Model-free Data-Driven Inference

Abstract

We present a model-free data-driven inference method that enables inferences on system outcomes to be derived directly from empirical data without the need for intervening modeling of any type, be it modeling of a material law or modeling of a prior distribution of material states. We specifically consider physical systems with states characterized by points in a phase space determined by the governing field equations. We assume that the system is characterized by two likelihood measures: one µD measuring the likelihood of observing a material state in phase space; and another µE measuring the likelihood of states satisfying the field equations, possibly under random actuation. We introduce a notion of intersection between measures which can be interpreted to quantify the likelihood of system outcomes. We provide conditions under which the intersection can be characterized as the athermal limit µ∞ of entropic regularizations µ β , or thermalizations, of the product measure µ = µD × µE as β → +∞. We also supply conditions under which µ∞ can be obtained as the athermal limit of carefully thermalized (µ h,β h ) sequences of empirical data sets (µ h ) approximating weakly an unknown likelihood function µ. In particular, we find that the cooling sequence β h → +∞ must be slow enough, corresponding to quenching, in order for the proper limit µ∞ to be delivered. Finally, we derive explicit analytic expressions for expectations E[f ] of outcomes f that are explicit in the data, thus demonstrating the feasibility of the model-free data-driven paradigm as regards making convergent inferences directly from the data without recourse to intermediate modeling steps.

Model-Free Data-Driven Inference in Computational Mechanics

“…The convergence properties of the method with respect to the empirical data have been assessed with the aid of a standard verification test based on Gaussian material data. Robust convergence of posterior distributions is obtained, consistent with quantitative error estimates derived from analysis [10,16]. It bears emphasis that, in these tests and in all subsequent calculations, the empirical material-data sets are the only input to the calculations and, once generated, the underlying distribution whence they are sampled is discarded altogether and plays no subsequent role.…”

“…The convergence properties of the method with respect to the empirical data are assessed with the aid of selected applications and benchmark tests, including a standard verification test based on Gaussian material data, brittle materials exhibiting random tensile strength, and a simple lightweight space structure. Robust convergence of posterior distributions is obtained in these tests, consistent with quantitative error estimates derived from analysis [10,16]. The ability of the proposed Data-Driven method of inference to deal effectively with general material data sets and complex behavior without need for models, hypotheses or assumptions is quite remarkable.…”

“…In this work, we present an alternative method of inference that determines likelihoods of outcomes directly from the empirical material data and requires no material or prior-distribution ansätze. We begin by 'thermalizing' the data, which effectively replaces the empirical material likelihood by a sum of Gaussians of a certain width [10]. The thermalized likelihoods also follow variationally by minimizing a regularized Kullback-Leibler divergence [11,12,13] from the empirical material likelihood.…”

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

“…Classically, deterministic problems in mathematical physics are closed by further restricting the states of the system to lie in a subset representing the material law of the system, i. e., the locus of states attainable by a specific material. In this paper, we work within a general framework [2] for systems in which the material law and the admissibility constraints are described by positive Radon likelihood measures µ D ∈ M(Z) and µ E ∈ M(Z), respectively, representing the likelihood of y ∈ Z being a (local) material state observed in the laboratory and of z ∈ Z being admissible. Before presenting our new contributions, the main ideas underlying the work may be summarized as follows.…”

“…The ansatz-free approach of [2] adopted here leads to a direct connection between data and inference and is therefore lossless and free of modeling bias. In addition, it allows to treat unbounded likelihoods, a setting where it is not clear how to set-up a Bayesian framework that is able to address the questions of inference and approximation.…”