Sparse Gaussian processes for solving nonlinear PDEs

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. Therefore, 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.