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

TL;DR

This paper develops a measure-theoretic framework for data-driven inference in physical systems, providing convergence rates and error bounds for approximations based on empirical measures, with applications to transportation networks.

Contribution

It introduces a novel convergence analysis for data-driven inference in physical systems using the flat metric, including error bounds and annealing rates, with an application to transportation networks.

Findings

Derived error bounds for empirical measure approximations.

Established convergence rates for annealing procedures.

Applied the theory to a transportation network example.

Abstract

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.