A new reliability-based data-driven approach for noisy experimental data with physical constraints

Abstract

Data Science has burst into simulation-based engineering sciences with an impressive impulse. However, data are never uncertainty-free and a suitable approach is needed to face data measurement errors and their intrinsic randomness in problems with well-established physical constraints. As in previous works, this problem is here faced by hybridizing a standard mathematical modeling approach with a new data-driven solver accounting for the phenomenological part of the problem, with the aim of finding a solution point, satisfying some constraints, that minimizes a distance to a given data-set. However, unlike such works that are established in a deterministic framework, we use the Mahalanobis distance in order to incorporate statistical second order uncertainty of data in computations, i.e. spread and correlations. We develop the underlying stochastic theoretical framework and establish the fundamental mathematical and statistical properties. The performance of the resulting reliability-based data-driven procedure performance is evaluated in a simple but illustrative unidimensional problem as well as in a more realistic solution of a 3D structural problem with a material with intrinsically random constitutive behavior as concrete. The results show, in comparison with other data-driven solvers, better * Corresponding author Email address: mdoblare@unizar.es (Manuel Doblaré) convergence, higher accuracy, clearer interpretation, and major flexibility besides the relevance of allowing uncertainty management, with low computational demand.

A physics-constrained data-driven approach based on locally convex reconstruction for noisy database

“…From the pure data-driven approach point of view (refer to [54]), LCDD is inspired by measuring the distance to a local convex set instead of from a single discrete data, aim to enhance the robustness against noise and prevent undesirable local minima. From the fitted datadriven (or linearization) approach point of view, on the other hand, LCDD relies on the approximation of locally linear material graph by the manifold learning methodologies [21,63] to capture the global structure via local data information.…”

“…where 1 {} m V  = are the quadrature weights associated with the m integration points, and . One approach for selecting the weighted coefficients is by computing the covariance of the material data set and using the so-called Mahalanobis distance for multivariate data, as proposed in [54]. Investigating the effect of weighted coefficient is out of the scope of this study.…”

“…This is due to fact that when the direction of gradient descent search (linearly scaled with M in this example) associated with certain data-driven iterations is nearly normal to the underlying material graph, the resulting displacement increment driven by the force residual (see (17a)) becomes negligible and would not change the computational state sufficiently to approach to other data points in closer to the physical manifold, and thus the data-driven iterations stop at this undesirable local data. This issue is attributed to the non-continuous nature of discrete data, resulting in the susceptibility of data-driven solvers to the selection of weighted coefficient M , the associated metric norm used to measure distance in the phase space, and the density and the underlying structure of data [54,60].…”

“…But the resulting optimization problem becomes intractable and need to employ simulated annealing schedule. Alternatively, Ayensa-Jimenez et al [54] modified the standard DMDD approach by explicitly incorporating statistical quantities into the cost function and defined a stochastic analogous problem.…”

Learning Corrections for Hyperelastic Models From Data

“…In the field of computational materials science, this approach seems to begin by the works of Ortiz (2016, 2017a). In it, and the subsequent works, they present a method in which the constitutive equation is substituted by experimental data, that could be possibly noisy (Kirchdoerfer and Ortiz, 2017b;Ayensa-Jiménez et al, 2018). In them, it is recognized that some equations (notably, equilibrium, compatibility) are of a higher epistemic nature, while constitutive equations-that are often phenomenological and, therefore, of lower epistemic valuecould easily be replaced by data (Latorre and Montáns, 2014).…”