We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel Hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints. We derive a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions. In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs. Finally, we demonstrate our method on several linear PDEs including the Poisson, Helmholtz, Schr\"{o}dinger, Fokker-Planck, and heat equation. We highlight its robustness to noise as well as its ability to generalize to new data with varying degrees of smoothness and mesh discretization without any additional training.

翻译：我们提出了一种新的数据驱动方法，用于在给定输入-输出函数样本对的情况下，学习各类线性偏微分方程的基本解（格林函数）。基于函数线性回归的理论，我们在再生核希尔伯特空间中估计最佳拟合的格林函数及基本解的偏置项，从而能够正则化其光滑性并施加多种结构约束。我们推导了算子再生核希尔伯特空间的一般表示定理，将原无限维回归问题近似为有限维问题，将搜索空间缩减为格林函数的参数化类。为研究格林函数估计量的预测误差，我们将标量输出的函数线性回归先前结果推广至函数输出情形。最后，我们以泊松方程、亥姆霍兹方程、薛定谔方程、福克-普朗克方程和热方程等若干线性偏微分方程为例验证了该方法，突显其对噪声的鲁棒性，以及无需额外训练即可泛化至不同光滑程度与网格离散化程度的新数据的能力。