A manifold learning approach to data-driven computational elasticity and inelasticity.

*(English)*Zbl 1390.74195Summary: Standard simulation in classical mechanics is based on the use of two very different types of equations. The first one, of axiomatic character, is related to balance laws (momentum, mass, energy,…), whereas the second one consists of models that scientists have extracted from collected, natural or synthetic data. Even if one can be confident on the first type of equations, the second one contains modeling errors. Moreover, this second type of equations remains too particular and often fails in describing new experimental results. The vast majority of existing models lack of generality, and therefore must be constantly adapted or enriched to describe new experimental findings. In this work we propose a new method, able to directly link data to computers in order to perform numerical simulations. These simulations will employ axiomatic, universal laws while minimizing the need of explicit, often phenomenological, models. This technique is based on the use of manifold learning methodologies,
that allow to extract the relevant information from large experimental datasets.

##### MSC:

74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |

74A20 | Theory of constitutive functions in solid mechanics |

68T05 | Learning and adaptive systems in artificial intelligence |

62-07 | Data analysis (statistics) (MSC2010) |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62P35 | Applications of statistics to physics |

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\textit{R. Ibañez} et al., Arch. Comput. Methods Eng. 25, No. 1, 47--57 (2018; Zbl 1390.74195)

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##### References:

[1] | Amsallem, D; Farhat, C, An interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA J, 46, 1803-1813, (2008) |

[2] | Brunton, SL; Proctor, JL; Kutz, JN, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc Natl Acad Sci, 113, 3932-3937, (2016) · Zbl 1355.94013 |

[3] | Darema, F, Grid computing and beyond: the context of dynamic data driven applications systems, Proc IEEE, 93, 692-697, (2005) |

[4] | González D, Aguado JV, Cueto E, Abisset-Chavanne E, Chinesta F (2016) kpca-based parametric solutions within the PGD framework. Arch Comput Methods Eng. doi:10.1007/s11831-016-9173-4 · Zbl 06887320 |

[5] | González, D; Cueto, E; Chinesta, F, Computational patient avatars for surgery planning, Ann Biomed Eng, 44, 35-45, (2015) |

[6] | Kirchdoerfer, T; Ortiz, M, Data-driven computational mechanics, Comput Methods Appl Mech Eng, 304, 81-101, (2016) |

[7] | Ladeveze, P, The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables, Comptes Rendus Académie des Sci Paris, 309, 1095-1099, (1989) · Zbl 0677.73060 |

[8] | Lee JA, Verleysen M (2007) Nonlinear dimensionality reduction. Springer, Berlin · Zbl 1128.68024 |

[9] | Liu, Z; Bessa, MA; Liu, WK, Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials, Comput Methods Appl Mech Eng, 306, 319-341, (2016) |

[10] | Lopez E, Gonzalez D, Aguado JV, Abisset-Chavanne E, Cueto E, Binetruy C, Chinesta F (2016) A manifold learning approach for integrated computational materials engineering. Arch Comput Methods Eng. doi:10.1007/s11831-016-9172-5 · Zbl 1390.74196 |

[11] | Michopoulos J, Farhat C, Houstis E (2004) Dynamic-data-driven real-time computational mechanics environment. In: Bubak M, van Albada GD, Sloot PMA, Dongarra J (eds) Computational science—ICCS 2004: 4th international conference, Kraków, Poland, June 6-9, 2004, proceedings, Part III, pp 693-700, Springer, Berlin |

[12] | Olson, GB, Designing a new material world, Science, 288, 993-998, (2000) |

[13] | Peherstorfer, B; Willcox, K, Dynamic data-driven reduced-order models, Comput Methods Appl Mech Eng, 291, 21-41, (2015) |

[14] | Peherstorfer, B; Willcox, K, Data-driven operator inference for nonintrusive projection-based model reduction, Comput Methods Appl Mech Eng, 306, 196-215, (2016) |

[15] | Polito M, Perona P (2001) Grouping and dimensionality reduction by locally linear embedding. In: Advances in neural information processing systems 14, pp 1255-1262. MIT Press · Zbl 1355.94013 |

[16] | Raghupathi, W; Raghupathi, V, Big data analytics in healthcare: promise and potential, Health Inf Sci Syst, 2, 1-10, (2014) |

[17] | Roweis, ST; Saul, LK, Nonlinear dimensionality reduction by locally linear embedding, Science, 290, 2323-2326, (2000) |

[18] | Tenenbaum, JB; Silva, V; Langford, JC, A global framework for nonlinear dimensionality reduction, Science, 290, 2319-2323, (2000) |

[19] | Wang Q (2012) Kernel principal component analysis and its applications in face recognition and active shape models. CoRR, abs/1207.3538 |

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