Model-Free and Prior-Free Data-Driven Inference in Mechanics

Conti, Sergio; Hoffmann, Franca; Ortiz, Michael

doi:10.1007/s00205-022-01836-7

Cited by 3 publications

(13 citation statements)

References 26 publications

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“…Our aim is to obtain quantitative estimates for the convergence of 𝜇 𝛽 to its limit 𝜇 ∞ and the convergence of 𝜇 ℎ,𝛽 to 𝜇 𝛽 , leading in particular to a prescription for the choice 𝛽 ℎ which ensures the desired convergence 𝜇 ℎ,𝛽 ℎ → 𝜇 ∞ . In order to make convergence quantitative, we work in a metric setting and not only in terms of weak convergence as in [4]. Our starting point is this observation:…”

Section: Resultsmentioning

confidence: 99%

“…We adopt, in this metric setting, the notion of transversality and of diagonal concentration for (possibly unbounded) measures via thermalization as developed in [4]. We introduce a weaker concept, weak transversality, which corresponds to transversality along subsequences, see Definition 3.1, and circumvents the need for regularity assumptions on the measure 𝜇.…”

Section: Resultsmentioning

confidence: 99%

“…(ii) By Prokhorov's theorem, if a sequence 𝜇 𝛽 ℎ is uniformly bounded and uniformly tight then it has a subsequence which converges narrowly and 𝜇 is weakly transversal. (iii) If a diagonal concentration exists, then it is supported on diag(𝑍 × 𝑍) (see [4,Lemma 4.3]). (iv) If 𝜇 𝛽 is uniformly bounded, uniformly tight and Cauchy with respect to the flat norm, then it is strongly transversal.…”

Section: General Settingmentioning

confidence: 99%

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

2023

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We study a Data‐Driven approach to inference in physical systems in a measure‐theoretic framework. The systems under consideration are characterized by two measures defined over the phase space: (i) A physical likelihood measure expressing the likelihood that a state of the system be admissible, in the sense of satisfying all governing physical laws; (ii) A material likelihood measure expressing the likelihood that a local state of the material be observed in the laboratory. We assume deterministic loading, which means that the first measure is supported on a linear subspace. We additionally assume that the second measure is only known approximately through a sequence of empirical (discrete) measures. We develop a method for the quantitative analysis of convergence based on the flat metric and obtain error bounds both for annealing and the discretization or sampling procedure, leading to the determination of appropriate quantitative annealing rates. Finally, we provide an example illustrating the application of the theory to transportation networks.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: General Settingmentioning

confidence: 99%

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

2023

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“…In the case of non-increasing accuracy, measurements for a given strain rate ∈ R d×d might be located in a neighbourhood of the exact value with a certain likelihood. In this case, the set of data converges in a weak sense to some distribution, see [4]. See also [28] for the analysis of single outliers in measurements.…”

Section: A Variational Data-driven Approachmentioning

confidence: 99%

“…In Sect. 4 we check the conditions (H2)-(H4) on the integrand f. To verify (H1), for a given sequence v n we need to construct a suitable ( p, q)-equi-integrable modification w n that conserves both the differential constraints and the boundary conditions. For this purpose we need the following two auxiliary results: Lemma 3.7.…”

Section: Proof (I)mentioning

confidence: 99%

A Data-Driven Approach to Viscous Fluid Mechanics: The Stationary Case

Lienstromberg

Schiffer

Schubert

2023

Arch Rational Mech Anal

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We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81–101, 2016; Conti et al. in Arch Ration Mech Anal 229:79–123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and $${\mathscr {A}}$$ A -quasiconvexity, we show a $$\Gamma $$ Γ -convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.

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