Optimal Linear Interpolation under Differential Information: application to the prediction of perfect flows

TL;DR

This paper extends Kriging interpolation methods to incorporate linear PDE information, improving predictions of physical flows with sparse data through novel constrained and co-Kriging approaches.

Contribution

It introduces two new methods for PDE-constrained Kriging, enhancing accuracy and efficiency in modeling physical phenomena with limited observations.

Findings

The Lagrangian approach offers strict PDE constraint satisfaction but is computationally intensive.

Co-Kriging effectively combines physical and observational data for improved interpolation.

Numerical experiments demonstrate the methods' applicability to ODEs, PDEs, and fluid flow simulations.

Abstract

Approximation of functions satisfying partial differential equations (PDEs) is paramount for simulation of physical fluid flows and other problems in physics. Recently, physics-informed machine learning approaches have proven useful as a data-driven complement to numerical models for partial differential equations, bringing faster responses and allowing us to capitalize on past observations. However, their efficiency and convergence depend on the availability of vast training datasets. For sparse observations, Gaussian process regression or Kriging has emerged as a powerful interpolation model, offering principled estimates and uncertainty quantification. Several attempts have been made to condition Gaussian processes on linear PDEs via artificial or collocation observations and kernel design.These methods suffer from scalability issues in higher dimensions and limited generalizability. The aim of this study is to explore the extension of the Kriging predictor in the presence of linear PDE information at a finite number of collocation points. Two approaches are proposed: 1) A collocated co-Kriging with primary observations of the physical field and auxiliary differential observations; 2) A constrained Kriging optimization problem strongly satisfying linear PDE constraints at the points of prediction through a Lagrangian formulation. Numerical experiments are given for ordinary differential equations, 2D harmonic PDEs and an application to perfect flows around a cylinder. This work highlights a trade-off between the computational efficiency of the Lagrange multipliers approach and the strict interpolation of observations.