# Geometric Deep Lean Learning: Deep Learning in Industry 4.0 Cyber–Physical Complex Networks

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## Abstract

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## 1. Introduction

## 2. Background

- Industry 4.0. The term Industry 4.0 has gained large traction since it was first publicized [6], stating the need for a paradigm shift towards a less centrally controlled manufacturing structure. It is seen as the Fourth Industrial Revolution, with the first three being mechanization through steam power, mass production through electrically operated engineering, and the digital revolution through the integration of electronics and IT. Industry 4.0 enables more production autonomy as technology becomes more interconnected, and machines are able to influence each other by creating a cyber–physical system.
- Cyber–Physical Systems. The term “cyber–physical system” in the context of Industry 4.0 refers to the tight conjoining of and co-ordination between computational and physical resources. The impact on the development of such systems is a new paradigm of technical systems based on collaborative embedded software systems [7].
- Lean Management. Lean-management systems in an Industry 4.0 cyber–physical context have been described as sociotechnical entities that aim to systematically reduce the variability of value-creation processes [8,9,10,11,12,13]. These two fundamental dimensions, the social and the technical, are subsequently meant to symbiotically support each other to maximize value creation through the systematic elimination of activities that do not add value for the client. A series of models were presented by scholars that allow the analysis and quantification of these systems as complex networks [14,15].
- Complex-Networked Organizational Design. Under the organizational-network paradigm, modern Industry 4.0 cyber–physical lean-management-oriented organizations can be understood as a symbiotic sociotechnical ecosystem of social networks [16] that interacts with increasingly complex-networked physically distributed interconnected sensors [17], whose readings are modeled as time-dependent signals on the vertices, human or cyber–physical, respectively. This means that, on the nodes of the network, attributes can be found that describe them as having the form of a given time series.Within this framework, a complex network is defined as a graph with nontrivial topological features that do not occur in simple graphs such as lattices and random networks [18]. For any given time t, lean complex cyber–physical networks can be formally described by time-dependent graphs $\Omega \left(t\right)=\left[N\right(t);E(t\left)\right]$ that can be understood as lists of $N\left(t\right)$ nodes and $E\left(t\right)\subset \left(N\right(t\left)xN\right(t\left)\right)$ edges that represent its human and cyber–physical nodes, and its standard communication edges [19]. Given the static graph in t, $\Omega \left(t\right)$, each node and edge can be characterized by a series of typically two-dimensional signals $x=[{x}_{1},\dots ,{x}_{n}]\in \left({\mathbb{R}}^{n}x{\mathbb{R}}^{m}\right)$, where n relevant parameters of the node or axis are described as the time series of m elements. In the case of nodes, signals typically represent demographic, sociological, or competence information. In case that the nodes are human, and in the case of a cyber–physical node, relevant information on the state of the cyber–physical node expressed in time series of several key performance indicators. In the case of edges, signals typically represent information referring to the quality of measurable relationships of the individual with other stakeholders of the organization; in the case of human–human or cyber–physical-to-human edges, of the time series associated with relevant key performance indicators being reported to other stakeholders. Specifically, snapshots for the time-dependent graph can be built, that is, the time-dependent graph is considered as an ordered pair of potentially different sets. A time-dependent graph considered as a sequence of static graphs is given by Expression 1.$$\Omega =[\Omega \left({t}_{1}\right),\Omega \left({t}_{2}\right),\dots ,\Omega \left({t}_{k}\right)]$$This method is most commonly used for modeling discrete time-dependent graphs, and is suitable for the time-dependent graph with a specific time structure, especially in real-time networks such as complex-networked cyber–physical systems [20]. This modelling method is assumed here, and the time sequence of static graphs is not explicitly mentioned when referring to time-dependent graphs.

## 3. Related Work

- Offline training, and decision-support learning and predicting from a global and integrated way, for example, by extracting relevant information from an organization by means of deep-learning algorithms that analyze previously labeled text in organizational categories [66]. Alternatively, by combining deep learning with other computing methods that allow for more balanced datasets and, hence, better deep-learning performance [67].
- Digital twin and augmented reality. Creating virtual environments that, by recording, visualization, and interaction with cyber–physical assets, are capable of generating necessary tagged information in real time that is fed to deep-learning algorithms [68]. The creation of digital twins in combination with deep-learning algorithms was also proposed to enable the parallel control of cyber–physical value-creating processes [69].

