# APIK: Active Physics-Informed Kriging Model with Partial Differential Equations

arXiv: https://arxiv.org/abs/2012.11798 · HTML: https://mlsanity.com/paper/2012.11798/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2020.

## Built on by (most cited first, 9 of 9)

- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priors](https://mlsanity.com/paper/2201.06463/) (2022)
- [Failure-averse Active Learning for Physics-constrained Systems](https://mlsanity.com/paper/2110.14443/) (2021)
- [Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2212.11880/) (2022)
- [PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2204.02583/) (2022)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis](https://mlsanity.com/paper/2305.13879/) (2023)
- [Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problems](https://mlsanity.com/paper/2505.21723/) (2025)

## Abstract (arXiv)

Kriging (or Gaussian process regression) is a popular machine learning method for its flexibility and closed-form prediction expressions. However, one of the key challenges in applying kriging to engineering systems is that the available measurement data is scarce due to the measurement limitations and high sensing costs. On the other hand, physical knowledge of the engineering system is often available and represented in the form of partial differential equations (PDEs). We present in this work a PDE Informed Kriging model (PIK), which introduces PDE information via a set of PDE points and conducts posterior prediction similar to the standard kriging method. The proposed PIK model can incorporate physical knowledge from both linear and nonlinear PDEs. To further improve learning performance, we propose an Active PIK framework (APIK) that designs PDE points to leverage the PDE information based on the PIK model and measurement data. The selected PDE points not only explore the whole input space but also exploit the locations where the PDE information is critical in reducing predictive uncertainty. Finally, an expectation-maximization algorithm is developed for parameter estimation. We demonstrate the effectiveness of APIK in two synthetic examples, a shock wave case study, and a laser heating case study.