Multifidelity Gaussian process regression for solving nonlinear partial differential equations

TL;DR

This paper introduces a multifidelity Gaussian process regression method using cokriging to efficiently solve nonlinear PDEs, demonstrated on Burgers' equation.

Contribution

It develops a kernel learning approach leveraging multifidelity simulations to improve PDE solving with Gaussian processes.

Findings

The method effectively integrates low- and high-fidelity data.

It provides a differentiable non-stationary kernel tailored to empirical data.

Demonstrated improved accuracy on Burgers' equation.

Abstract

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.