Sparse Gaussian processes for solving nonlinear PDEs

Rui Meng

doi:10.1016/j.jcp.2023.112340

Return

Sparse Gaussian processes for solving nonlinear PDEs

2023-10-01

Rui Meng

Xianjin Yang *

DOI：10.1016/j.jcp.2023.112340

Original

Save

View PDF

Abstract

En 中文

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. We conjecture that a solution exists on a condensed subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the condensed subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost. Our contributions include formulating the nonlinear PDE problem as an optimization problem on a condensed subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels. (c) 2023 Elsevier Inc. All rights reserved.

Keywords:

Partial differential equations

Sparse Gaussian process

AI Summary

Key information extracted from the uploaded paper, including a brief overview, abstract, background, key highlights, visual analysis, and future outlook.