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.

翻译：本文提出了一种基于稀疏高斯过程（SGPs）的高效数值方法，用于求解非线性偏微分方程（PDEs）。高斯过程（GPs）已被广泛研究用于求解PDEs，其核心思想是将问题表述为在再生核希尔伯特空间（RKHS）中寻找近似PDE解的逼近函数。该近似解位于由样本点处不同阶核函数导数求值生成的基函数张成的空间中。然而，由GPs定义的RKHS因矩阵求逆的三次计算复杂度而导致高昂的计算代价。因此，我们猜想解存在于一个能够实现相似逼近性能的"压缩"子空间中，并提出了一种基于SGP的方法，将优化问题重新表述于这个"压缩"子空间内。这显著降低了计算负担，同时保持了期望的精度。本文严格阐述了该问题，并提供了误差分析和数值实验以证明该方法的有效性。数值实验表明，SGP方法使用的诱导点数量不到均匀采样点的一半，却能达到与使用相同数量均匀采样点的GP方法相当的精度，从而大幅降低了计算成本。我们的贡献包括：利用SGP将非线性PDE问题转化为RKHS"压缩"子空间上的优化问题，同时提供了存在性证明和严格的误差分析。此外，该方法可视为GP方法在处理一般正定核时的推广形式。