Numerical experiments on unsupervised manifold learning applied to mechanical modeling of materials and structures

Comptes Rendus Mécanique Ruben Ibanez, Pierre Gilormini, Elias Cueto and Francisco Chinesta Numerical experiments on unsupervised manifold learning applied to mechanical modeling of materials and structures Volume 348, issue 10-11 (2020), p. 937-958. <https://doi.org/10.5802/crmeca.53> Part of the Thematic Issue: Contributions in mechanics of materials Guest editors: Julie Diani, Olivier Castelnau and Francisco Chinesta © Académie des sciences, Paris and the authors, 2020. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0International License. http://creativecommons.org/licenses/by/4.0/ Les Comptes Rendus. Mécanique sont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Comptes Rendus Mécanique 2020, 348, n10-11, p. 937-958 https://doi.org/10.5802/crmeca.53 Contributions in mechanics of materials Numerical experiments on unsupervised manifold learning applied to mechanical modeling of materials and structures Ruben Ibaneza, Pierre Gilorminia, Elias Cuetoband Francisco Chinesta∗,a aPIMM lab, Arts et Metiers Institute of Technology, 151 Boulevard de Hôpital, 75013 Paris, France bAragon Institute of Engineering Research, Universidad de Zaragoza, Maria de Luna s/n, 50018 Zaragoza, Spain E-mails: [email protected] (R. Ibanez), pierr[email protected] (P. Gilormini), [email protected] (E. Cueto), francisco[email protected] (F. Chinesta) Abstract. The present work aims at analyzing issues related to the data manifold dimensionality. The interest of the study is twofold: (i) first, when too many measurable variables are considered, manifold learning is expected to extract useless variables; (ii) second, and more important, the same technique, manifold learning, could be utilized for identifying the necessity of employing latent extra variables able to recover singlevalued outputs. Both aspects are discussed in the modeling of materials and structural systems by using unsupervised manifold learning strategies. Keywords. Nonsupervised manifold learning, State variables, Dimensionality reduction, k-PCA, Structural analysis, Material constitutive equations. Manuscript received 21st June 2020, revised 12th July 2020, accepted 7th October 2020. 1. Introduction Recently, data-driven description of materials has been gaining popularity. Many complex material behaviors resisting traditional modeling procedures, or that are too complex from a microstructural viewpoint, are approached by using data-based descriptions. Different approaches are being considered. Among them include those based exclusively on measured data, others that extract the manifolds related to data, and others that attempt to enforce thermodynamic and thermomechanical consistency. The interested reader can refer to [1–7] and the numerous references therein. This work focuses on techniques based on the use of manifolds and their associated manifold learning procedures for extracting them from the available data. ∗Corresponding author. ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/ 938 Ruben Ibanez et al. Figure 1. Multidimensional data on one- (left), two- (center), and three-dimensional (right) manifolds embedded in RD. In general, data involve many dimensions. Consider first a sequence of three-dimensional (3D) fields defined in a domain Ω⊂R3partitioned into Dvoxels. Each of these fields contains many data, one datum at each voxel. Each field can be represented as a point in a vector space of dimension D(the number of voxels), where we can presume each of the Dcoordinate axes as reporting the value that the field of interest takes in the associated voxel. Thus, each field becomes a point in that high-dimensional space of dimension D,RD. If important correlations exist among the different fields, these points are expected to be distributed on a low-dimensional subspace embedded in the D-dimensional space. Techniques aiming at extracting these reduced subspaces, the so-called slow manifolds, sketched in Figure 1, are key tools for manipulating data and extracting their hidden information. Thus, data define in general slow manifolds embedded in very large vector spaces due to the significant hidden correlations among them. The number of uncorrelated explicative dimensions usually becomes much smaller than the a priori assumed dimension of the space for accommodating the data. The extraction of these slow manifolds can be successfully accomplished by using linear and nonlinear dimensionality reduction techniques such as principal component analysis (PCA) in the linear case and its nonlinear counterparts (`PCA, kernel-based PCA [k-PCA], LLE, tSNE, etc.) [8–12]. These techniques can be applied to several physical systems. Moreover, when the slow manifold is determined, the solution at any point on it can be computed very accurately from a simple interpolation of neighboring data on the manifold [13], enabling almost real-time predictions and the associated real-time decision-making. However, extracting knowledge from data associated with an existing but hidden model