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.

Engineering is evolving in the same way than society is doing. Nowadays, data is acquiring a prominence never imagined. In the past, in the domain of materials, processes and structures, testing machines allowed extract data that served in turn to calibrate state-of-the-art models. Some calibration procedures were even integrated within these testing machines. Thus, once the model had been calibrated, computer simulation takes place. However, data can offer much more than a simple state-of-the-art model calibration, and not only from its simple statistical analysis, but from the modeling and simulation viewpoints. This gives rise to the the family of so-called twins: the virtual, the digital and the hybrid twins. Moreover, as discussed in the present paper, not only data serve to enrich physically-based models. These could allow us to perform a tremendous leap forward, by replacing big-data-based habits by the incipient smart-data paradigm.

International audienceThe use of constitutive equations calibrated from data has been implemented into standard numerical solvers for successfully addressing a variety problems encountered in simulation-based engineering sciences (SBES). However, the complexity remains constantly increasing due to the need of increasingly detailed models as well as the use of engineered materials. Data-Driven simulation constitutes a potential change of paradigm in SBES. Standard simulation in computational 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, either natural or synthetic, data. Data-driven (or data-intensive) simulation consists of directly linking experimental data to computers in order to perform numerical simulations. These simulations will employ laws, universally recognized as epistemic, while minimizing the need of explicit, often phenomenological, models. The main drawback of such an approach is the large amount of required data, some of them inaccessible from the nowadays testing facilities. Such difficulty can be circumvented in many cases, and in any case alleviated, by considering complex tests, collecting as many data as possible and then using a data-driven inverse approach in order to generate the whole constitutive manifold from few complex experimental tests, as discussed in the present work

Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well-known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low-data limit. We provide examples on the performance of the technique in up to ten dimensions.