Neural-network-based surrogate models are widely used to improve computational efficiency. Incorporating theoretical guidance into data-driven neural networks has improved their generalizability and accuracy. However, neural networks with strong form (partial differential equations) theoretical guidance have limited performance when strong discontinuity exists in the solution spaces, such as pressure discontinuity at sources/sinks in subsurface flow problems. In this study, we take advantage of weak form formulation and domain decomposition to deal with such difficulties. We propose two strategies based on our previously developed weak form Theory-guided Neural Network (TgNN-wf) to solve diffusivity equations with point sinks of either Dirichlet or Neumann type. Surrogate models are trained for well placement optimization and uncertainty analysis. Good agreement with numerical results is observed at lower computational costs, whereas strong form TgNN fails to provide satisfactory results, indicating the superiority of weak form formulation when solving discontinuous problems.

基于神经网络的代理模型被广泛用于提高计算效率。将理论指导整合到数据驱动的神经网络中，可以提高其通用性和准确性。但是，当溶液空间中存在强不连续性时，例如地下流问题中源/汇的压力不连续性，具有强形式（偏微分方程）理论指导的神经网络的性能会受到限制。在这项研究中，我们利用弱形式表述和域分解来解决此类难题。我们基于以前开发的弱形式理论指导神经网络（TgNN-wf）提出了两种策略，以解决Dirichlet或Neumann型点汇的扩散方程。替代模型经过训练，可以优化井位和不确定性分析。