Weak form theory-guided neural network (TgNN-wf) for deep learning of subsurface single- and two-phase flow

• Published in: Journal of computational physics Vol. 436; p. 110318
• Main Authors: Xu, Rui, Zhang, Dongxiao, Rong, Miao, Wang, Nanzhe
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.07.2021; Elsevier Science Ltd
• Subjects: Artificial neural networks; Boundary conditions; Computational physics; Conservation laws; Deep learning; Discontinuity; Domain decomposition methods; Lagrangian duality; Machine learning; Neural networks; Partial differential equations; Robustness (mathematics); Single-phase flow; Solution space; Theory-guided neural network; Two dimensional flow; Two phase flow; Weak form; Lagrangian duality; Weak form; Theory-guided neural network; Single-phase flow; Two-phase flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110318

•Weak form physics constraints are incorporated into a fully-connected neural network to predict future responses.•Domain decomposition reduces computational cost and captures local discontinuity.•Our model shows improved accuracy and robustness to noises compared to strong form theory-guided neural networks. 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 (<10,000). Furthermore, TgNN-wf is more robust to noise. Consequently, the proposed TgNN-wf paves the way for which a variety of deep learning problems in small data regimes can be solved more accurately and efficiently.