- Graphwise classification. For instance, in the classification of molecules [77]. In this model, atoms represent the nodes, and chemical bonds are the edges of a graph. Research aims to extract certain features that predict certain properties of the molecule. This is relevant, for instance, to the pharmaceutical companies that are in the business of drug design. Some of these properties are toxicity and water solubility. Given a graph, researchers aim to classify a molecule graph. This is analogous to classical deep-learning-based visual image classification [78].
- Vertexwise classification. For example, in a social-network domain in which nodes are people of which we have certain demographic information, a researcher aims to predict how these people will vote in the next election. The analogy in computer vision is semantic image segmentation [79] in which the pixels of an image are labeled as belonging to a certain category.
- Graph dynamics. There are also domains that are described by fixed graphs, and others in which the graph changes with time [70]. Complex-networked cyber–physical systems belong to the second class.
- Known vs. unknown domain. In some cases, the graph can be known; in others, it is only partially known, noisy, or not known at all and needs to be learned. In these cases, the researcher aims to not only learn the graph features, but also the graph itself [80].

- Spectral filtering methods.Spectral filtering methods make use of the spectral eigendecomposition of the Laplacian graph to elegantly mathematically define convolution-like operators. The fundamental limitation of the spectral construction is that it can only be used to single and static domains. This is because filter coefficients are dependent on the eigenvector- and eigenvalue-decomposition basis of the Laplacian graph, which is highly dependent on network architecture [70]. This approach is not suitable for our needs because of the dynamic characteristics of Industry 4.0 lean-management cyber–physical complex systems and their associated complex networks.
- Local filtering methods.

## 4. Geometric Deep Lean Learning Over Industry 4.0 Lean-Management Complex-Networked Cyber–Physical Systems

- Strategic organizational design. Performing classical inference problems [76].Recently, it has been shown that this classification can be considerably improved by using information about the proximity environment [83,84]. Analyzing signals on graph vertices and edges could potentially help to learn inherent structures of the graphs, such as organizational clusters, with better accuracy than that provided by topological information alone—this is a strategic challenge to which organizational design tries to respond.
- Trust and power structures. Learning hidden organizational properties.Although deep learning has been employed in a wide variety of fields of knowledge, such as modeling social influence [85] and computer vision [86,87,88], it is important to incorporate knowledge about the domain to be treated in the model. For example, in order to build a deep-learning model for the study of a network of sensors in a cyber–physical system of industry 4.0, it might be useful, in a first approximation, to choose the edge weights of the graph as a decreasing function of the distance between nodes, as this would lead to a smooth graph signal model [89]; however, this would not be suitable for a lean structural network, because adjacency does not necessarily mean similarity [14]. For this reason, the model of the graph to be used can be superimposed on other structures, instead of being a pure unconnected abstraction. In other words, the graph that represents the complex-networked cyber–physical system in an Industry 4.0 context, can be studied from different perspectives, superimposing it to a specific sociotechnical environment that helps to better understand the statistical information that it contains. As a consequence, the integration of these priors is a fundamental challenge for the success of geometric deep lean learning. Some examples are the structures of power or trust between the different actors of an organization that are fundamental variables that influence the success of an organization, but remain elusive, since they often cannot be directly measured. Geometric deep lean learning could be applied to learn these parameters as weights between the nodes of the complex organizational network.

