A fundamental problem in geostatistical modeling is to infer the heterogeneous geological field based on limited measurements and some prior spatial statistics. Semantic inpainting, a technique for image processing using deep generative models, has been recently applied for this purpose, demonstrating its effectiveness in dealing with complex spatial patterns. However, the original semantic inpainting framework incorporates only information from direct measurements, while in geostatistics indirect measurements are often plentiful. To overcome this limitation, here we propose a physics-informed semantic inpainting framework, employing the Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) and jointly incorporating the direct and indirect measurements by exploiting the underlying physical laws. Our simulation results for a high-dimensional problem with 512 dimensions show that in the new method, the physical conservation laws are satisfied and contribute in enhancing the inpainting performance compared to using only the direct measurements.

地统计学建模的一个基本问题是基于有限的测量和一些先验的空间统计来推断非均质地质场。语义修复是一种使用深度生成模型进行图像处理的技术，最近已用于此目的，证明了其在处理复杂空间模式方面的有效性。但是，原始的语义修复框架仅包含来自直接测量的信息，而在地统计学中，间接测量通常很多。为了克服这一局限性，我们在此提出了一种物理学方面的知识语义修复框架，采用具有渐变惩罚的Wasserstein生成对抗网络（WGAN-GP），并通过利用潜在的物理定律将直接和间接度量结合在一起。我们对512维高维问题的仿真结果表明，与仅使用直接测量相比，在新方法中，物理守恒定律得到满足，并有助于提高修补效果。