Multifidelity Gaussian process regression for solving nonlinear partial differential equations

2026-05-11

AI Summary

This work addresses the sensitivity to kernel selection and reliance on costly high-fidelity data in solving nonlinear partial differential equations (PDEs) by proposing a multi-fidelity Gaussian process regression framework based on co-kriging. The method leverages low-fidelity simulations to construct an empirical kernel, automatically learns a differentiable non-stationary kernel structure, and derives a high-fidelity kernel along with its mean function, both incorporating hyperparameter estimation and embedded physical constraints to guide PDE solution. Innovatively, it integrates multi-fidelity information directly into the kernel learning process, enabling adaptive construction of non-stationary kernels and effective cross-fidelity knowledge transfer. Experiments on the Burgers equation demonstrate that the proposed approach significantly outperforms conventional kernel methods, achieving notable improvements in accuracy, generalization capability, and computational efficiency.

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.

Problem

• nonlinear partial differential equations
• multifidelity
• kernel selection
• Gaussian process regression
• Burgers' equation

Innovation

• multifidelity
• Gaussian process regression
• non-stationary kernel
• cokriging
• physics-informed