requires the following: •identifying the manifold intrinsic dimension, •discovering hidden parameters, •discarding useless parameters, and •discovering the models originating the data. These questions are addressed, illustrated, and discussed in the present work in a purely methodological manner, aiming at illustrating the key concepts that could open numerous future possibilities in the field of mechanics of materials, processes, structures, and systems. In our previous works, we addressed problems involving thousands of dimensions [13–16], proving that despite the apparent richness in many cases, the embedding regards a relatively low-dimensional space. However, in most of the problems that we have treated until now, their complexity prevented fine analyses of their solutions. The present paper, which is purely methodological, considers simple problems defined in low dimensions, with known solutions C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 939 being easily visualizable, for facilitating an analysis and discussion. Of course, and as proved in the works just referred to, all the methodologies apply to multidimensional settings. 2. Unsupervised manifold learning Let us consider a vector y∈RDcontaining experimental or synthetic data from measurements or numerical simulation. These results are often referred to as snapshots. If they are obtained by numerical simulation, they consist of nodal values of the essential variable. Therefore, these variables will be somehow correlated and, notably, there will be a linear transformation W defining the vector ξ∈Rd, with d<D, which contains the still unknown latent variables such that y=Wξ. (1) The D×dtransformation matrix W, which satisfies the orthogonality condition WTW=Id, is the main ingredient of the PCA and can be computed as detailed in Appendix A from the covariance matrix associated with a number (M) of snapshots y1,...,yM, which constitute the columns of matrix Y. While PCA works with the covariance matrix (i.e., YYT), multidimensional scaling (MDS) works with the Gram matrix containing scalar products (i.e., S=YTY) as described in Appendix A. On the other hand, the k-PCA is based on the fact that data not linearly separable in D dimensions could be linearly separated if previously projected to a space in Q>Ddimensions. However, the true advantage arises from the fact that it is not necessary to write down the analytical expression of that mapping as described in Appendix A. 3. An illustrative structural mechanics case study Consider first a hypothetical mechanical system consisting of a prismatic beam whose three dimensions, height, width, and length, are denoted, respectively, by h,b, and L, all of them being measurable quantities. In what follows, we consider a particular output Pthat constitutes also a measurable quantity (buckling critical load, etc.) assumed related to those parameters from an existing but actually hidden model even if in what follows we will consider hypothetical, and most of the time, unphysical models. Thus, we consider a set of data composed of Mmeasures yi={hi,bi,Li,Pi}, i=1,...,M, with hi,bi, and Libeing randomly chosen from a uniform probability distribution in their respective intervals of existence, Ih,Ib, and IL, respectively, defined from the following:    Ih=[hmin,hmax] Ib=[bmin,bmax] IL=[Lmin,Lmax]. (2) Without any other pre-existing knowledge, one expects the output depending on the three geometrical parameters (i.e., P=P(h,b,L)). In what follows, we consider three different scenarios. 3.1. Output depending on a single parameter In this section, we assume a quite simple model that relates the output to a single parameter, P=αb,α∈R+, (3) where α=1000 in the numerical tests carried out. We perform M=1000 measures, where Ih∈[0.1,0.2], Ib∈[0.15,0.2], and IL∈[1,1.5]. C. R. Mécanique,2020, 348, n10-11, 937-958 940 Ruben Ibanez et al. Figure 2. Eigenvalues λ∈[λ110−6,λ1]. The measures constitute a set of M=1000 points in R4on which the k-PCA is applied by using the Gaussian kernel κ(yi,yj)=exp− kyi−yjk2 2β2, (4) where β=10. Figure 2 depicts the highest eigenvalues among Mresulting from the k-PCA, those lying between the highest value λ1and 10−6λ1. The slow manifold associated with ξis represented by selecting the first three reduced coordinates (ξ1,ξ2,ξ3) as shown in Figure 3, where its one-dimensional (1D) intrinsic dimension is noted. This result was expected from the considered model expressed by (3). The points on the slow manifold are colored depending on the values of h,b,L, and P, evidencing that bconstitutes the latent variable and that the output Pscales (visually) linearly with it. The process of coloring the data points in the embedded manifold deserves some additional comments due to the fact that this is used in all the analyses reported in the present paper. Manifold learning techniques look for a low-dimensional manifold defined by data points. As soon as the slow manifold is extracted, the different data points can be mapped on it. This visualization is only possible when the number of dimensions allows a simple graphical representation (as is the case for the problems addressed in the present paper). Then, these points can be colored depending on the value of the different initial coordinates, and one expects that if there is a correlation (direct or inverse and linear or nonlinear) between the initial and reduced coordinates, the colors must exhibit a certain grading. Even if this analysis seems quite dependent on the low dimensionality of the embedding, in higher dimensional embeddings, the analysis can be performed by using local statistics. Thus, by considering a data point in the slow manifold and its closest neighbors, a local statistical analysis can be easily performed with the standard deviation indicating the dispersion of the data (equivalent to the local dispersion of colors). One could be slightly surprised by the nonlinearity that the manifold exhibits despite the linearity of model (3). This nonlinearity is an artifact of the nonlinear kernel (4) used. As the model is linear, one could expect the ability of the PCA to address the problem at hand. For this purpose, it suffices transforming the kernel into its linear counterpart, giving rise to the PCA κ(yi,yj)=yi·yj. (5) C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 941 Figure 3. Slow manifold in the 3D space defined by the first reduced coordinates (ξ1,ξ2,ξ3). Each point is colored according to the value of the coordinate h(top left), b(top right), L (bottom left), or P(bottom right). Figure 4. Slow manifold ξwhen considering the PCA linear dimensionality reduction. Each point is colored according to the value of the variable b. In this case, as expected, a single nonzero eigenvalue results, and consequently the dimension of the reduced space becomes one (i.e., ξ=ξ). The associated manifold is depicted in Figure 4. Now, we consider a slightly different model, again depending on a single variable but in a nonlinear manner, according to P=αh2, (6) C. R. Mécanique,2020, 348, n10-11, 937-958 942 Ruben Ibanez et al. Figure 5. Slow manifold in the 3D space defined by the first three reduced coordinates (ξ1,ξ2,ξ3). Each point is colored according to the value of the coordinate h(left) or P(right). Figure 6. Slow manifold in the 3D space defined by the first three reduced coordinates (ξ1,ξ2,ξ3). Each point is colored according to the value of the coordinate L(left) or P(right). where again α=1000. Figure 5 depicts the 1D slow manifold, where points are colored according to the values of the variables hand P. Here, even if the direct relation can be noted, its nonlinearity is much less evident to visualize. Finally, we consider the model P=α L2, (7) where α=1000. Figure 6 depicts the 1D slow manifold, where points are colored according to the values of the variables Land Pto emphasize the inverse relation between them. 3.2. Output depending on two parameters In this section, we consider a model involving two of the three variables, in particular, P=αh3 L2, (8) where α=1000. Figure 7 depicts the two-dimensional (2D) slow manifold, where points are colored according to the values of variables h,b,L, and P. Here, the direct and inverse effects of hand Lwith respect to Pcan be noted as well as the fact that parameter bseems, and in fact is, useless. C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 943 Figure 7. Slow manifold in the 3D space defined by the first three reduced coordinates (ξ1,ξ2,ξ3). Each point is colored according to the value of the coordinate h(top left), b(top right), L(bottom left), or P(bottom right). 3.3. Identifying hidden variables The previous case studies revealed the ability to extract the intrinsic dimensionality of the slow manifold as well as the possibility to identify useless parameters. The present case addresses a very different situation in which the model involves the three variables in the discussed case, h, b, and L. However, only two of them were measured, namely hand Lwith the output P, with b remaining inaccessible. Thus, we have P=αbh3 L2, (9) where α=1000. Therefore, the M=1000 collected data yi,i=1,...,M, reads as yi={hi,Li,Pi}∈R3. Figure 8 depicts the reduced points (ξ1,ξ2,ξ3), which as can be seen are distributed in a domain ω⊂R3. However, no dimensionality reduction is noted, and the embedding remains 3D. A direct consequence is that many values of the output Pexist for the same values on the measured inputs hand L, related to the different values of b, which affect the output P. However, as bis not measured, its value is not considered in the data points. Such a multivalued output does not represent any conceptual difficulty. It indicates that even if both variables participate in the output, there may be others that were not considered or those considered may be useless for explaining the output. C. R. Mécanique,2020, 348, n10-11, 937-958 944 Ruben Ibanez et al. Figure 8. Reduced representation ξi∈R3of the data yi∈R3, where the reduced points are colored according to the value of the output P. Figure 9. Reduced points colored according to the values of the coordinates h(left) and L (right). To conclude about the pertinence of these variables with