# Physics-Information-Aided Kriging: Constructing Covariance Functions using Stochastic Simulation Models

arXiv: https://arxiv.org/abs/1809.03461 · HTML: https://mlsanity.com/paper/1809.03461/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2018.

## Built on by (most cited first, 3 of 3)

- [Failure-averse Active Learning for Physics-constrained Systems](https://mlsanity.com/paper/2110.14443/) (2021)
- [Physics-Coupled Spatio-Temporal Active Learning for Dynamical Systems](https://mlsanity.com/paper/2108.05385/) (2021)
- [Multifidelity Gaussian process regression for solving nonlinear partial differential equations](https://mlsanity.com/paper/2605.10383/) (2026)

## Abstract (arXiv)

In this work, we propose a new Gaussian process regression (GPR) method: physics information aided Kriging (PhIK). In the standard data-driven Kriging, the unknown function of interest is usually treated as a Gaussian process with assumed stationary covariance with hyperparameters estimated from data. In PhIK, we compute the mean and covariance function from realizations of available stochastic models, e.g., from realizations of governing stochastic partial differential equations solutions. Such constructed Gaussian process generally is non-stationary, and does not assume a specific form of the covariance function. Our approach avoids the optimization step in data-driven GPR methods to identify the hyperparameters. More importantly, we prove that the physical constraints in the form of a deterministic linear operator are guaranteed in the resulting prediction. We also provide an error estimate in preserving the physical constraints when errors are included in the stochastic model realizations. To reduce the computational cost of obtaining stochastic model realizations, we propose a multilevel Monte Carlo estimate of the mean and covariance functions. Further, we present an active learning algorithm that guides the selection of additional observation locations. The efficiency and accuracy of PhIK are demonstrated for reconstructing a partially known modified Branin function, studying a three-dimensional heat transfer problem and learning a conservative tracer distribution from sparse concentration measurements.