We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

翻译：我们针对使用高斯过程和基于核的方法求解非线性（可能含参数）的偏微分方程，引入了先验的索伯列夫空间误差估计。主要假设为：(1) 核的再生核希尔伯特空间连续嵌入到具有足够正则性的索伯列夫空间；(2) 微分算子的稳定性以及偏微分方程解映射在相应索伯列夫空间之间的稳定性。证明过程围绕核插值的索伯列夫范数误差估计展开，并依赖于解的极小范数性质。若偏微分方程的解空间足够光滑，该误差估计展示出维度有利的收敛速率。我们通过高维非线性椭圆型偏微分方程及参数化偏微分方程的实例验证了上述结论。尽管近期某些机器学习方法声称可突破维数灾难以求解高维偏微分方程，但我们的分析揭示了更微妙的图景：解的正则性与维数灾难的出现之间存在权衡关系。因此，我们的结果与以下认知一致——当解足够正则时，维数灾难可被消除。