- Convolution on non-Euclidean complex-networked cyber–physical graph time-dependent signals.As expressed in Expression 1, for weighted time-dependent directed graph $\Omega \left(t\right)$, a series of signals $x=[x\left(1\right),\dots ,x\left(n\right)]\in \left({\mathbb{R}}^{n}x{\mathbb{R}}^{m}\right)$ expressed on its human and cyber–physical nodes, and on its standard communication edges, are considered, in which components of ${x}_{a}$ reside in or are protruding from node a.For each node, we define a proximity environment given by group ${N}_{a}=\{b:(b,a)\subset E\}$ that represents set of nodes b connected with a. This ${N}_{a}$ set is characterized by an ${\mathbb{R}}^{NxN}$ matrix S called the network-translation matrix operator that defines the manifold metric. We defined S as the graph adjacency matrix, the Laplacian of the graph, or any other normalization of it, as a linear transformation to encode the structure of a graph. Without loss of generality, the singularity problem of the adjacency matrix, which is nontrivial, was not considered in this work [91]. As shown in Figure 2, group ${N}_{a}$ represents the manifold upon which the convolution acts.The Fourier decomposition of graph $\Omega \left(t\right)$ is expressed by $\widehat{x}={U}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}x$, where $S=\mathcal{U}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\Lambda \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathcal{U}}^{-1}$ and autovalues $\Lambda $ describe the frequencies of the graph [92]. Now, we can directly filter x from the spectral domain by means of function $f:\mathbb{C}\to \mathbb{R}$ that allows to compute convolution $\widehat{z}=f(\Lambda )\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\widehat{x}$ by means of point-by-point multiplication in the spectral domain between filter $f(\Lambda )$ and the Fourier transform of the graph in x. Therefore, by inverting the Fourier transform of the graph, we obtain the extension of the convolutional operation to the non-Euclidean time-dependent graph in Equation (2).$$\begin{array}{c}\hfill z=\mathcal{P}\left(S\right)x\text{}\mathrm{and}\text{}\mathcal{P}\left(S\right)=\mathcal{U}f(\Lambda ){\mathcal{U}}^{-1}\end{array}$$The filter operation can be directly described on the node, resulting in an alternative formulation given by Equation (3), where scalar parameter ${\varphi}_{a,b}$ is a representation of the information weights coming from neighbour node b into or from node a.$$\begin{array}{c}\hfill {z}_{a}=\sum _{b\in {N}_{a}\cup a}^{}{\varphi}_{a,b}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{x}_{b}\end{array}$$Due to the local properties of S, ${z}_{a}$ can be obtained in the domain of the node through local-information exchange. This means that the initial signal on the node is recursively transformed by S a K number of times until decomposition is obtained that determines ${z}_{a}$ as the convolution between the network filter with a polynomial transfer function and ${x}_{b}$.By means of the Fourier transform of the network, the screening operation of Equation (3) has the transfer function given by Equation (4):$$\begin{array}{c}\hfill h(\Lambda )=\sum _{k=0}^{\kappa}{\varphi}_{k}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\Lambda}^{k}\end{array}$$This filter, based on local-information exchanges, captures information in K-radius proximity from the node representing the depth of the geometric-deep-lean-learning algorithm.Taking into account this convolutional operation given by Equation (3), we are able to compute the fth level feature produced as output of the lth layer:$$\begin{array}{c}\hfill {y}_{f}^{l}={\sigma}^{l}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\sum _{g=1}^{l-1}{\mathcal{P}}_{f,g}^{l}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{y}_{g}^{l-1}\end{array}$$
- -
- ${\sigma}^{l}$ represents the nonlinear activation function (i.e., ReLU); and
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- ${\mathcal{P}}_{f,g}^{l}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{y}_{g}^{l-1}$ indicates the graph structure relating the gth input ${y}_{g}^{l-1}$ to the fth output ${y}_{f}^{l}$.

Now, we simply combine two cases to model the mechanism of a convolutional network applied to a non-Euclidean graph in each time slot: the case in which edges vary, and that in which nodes vary. This can be combined into a single expression to describe ${\mathcal{P}}_{f,g}^{l}$ given by Equation (5):$$\begin{array}{c}\hfill {\mathcal{P}}_{f,g}^{l}\left(S\right)=\sum _{k=1}^{K}{{\rm Y}}_{f,g}^{l,(k:1)}+\sum _{k=0}^{K}\left(\right)open="("\; close=")">\prod _{m=0}^{k}{{\rm Y}}_{d}^{\left(m\right)}+{\varphi}_{k}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\tau}^{k}\end{array}$$- -
- ${\sum}_{k=1}^{K}{{\rm Y}}_{f,g}^{l,(k:1)}$ represents the edge-varying case, in which
- *
- ${{\rm Y}}_{f,g}^{l,(k:1)}$ acts as a shift operator, and therefore represents a learning paradigm for data embedded within complex graphs, whose weights are known to some degree of ambiguity, are only partially known, or are unknown.

- -
- ${\sum}_{k=0}^{K}\left(\right)open="("\; close=")">{\prod}_{m=0}^{k}{{\rm Y}}_{d}^{\left(m\right)}+{\varphi}_{k}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\tau}^{k}$ represents the node-varying case, in which
- *
- $d\subset E$ is a special set of nodes (i.e., nodes with a degree above a certain threshold, nodes with a certain level of hierarchy in the organization, or any other relevant feature),
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- ${\varphi}_{k}\in {[0,1]}^{Nxd}$ is a binary matrix, and
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- ${\tau}^{k}$ is a vector describing the node parameters in d.

- Pooling in non-Euclidean complex-networked cyber–physical graph time-dependent signals.As introduced earlier, downsampling pooling layers in classical deep-learning architectures that extract information from Euclidean domains such as speech, images, or videos typically report the maximal output within rectangular proximity [93]. In this way, it is possible to extract local characteristics that are shared by other areas of the images, thus considerably reducing the number of parameters that the deep network has to learn without sacrificing its learning capacity. Pooling can be described as a progressive coarsening of the graph. A simple way to do this is to collapse edges and reduce the size of the graph through a standard max-pooling operation on the nodes by just taking the maximum of each one of the feature tensors on each of the nodes being coarsened. This can be represented as a binary-tree structure of node indices. These pooling modules on graphs can be inserted between the convolutional modules in order to extract high-level graph representations, and thus be able to perform effective graph classification.Some alternatives in this field have not been to try to pool the whole network, but different hierarchies of the complex network in order to be able to learn which node groups have similar characteristics [94]. Once these groups are learned, clusters are made, and network pooling is carried out as described above or with an alternative method. This process is repeated for each of the network layers; thus, its classification is obtained. This presupposes, however, prior knowledge of the network structure.The extraction of shared local characteristics is not possible through this method in time-varying non-Euclidean domains, i.e., complex-networked cyber–physical graphs, because no stationarity or shift invariance can be found within these domains. Wu et al. [95], and Lee et al. [96] provided state-of-the-art surveying overview of this interesting open research question.

