in blockchain-empowered digital twin edge networks. IEEE Internet Things J 2022;9(13):11154-67.
[23] Jiang L, Zheng H, Tian H, Xie S, Zhang Y.区块链赋能的数字孪生边缘网络中的合作联合学习与模型更新验证。IEEE Internet Things J 2022;9(13):11154-67.
[24] Zhang H, Qi Q, Tao F. A consistency evaluation method for digital twin models. J Manuf Syst 2022;65:158-68.
[24] Zhang H, Qi Q, Tao F. 数字孪生模型一致性评价方法.J Manuf Syst 2022;65:158-68.
[25] Wang Y, Wang S, Yang B, Zhu L, Liu F. Big data driven Hierarchical Digital Twin Predictive Remanufacturing paradigm: architecture, control mechanism, application scenario and benefits. J Clean Prod 2020;248:1-15.
[25] Wang Y, Wang S, Yang B, Zhu L, Liu F. Big data driven Hierarchical Digital Twin Predictive Remanufacturing paradigm: architecture, control mechanism, application scenario and benefits.J Clean Prod 2020;248:1-15.
[26] Liu S, Shen H, Li J, Lu Y, Bao J. An adaptive evolutionary framework for the decision-making models of digital twin machining system. 17th IEEE Int Conf Autom Sci Eng 2021.
[26] Liu S, Shen H, Li J, Lu Y, Bao J. 数字孪生加工系统决策模型的自适应进化框架。17th IEEE Int Conf Autom Sci Eng 2021.
[27] Liu S, Lu Y, Li J, Song D, Sun X, Bao J. Multi-scale evolution mechanism and knowledge construction of a digital twin mimic model. Robot Comput-Integr Manuf 2021;71:102123.
[27] Liu S, Lu Y, Li J, Song D, Sun X, Bao J. 数字孪生仿真模型的多尺度演化机制与知识构建.Robot Comput-Integr Manuf 2021;71:102123.
[28] Grieves M, Vickers J. Digital Twin: Mitigating Unpredictable, Undesirable Emergent Behavior in Complex Systems. Springer International Publishing; 2017.
[28] Grieves M, Vickers J. Digital Twin: Mitigating Unpredictable, Undesirable Emergent Behavior in Complex Systems.Springer International Publishing; 2017.
[29] Saddik AE. Digital twins: the convergence of multimedia technologies. IEEE Multimed 2018;25(2):87-92.
[29] Saddik AE.数字双胞胎：多媒体技术的融合。IEEE Multimed 2018;25(2):87-92.
[30] Bolton RN, Mccoll K, Cheung L, Gallan A, Orsingher C, Witell L, et al. Customer experience challenges: bringing together digital, physical and social realms. J Serv Manag 2018;29(5):776-808.
[30] Bolton RN, Mccoll K, Cheung L, Gallan A, Orsingher C, Witell L, et al. Customer Experience challenges: bringing together digital, physical and social realms.J Serv Manag 2018;29(5):776-808.
[31] Grieves M. Product lifecycle management: driving the next generation of lean thinking. New York, USA, 2006.
[31] Grieves M. Product lifecycle management: driving the next generation of lean thinking.美国纽约，2006 年。
[32] ISO 23247-1:2021, Automation systems and integration-Digital twin framework for manufacturing-Part 1: Overview and general principles.
[32] ISO 23247-1:2021，自动化系统与集成-制造业数字孪生框架-第 1 部分：概述和一般原则。
[33] Tao F, Qi Q, Wang L, Nee A. Digital twins and cyber-physical systems toward smart manufacturing and industry 4.0: correlation and comparison. Engineering 2019;5: 653-61.
[33] Tao F, Qi Q, Wang L, Nee A. 面向智能制造和工业 4.0 的数字孪生与网络物理系统：关联与比较。工程学 2019; 5: 653-61.
[34] Long M, Yue C, Cao Z, Wang J, Jordan M. Transferable representation learning with deep adaptation networks. IEEE Trans Pattern Anal Mach Intell 2019;41(12): 3071-85.
[34] Long M, Yue C, Cao Z, Wang J, Jordan M. 用深度适应网络进行可迁移表征学习。IEEE Trans Pattern Anal Mach Intell 2019;41(12)：3071-85.
[35] Wu Z, Pan S, Chen F, Long G, Zhang C, Yu P. A comprehensive survey on graph neural networks. IEEE Trans Neural Netw Learn Syst 2021;32(1):4-24.
[35] Wu Z, Pan S, Chen F, Long G, Zhang C, Yu P.图神经网络综述.IEEE Trans Neural Netw Learn Syst 2021;32(1):4-24.
[36] Liu Y, Dong H, Wang X, Han S. Time series prediction based on temporal convolutional network. 18th Int Conf Comput Inf Sci 2019.
[36] Liu Y, Dong H, Wang X, Han S. 基于时序卷积网络的时间序列预测。18th Int Conf Comput Inf Sci 2019.
[37] Huang S, Guo Y, Yang N, Zha S, Liu D, Fang W. A weighted fuzzy C-means clustering method with density peak for anomaly detection in IoT-enabled manufacturing process. J Intell Manuf 2021;32:1845-61.
[37] Huang S, Guo Y, Yang N, Zha S, Liu D, Fang W. A weighted fuzzy C-means clustering method with density peak for anomaly detection in IoT-enabled manufacturing process.J Intell Manuf 2021;32:1845-61.

