Solution of diffusivity equations with local sources/sinks and surrogate modeling using weak form Theory-guided Neural Network

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.