Weak form Theory-guided Neural Network (TgNN-wf) for deep learning of subsurface single- and two-phase flow

Abstract Deep neural networks (DNNs) are widely used as surrogate models, and incorporating theoretical guidance into DNNs has improved generalizability. However, most such approaches define the loss function based on the strong form of conservation laws (via partial differential equations, PDEs), which is subject to diminished accuracy when the PDE has high-order derivatives or the solution has strong discontinuities. Herein, we propose a weak form theory-guided neural network (TgNN-wf), which incorporates the weak form residual of the PDE into the loss function, combined with data constraint and initial and boundary condition regularizations, to overcome the aforementioned difficulties. The original loss minimization problem is reformulated into a Lagrangian duality problem, so that the weights of the terms in the loss function are optimized automatically. We use domain decomposition with locally-defined test functions, which captures local discontinuity effectively. Two numerical cases demonstrate the superiority of the proposed TgNN-wf over the strong form TgNN, including hydraulic head prediction for unsteady-state 2D single-phase flow problems and saturation profile prediction for 1D two-phase flow problems. Results show that TgNN-wf consistently achieves higher accuracy than TgNN, especially when strong discontinuity in the parameter or solution space is present. TgNN-wf also trains faster than TgNN when the number of integration subdomains is not too large (