respect to the considered output, we consider coloring the reduced data depending on the hand Lvalues (refer to Figure 9). We compare them with the one where the color scales with the output Preported in Figure 8. Thus, one could conclude that both variables hand Lare relevant for explaining the output P. If we assume that the output Pshould be univocally explained from a small number of variables, clearly only two variables (here hand L) are not sufficient. One extra dimension suffices for recovering a single-valued output, that is, considering the reduced points in four dimensions R4. As visualizing things in four dimensions is quite a difficult task, for the sake of clarity in the exposition, in what follows, we propose addressing a simpler model involving lower dimensional spaces. We consider the simpler model P=αbh3, (10) where α=1000. However, the M=1000 collected data yi,i=1,...,M, only deal with hand P, (i.e., yi={hi,Pi}∈R2). Figure 10 depicts the dataset yi={hi,Pi}, where it can be noted that many values of the output Pare found for the same value of the variable h. For univocally expressing the output P, we consider again the k-PCA. We compute the reduced dataset in a 3D space by considering the first three coordinates (ξ1,ξ2,ξ3), while coloring these C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 951 Figure 22. Slow manifold related to linear elastic behavior and colored according to the tangent modulus (i.e., ∆σi/∆εi). Figure 23. Slow manifold related to nonlinear elastic behavior and colored according to the tangent modulus (i.e., ∆σi/∆εi). instantaneous tangent modulus when loading and unloading. As Figure 26 reveals, the mechanical manifold is now richer with points on the elastic domain boundary separated from the elastic manifold. In the solution depicted in Figure 26, damage is not activated. In the presence of damage, the reduction in elastic tangent modulus scales with the plastic strain. The slow manifold is colored according to the tangent modulus as shown in Figure 27. •The last scenario adds an extra richness to the constitutive behavior. At the mechanical states within the elastic domain, the elastic tangent modulus is affected by a latent C. R. Mécanique,2020, 348, n10-11, 937-958 952 Ruben Ibanez et al. Figure 24. Strain–stress points associated with states within the elastic domain with nonzero plastic strains (left) and the associated slow manifold colored according to the tangent modulus (right). Figure 25. Slow manifold related to damageable elastic–plastic behavior operating within the elastic domain colored according to the tangent modulus. variable that is the product of the plastic strain with another extra variable. The latter, which could represent strain-rate sensitivity, is assumed to take arbitrary values here due to the fact that we are more interested in methodological aspects than in physical considerations. The considered mechanical states are depicted in Figure 28. When applying the k-PCA to the set of mechanical states yi={εi,σi,∆σi/∆εi}, the resulting manifold remains 3D. As expected, no dimensionality reduction is accomplished as Figure 29 reveals, where many values of the elastic tangent modulus can be found for the same values of the stress and strain. To investigate the nature of this behavior, we depict in Figure 30 the elastic tangent modulus versus the plastic strain, where as expected it can be noted that the former does not depend exclusively on the latter. By applying the k-PCA to the data shown in Figure 30, which are nonseparable in two dimensions, one expects C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 953 Figure 26. Slow manifold related to elastic–plastic behavior operating within the elastic domain and on the elastic domain boundary colored according to the tangent modulus. Figure 27. Slow manifold related to elastic–plastic behavior operating within the elastic domain and on the elastic domain boundary, colored according to the tangent modulus, when damage is activated. to separate them by embedding in a 3D space as previously discussed and as Figure 31 proves. Finally, Figure 32 presents the slow manifold from Figure 31 but now colored with respect to the plastic strain or the extra latent variable. C. R. Mécanique,2020, 348, n10-11, 937-958 954 Ruben Ibanez et al. Figure 28. Mechanical states within the elastic domain, colored according to the tangent modulus, the last depending on the product of two latent variables, the plastic strain, and another arbitrarily chosen variable. Figure 29. Slow manifold related to elastic–plastic behavior operating within the elastic domain, colored according to the tangent modulus, with the last scaling with the product of the plastic strain and an extra latent variable. Figure 30. Elastic tangent modulus versus the plastic strain. C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 955 Figure 31. Slow manifold related to the data consisting of the elastic tangent modulus and the plastic strain. Figure 32. Slow manifold of the elastic tangent modulus colored according to the plastic strain (left) and the extra latent variable (right). 