Weak Form Theory-guided Neural Network (TgNN-wf) for Deep Learning of Subsurface Single and Two-phase Flow
arXiv:2009.04543 · doi:10.1016/j.jcp.2021.110318
Abstract
Deep neural networks (DNNs) are widely used as surrogate models in geophysical applications; incorporating theoretical guidance into DNNs has improved the generalizability. However, most of such approaches define the loss function based on the strong form of conservation laws (via partial differential equations, PDEs), which is subject to deteriorated 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 formulation of the PDE into the loss function combined with data constraint and initial and boundary conditions regularizations to tackle the aforementioned difficulties. In the weak form, high order derivatives in the PDE can be transferred to the test functions by performing integration-by-parts, which reduces computational error. 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 the hydraulic head prediction for unsteady-state 2D single-phase flow problems and the saturation profile prediction for 1D two-phase flow problems. Results show that TgNN-wf consistently has higher accuracy than TgNN, especially when strong discontinuity in the solution is present. TgNN-wf also trains faster than TgNN when the number of integration subdomains is not too large (<10,000). Moreover, TgNN-wf is more robust to noises. Thus, the proposed TgNN-wf paves the way for which a variety of deep learning problems in the small data regime can be solved more accurately and efficiently.
35 pages, 10 figures, and 6 tables
References in corpus (13)
- PyTorch: An Imperative Style, High-Performance Deep Learning Library
- Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
- Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
- Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
- hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
- Deep Learning of Subsurface Flow via Theory-guided Neural Network
- Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
- Using Noisy or Incomplete Data to Discover Models of Spatiotemporal Dynamics
- Deep-Learning based Inverse Modeling Approaches: A Subsurface Flow Example
- Efficient Uncertainty Quantification for Dynamic Subsurface Flow with Surrogate by Theory-guided Neural Network
- A Lagrangian Dual-based Theory-guided Deep Neural Network
- The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
- Deep Neural Network Approach to Forward-Inverse Problems