## 5. Conclusions and Management Implications

- Graphwise classification. The classification of complex cyber–physical graphs by deep lean learning, thus creating product families and allowing automated decision making in real time in which products are developed, produced, and channeled to the final customer.
- Vertexwise classification. The classification of certain crucial nodes in the value-creation process by means of deep-lean-learning models that allows an improvement of organizational design to assure an increase of overall process performance.
- Graph dynamics. Learning complex-networked cyber–physical graph dynamics is of great interest when dealing with change management within non-Euclidean sociotechnical systems.
- Known vs. unknown domains. The learning, generation, and semisupervised design of value streams by learning the most suitable complex cyber–physical graphs for certain types of products, thus potentially generating high customization with high efficiency and effectiveness in resource use.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IIoT | Industrial Internet of Things |

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**Figure 1.**Macroscopic, mesoscopic, and microscopic levels of organizational sociotechnical complexity.

Social | Technical | Socio Technical | |
---|---|---|---|

Micro | Imai, 2012 [26]; Stock and Seliger, 2016 [27] | Takeda, 2009 [28]; Francis and Bian, 2019 [29]; Jabeur et al., 2015 [17]; Li et al., 2017 [30]; Aazam et al., 2018 [31]; Tao et al., 2018 [32]; Mushtaq and Haq, 2019 [33]; Shevchik et al., 2019 [34]; Al-Jaroodi and Mohamed, 2019 [35]; Sun et al., 2019 [36] | Villalba-Diez et al., 2015 [23]; Villalba-Diez et al., 2019 [24]. |

Meso | Rother, 2010 [37]; Villalba et al., 2018 [13]; Birkel et al., 2019 [38] | Takeda, 2011 [39]; Davis et al., 2012 [40]; Gomez et al., 2015 [41]; Culot, 2019 [42]; Jimenez et al., 2016 [43]; Wang et al., 2018 [44]; Villalba-Diez et al., 2019 [25]; Jang et al., 2019 [45]; Ordieres-Mere et al., 2019 [46] | Villalba-Diez and Ordieres-Mere, 2015 [10]; Villalba-Diez et al., 2015 [9]; Villalba-Diez and Ordieres-Mere, 2016 [11]; Villalba-Diez et al., 2017 [47]; Davies et al., 2017 [48]; Kumar et al., 2019 [49]. |

Macro | Womack and Jones, 2003 [50]; Toyota, 2014 [51]; Burton et al., 2015 [52]; Covey, 2004 [53]; Rabelo et al., 2019 [54]; Romero et al., 2017 [55]; Wang et al., 2019 [56]; Guo and Jyang, 2019 [57] | Lee et al., 2015 [58]; Wang et al., 2015 [59]; Goodfellow et al., 2016 [5]; Sisini et al., 2018 [60]; Zheng et al., 2018a [61]; Lu and Xu, 2019 [62] | Stock and Seliger, 2016 [27]; Villalba-Diez, 2017 [15]; Villalba-Diez, 2017 [14]; Kiel et al., 2017 [63]; Stock et al., 2016 [64]; Shang et al., 2019 [65]. |

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**MDPI and ACS Style**

Villalba-Díez, J.; Molina, M.; Ordieres-Meré, J.; Sun, S.; Schmidt, D.; Wellbrock, W.
Geometric Deep Lean Learning: Deep Learning in Industry 4.0 Cyber–Physical Complex Networks. *Sensors* **2020**, *20*, 763.
https://doi.org/10.3390/s20030763

**AMA Style**

Villalba-Díez J, Molina M, Ordieres-Meré J, Sun S, Schmidt D, Wellbrock W.
Geometric Deep Lean Learning: Deep Learning in Industry 4.0 Cyber–Physical Complex Networks. *Sensors*. 2020; 20(3):763.
https://doi.org/10.3390/s20030763

**Chicago/Turabian Style**

Villalba-Díez, Javier, Martin Molina, Joaquín Ordieres-Meré, Shengjing Sun, Daniel Schmidt, and Wanja Wellbrock.
2020. "Geometric Deep Lean Learning: Deep Learning in Industry 4.0 Cyber–Physical Complex Networks" *Sensors* 20, no. 3: 763.
https://doi.org/10.3390/s20030763