Model-Free Data-Driven Inference in Computational Mechanics

Abstract

We extend the model-free Data-Driven computing paradigm to solids and structures that are stochastic due to intrinsic randomness in the material behavior. The behavior of such materials is characterized by a likelihood measure instead of a constitutive relation. We specifically assume that the material likelihood measure is known only through an empirical point-data set in material or phase space. The state of the solid or structure is additionally subject to compatibility and equilibrium constraints. The problem is then to infer the likelihood of a given structural outcome of interest.In this work, we present a Data-Driven method of inference that determines likelihoods of outcomes from the empirical material data and that requires no material or prior modeling. In particular, the computation of expectations is reduced to explicit sums over local material data sets and to quadratures over admissible states, i. e., states satisfying compatibility and equilibrium. The complexity of the material data-set sums is linear in the number of data points and in the number of members in the structure. Efficient population annealing procedures and fast search algorithms for accelerating the calculations are presented. The scope, cost and convergence properties of the method are assessed with the aid selected applications and benchmark tests.

Convergence rates for ansatz‐free data‐driven inference in physically constrained problems

“…We do not consider the definition of the constraint set 𝐸 as a modeling step given that it encodes the governing physical laws. Datadriven inference rules such as ( 7) are amenable to efficient numerical implementation in combination with stochastic quadrature formulas for the evaluation of the integrals [5].…”

“…In this paper, we work within a general framework [4] for systems in which the material law and the admissibility constraints are described by positive Radon likelihood measures 𝜇 𝐷 ∈ (𝑍) and 𝜇 𝐸 ∈ (𝑍), respectively, representing the likelihood of 𝑦 ∈ 𝑍 being a (local) material state observed in the laboratory and of 𝑧 ∈ 𝑍 being admissible. The goal of this line of work is to advance the theory for this new framework, as this setting is opening doors to tackle problems that are not accessible with more classical approaches, such as the practical problem addressed in [5] which poses challenges for example if addressed with a Bayesian approach. Before presenting our new contributions, the main ideas underlying the work may be summarized as follows.…”

Convergence rates for ansatz-free data-driven inference in physically constrained problems

“…We remark that ( 13) is the narrow convergence normally used for probability measures, which is stronger than weak convergence of finite Radon measures usually defined testing with f ∈ C 0 (R n ). Condition (13) in particular (testing with f = 1) implies µ k (R n ) → µ(R n ). Indeed, it is possible to show that, for nonnegative measures, condition ( 13) is equivalent to (12) and…”

“…We do not consider the definition of the constraint set E as a modelling step given that it encodes the governing physical laws. Data-driven inference rules such as (7) are amenable to efficient numerical implementation in combination with stochastic quadrature formulas for the evaluation of the integrals [13].…”

“…This epistemic bottleneck notwithstanding, we note that the convergence of the thermalized approximations µ h,β h can in principle be assessed by determining if the sequence is Cauchy, since that criterion does not require explicit knowledge of the limit. A detailed assessment of matters of implementation is beyond the scope of the present paper and may be found in [13].…”