Variational Embedding of Measured Data in Physics-Constrained Data-Driven Modeling

Arif Masud, Shoaib Goraya

Research output: Contribution to journalArticlepeer-review


This paper presents a physics-constrained data-driven method that variationally embeds measured data in the modeling and analysis framework. The physics-based model is augmented with sparse but high-fidelity 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 high-fidelity datasets while preserving conservation properties of the balance laws. Method is applied to a non-smooth model problem and the mathematical attributes of the formulation are investigated in the context of high-fidelity sparse datasets.

Original languageEnglish (US)
Article number111001
JournalJournal of Applied Mechanics, Transactions ASME
Issue number11
StatePublished - Nov 1 2022
Externally publishedYes


  • Data-driven modeling
  • Elasticity
  • Physicsconstrained modeling
  • Variational multiscale method

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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