[Paper Review] When Bifidelity Meets CoKriging: An Efficient Physics-Informed Multifidelity Method

Xiu Yang, Xueyu Zhu+1 more|arXiv (Cornell University)|Dec 7, 2018

Probabilistic and Robust Engineering DesignDecision Sciences51 references3 citations

TL;DR

This paper proposes a physics-informed multifidelity method that combines bifidelity approximation with CoKriging to efficiently reconstruct physical states using sparse observations and stochastic simulations. By leveraging low-fidelity model statistics to construct a Gaussian process in CoKriging, the method achieves accurate predictions while preserving linear physical constraints within a quantifiable error bound, significantly reducing computational cost.

ABSTRACT

In this work, we propose a framework that combines the approximation-theory-based multifidelity method and Gaussian-process-regression-based multifidelity method to achieve data-model convergence when stochastic simulation models and sparse accurate observation data are available. Specifically, the two types of multifidelity methods we use are the bifidelity and CoKriging methods. The new approach uses the bifidelity method to efficiently estimate the empirical mean and covariance of the stochastic simulation outputs, then it uses these statistics to construct a Gaussian process (GP) representing low-fidelity in CoKriging. We also combine the bifidelity method with Kriging, where the approximated empirical statistics are used to construct the GP as well. We prove that the resulting posterior mean by the new physics-informed approach preserves linear physical constraints up to an error bound. By using this method, we can obtain an accurate construction of a state of interest based on a partially correct physical model and a few accurate observations. We present numerical examples to demonstrate performance of the method.

Motivation & Objective

• To address the high computational cost of physics-informed Gaussian process regression when relying on expensive high-fidelity simulations.
• To improve efficiency in uncertainty quantification by integrating low-fidelity models into physics-informed Kriging frameworks.
• To ensure that the reconstructed physical states satisfy linear physical constraints up to a bounded error.
• To develop a scalable, multifidelity framework that leverages empirical statistics from bifidelity approximations to inform CoKriging without requiring full high-fidelity sampling.

Proposed method

• The method uses the bifidelity approach to estimate empirical mean and covariance of stochastic simulation outputs using a small number of high-fidelity and many low-fidelity samples.
• These estimated statistics are used to construct a Gaussian process (GP) for the low-fidelity model in a CoKriging framework, replacing traditional parametric kernel estimation.
• The framework integrates realizations from a partially correct physical model with sparse observation data to produce a posterior predictive distribution on the entire domain.
• It proves that the resulting posterior mean preserves linear physical constraints up to an error bound dependent on the fidelity gap and statistical approximation error.
• The approach combines bifidelity with Kriging by using approximated statistics to define the GP prior, enabling efficient inference.
• The method is extendable to multifidelity settings beyond two fidelities, supporting hierarchical model hierarchies.

Experimental results

Research questions

• RQ1 Can bifidelity approximation be effectively combined with CoKriging to reduce computational cost in physics-informed modeling?
• RQ2 How can empirical statistics from low-fidelity simulations be used to construct a reliable GP prior in CoKriging without full high-fidelity sampling?
• RQ3 To what extent does the proposed method preserve linear physical constraints in the reconstructed state, and what is the associated error bound?
• RQ4 How does the method perform in terms of accuracy and efficiency compared to standard physics-informed Kriging with full high-fidelity sampling?
• RQ5 Can the framework be generalized to multifidelity settings beyond two levels of fidelity?

Key findings

• The proposed method achieves accurate state reconstruction with significantly reduced computational cost by replacing high-fidelity sampling with bifidelity-estimated statistics.
• The posterior mean of the method preserves linear physical constraints within an error bound proportional to the fidelity gap and statistical approximation error.
• The error bound is derived explicitly in terms of the number of high-fidelity samples, the number of low-fidelity samples, and the variance of the model outputs.
• Numerical examples demonstrate that the method maintains high accuracy even with a small number of high-fidelity samples and sparse observations.
• The framework is robust and scalable, with theoretical guarantees on constraint preservation and convergence under mild assumptions.
• The method outperforms standard physics-informed Kriging in computational efficiency while maintaining comparable or better accuracy.