- Corresponding authors. 通讯作者：
E-mail addresses: guoyu@nuaa.edu.cn (Y. Guo), shaohuah@nuaa.edu.cn (S. Huang).
电子邮件地址：guoyu@nuaa.edu.cn (Y. Guo), shaohuah@nuaa.edu.cn (S. Huang)。

Towards discrete manufacturing workshop-oriented digital twin model: Modeling, verification and evolution 面向离散制造车间的数字孪生模型：建模、验证和演化

A R TICLE IN F O文章

Keywords: 关键词：

Abstract 摘要

1. Introduction 1.导言

2. Related work 2.相关工作

2.1. Digital twin model modeling2.1.数字孪生模型建模

2.2. Digital twin model verification2.2.数字孪生模型验证

2.3. Digital twin model evolution2.3.数字孪生模型的演变

3. Connotation and characteristics of digital twin in discrete manufacturing workshop3.离散制造车间数字孪生的内涵和特点

3.1. Concept of digital twin in discrete manufacturing workshop3.1.离散制造车间数字孪生的概念

3.2. Characteristics of digital twin in discrete manufacturing workshop3.2.离散制造车间数字孪生的特点

4. Key technologies of digital twin modeling in discrete manufacturing workshop4.离散制造车间数字孪生建模的关键技术

4.1. Model migration based matching modeling technology4.1.基于匹配建模技术的模型迁移

4.2. GCN-TCN based digital twin model verification4.2.基于 GCN-TCN 的数字孪生模型验证

4.3. Digital twin model synchronous evolution4.3.数字孪生模型同步演化

5. Experiments and discussion5.实验和讨论

5.1. Analysis of model modeling5.1.模型建模分析

5.2. Analysis of model verification5.2.模型验证分析

5.3. Analysis of model evolution5.3.模型演变分析

5.4. Discussion 5.4.讨论情况

6. Conclusion 6.结论

Declaration of Competing Interest竞争利益声明

Acknowledgment 鸣谢

References 参考资料

Towards discrete manufacturing workshop-oriented digital twin model: Modeling, verification and evolution
面向离散制造车间的数字孪生模型：建模、验证和演化

A R TICLE IN F O
文章

2.1. Digital twin model modeling
2.1.数字孪生模型建模

2.2. Digital twin model verification
2.2.数字孪生模型验证

2.3. Digital twin model evolution
2.3.数字孪生模型的演变

3. Connotation and characteristics of digital twin in discrete manufacturing workshop
3.离散制造车间数字孪生的内涵和特点

3.1. Concept of digital twin in discrete manufacturing workshop
3.1.离散制造车间数字孪生的概念

3.2. Characteristics of digital twin in discrete manufacturing workshop
3.2.离散制造车间数字孪生的特点

4. Key technologies of digital twin modeling in discrete manufacturing workshop
4.离散制造车间数字孪生建模的关键技术

4.1. Model migration based matching modeling technology
4.1.基于匹配建模技术的模型迁移

4.2. GCN-TCN based digital twin model verification
4.2.基于 GCN-TCN 的数字孪生模型验证

4.3. Digital twin model synchronous evolution
4.3.数字孪生模型同步演化

5. Experiments and discussion
5.实验和讨论

5.1. Analysis of model modeling
5.1.模型建模分析

5.2. Analysis of model verification
5.2.模型验证分析

5.3. Analysis of model evolution
5.3.模型演变分析

Declaration of Competing Interest
竞争利益声明