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

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 μ_D measuring the likelihood of observing a material state in phase space; and another μ_E 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_B, or thermalizations, of the product measure μ = μ_D x μ_E 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 annealing, in order for the proper limit μ_∞ to be delivered. Finally, we derive explicit analytic expressions for expectations E[⨍] of outcomes ⨍ 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.