Optimal Linear Interpolation under Differential Information: application to the prediction of perfect flows
For researchers in physics-informed machine learning and surrogate modeling, this work offers a principled way to integrate sparse PDE information into Gaussian process regression, though the methods are incremental extensions of existing Kriging techniques.
This paper extends Kriging to incorporate linear PDE constraints at collocation points, proposing collocated co-Kriging and constrained Kriging via Lagrange multipliers. Numerical experiments on ODEs, 2D harmonic PDEs, and perfect flows show a trade-off between computational efficiency and strict interpolation.
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.