We present a data-driven approach to mathematically model physical systems whose governing partial differential equations are unknown, by learning their associated Green's function. The subject systems are observed by collecting input-output pairs of system responses under excitations drawn from a Gaussian process. Two methods are proposed to learn the Green's function. In the first method, we use the proper orthogonal decomposition (POD) modes of the system as a surrogate for the eigenvectors of the Green's function, and subsequently fit the eigenvalues, using data. In the second, we employ a generalization of the randomized singular value decomposition (SVD) to operators, in order to construct a low-rank approximation to the Green's function. Then, we propose a manifold interpolation scheme, for use in an offline-online setting, where offline excitation-response data, taken at specific model parameter instances, are compressed into empirical eigenmodes. These eigenmodes are subsequently used within a manifold interpolation scheme, to uncover other suitable eigenmodes at unseen model parameters. The approximation and interpolation numerical techniques are demonstrated on several examples in one and two dimensions.

翻译：我们提出了一种数据驱动方法，通过学习未知偏微分方程控制下的物理系统对应的格林函数，来对其建立数学模型。通过收集高斯过程激励下系统响应的输入-输出对来观测目标系统。本文提出了两种学习格林函数的方法。第一种方法利用系统的本征正交分解模态作为格林函数特征向量的替代，随后基于数据拟合其特征值。第二种方法将随机奇异值分解推广到算子层面，从而构建格林函数的低秩近似。进一步地，我们提出了一种流形插值方案，适用于"离线-在线"场景：在特定模型参数实例下采集的离线激励-响应数据被压缩为经验本征模态，这些模态随后通过流形插值方案，用于揭示未见模型参数下的其他合适本征模态。我们通过一维和二维的多个算例验证了近似与插值数值技术的有效性。