Variational Embedding of Measured Data in PhysicsConstrained DataDriven Modeling

Abstract

This paper presents a physicsconstrained datadriven method that variationally embeds measured data in the modeling and analysis framework. The physicsbased model is augmented with sparse but highfidelity data through a variationally derived loss function. The structure of the loss function is analyzed in the context of variational correction to the modeled response wherein loss function penalizes the difference in the modeled response from the measured data that represents the local response of the system. The variationally embedded measured data (VEMD) method results in forward simulations that are not only driven by boundary and initial conditions but are also augmented by real measurements taken at only a small number of observation points. In the context of forward simulations, the proposed approach can be seen as inducing inductive biases that exploit the difference between the computed and measured quantities in the parametric space. With the help of a model problem, we show that the proposed method learns from the sparse highfidelity datasets while preserving conservation properties of the balance laws. Method is applied to a nonsmooth model problem and the mathematical attributes of the formulation are investigated in the context of highfidelity sparse datasets.