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 $\mu _D$ measuring the likelihood of observing a material state in phase space; and another $\mu _E$ 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 $\mu _\infty$ of entropic regularizations $\mu _\beta$, or thermalizations, of the product measure $\mu = \mu _D\times \mu _E$ as $\beta \rightarrow +\infty$. We also supply conditions under which $\mu _\infty$ can be obtained as the athermal limit of carefully thermalized $(\mu _{h,\beta _h})$ sequences of empirical data sets $(\mu _h)$ approximating weakly an unknown likelihood function $\mu$. In particular, we find that the cooling sequence $\beta _h \rightarrow +\infty$ must be slow enough, corresponding to annealing, in order for the proper limit $\mu _\infty$ to be delivered. Finally, we derive explicit analytic expressions for expectations $\mathbb {E}[f]$ 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.