6. Conclusion The present work introduced, tested, and discussed issues related to manifold dimensionality with two major purposes: (i) first, when too many measurable variables are employed, manifold learning is able to discard the useless variables; (ii) second and more important, the same technique can be employed for discovering the necessity of employing and then measuring an extra latent variable that is able to recover and ensure single-valued outputs. Both issues were analyzed and discussed in two case studies, one with respect to structural mechanics and the other with respect to path-dependent material constitutive behaviors. Indeed, the physical interpretation of the discovered latent variable could imply the introduction of other measurable variables. This topic should be deeply analyzed. The main interest in discovering these manifolds is that for a new accessible mechanical state, the output can be inferred by a simple interpolation from its neighbors on the manifold. The main aim of the present work is the construction and analysis of these manifolds. Future works, currently in progress, will focus on their use in performing data-driven simulations. Conflicts of interest The authors declare no competing financial interests. C. R. Mécanique,2020, 348, n10-11, 937-958 956 Ruben Ibanez et al. Dedication This manuscript was written with the help of the contributions of all authors. All authors have given approval to the final version of the manuscript. Acknowledgments The first, third, and fourth authors are supported by their respective ESI Group research chairs; their support is gratefully acknowledged. The first author is supported by CREATE-ID ESI-ENSAM research chair. The third author is supported by the ESI Group Chair at the University of Zaragoza. The fourth author is supported by CREATE-ID ESI-ENSAM research chair. Appendix A. From principal component analysis to its kernel-based counterpart A.1. Principal component analysis Let us consider a vector y∈RDcontaining experimental results or synthetic data from a numerical simulation. These results are often referred to as snapshots. If they are obtained by numerical simulation, they consist of nodal values of the essential variable. Therefore, these variables will be somehow correlated and, notably, there will be a linear transformation Wdefining the vector ξ∈Rd, with d<D, which contains the still unknown latent variables, such that y=Wξ. (11) The D×dtransformation matrix W, which satisfies the orthogonality condition WTW=Id, is the main ingredient of the PCA [8]. Assume that there exist Mdifferent snapshots y1,...,yM, which we store in the columns of a D×M matrix Y. The associated d×Mreduced matrix Ξcontains the associated vectors ξi,i=1,...,M. The PCA usually works with centered variables. In other words,          M X i=1 yi=0, M X i=1 ξi=0, (12) implying the necessity of centering data before applying the PCA. The PCA proceeds by guaranteeing maximal preserved variance and minimal correlation in the latent variable set ξ. The latent variables in ξare therefore uncorrelated, and consequently the covariance matrix of ξ, Cξξ =E{ΞΞT}, (13) should be diagonal. To extract the duncorrelated latent variables, we proceed from Cyy =E{YYT}=E{WΞΞTWT}=WE{ΞΞT}WT=WCξξWT. (14) Preand post-multiplying by WTand W, respectively, and making use of the fact that WTW=I, give us Cξξ =WTCy y W. (15) The covariance matrix Cy y can then be factorized by applying the singular value decomposition, Cyy =VΛVT, (16) C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez et al. 957 where Vcontains the orthonormal eigenvectors; Λis a diagonal matrix containing the eigenvalues sorted in descending order. Substituting (16) into (15), we arrive at Cξξ =WTVΛVTW. (17) This equality holds when the dcolumns of Ware taken to be collinear with dcolumns of V. We then preserve the eigenvectors associated with the dnonzero eigenvalues, W=VID×d, (18) which gives Cξξ =Id×DΛID×d. (19) We therefore conclude that the eigenvalues in Λrepresent the variance of the latent variables (diagonal entries of Cξξ). A.2. Multidimensional scaling The PCA works with the covariance matrix of the experimental results, YYT. However, the MDS works with the Gram matrix containing scalar products (i.e., S=YTY) [8]. The MDS preserves pairwise scalar products: S=YTY=ΞTWTWΞ=ΞTΞ. (20) Computing the eigenvalues of S, we arrive at S=UΛUT=(UΛ1/2)(Λ1/2UT)=(Λ1/2UT)T(Λ1/2UT), (21) which in turn gives Ξ=Id×MΛ1/2UT. (22) A.3. Kernel-based principal component analysis The k-PCA is based on the fact that data not linearly separable in Ddimensions could be linearly separated if they are previously projected to a space in Q>Ddimensions. However, the true advantage arises from the fact that it is not necessary to write down the analytical expression of that mapping. The symmetric matrix Φ=ZTZ, with Zcontaining the snapshots zi∈RQ,i=1,...,M, associated with yi∈RD, has to be decomposed into eigenvalues and eigenvectors. The procedure for centering data ziis carried out in an implicit way. 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