Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training (2110.03049v1)

Abstract: Physics-informed neural networks (PINNs) have received significant attention as a unified framework for forward, inverse, and surrogate modeling of problems governed by partial differential equations (PDEs). Training PINNs for forward problems, however, pose significant challenges, mainly because of the complex non-convex and multi-objective loss function. In this work, we present a PINN approach to solving the equations of coupled flow and deformation in porous media for both single-phase and multiphase flow. To this end, we construct the solution space using multi-layer neural networks. Due to the dynamics of the problem, we find that incorporating multiple differential relations into the loss function results in an unstable optimization problem, meaning that sometimes it converges to the trivial null solution, other times it moves very far from the expected solution. We report a dimensionless form of the coupled governing equations that we find most favourable to the optimizer. Additionally, we propose a sequential training approach based on the stress-split algorithms of poromechanics. Notably, we find that sequential training based on stress-split performs well for different problems, while the classical strain-split algorithm shows an unstable behaviour similar to what is reported in the context of finite element solvers. We use the approach to solve benchmark problems of poroelasticity, including Mandel's consolidation problem, Barry-Mercer's injection-production problem, and a reference two-phase drainage problem. The Python-SciANN codes reproducing the results reported in this manuscript will be made publicly available at https://github.com/sciann/sciann-applications.

• The paper introduces a novel PINN framework with stress-split sequential training that improves simulation stability for coupled poromechanics problems.
• Numerical results show high fidelity in benchmark tests like Mandel's consolidation and Barry-Mercer's injection-production, outperforming traditional methods.
• A dimensionless formulation streamlines neural network training for both single- and multiphase flow, setting the stage for further adaptive and hybrid modeling research.

Physics-Informed Neural Network Simulation of Multiphase Poroelasticity Using Stress-Split Sequential Training

The recent work focusing on the simulation of multiphase poroelasticity using physics-informed neural networks (PINNs) presents a promising approach to handle complex coupled problems characterized by partial differential equations (PDEs). The authors address the inherent challenge of training PINNs for such forward problems, where traditional formulations may lead to unstable optimization due to the multi-objective and non-convex nature of the associated loss functions. They propose a novel dimensionless formulation alongside a sequential training approach based on stress-split algorithms, which they found provides more stable results compared to traditional strain-split techniques.

Core Contributions and Methodology

The paper initiates by outlining the context of poroelasticity in engineering systems such as geotechnical and reservoir engineering, highlighting the importance of improved computational models that can integrate real-world data. This is a critical step forward given the limitations of classical techniques which often require complex external optimization loops to align outputs with empirical data.

The main contribution of this paper is the development of a PINN framework tailored specifically for coupled flow and deformation in porous media. The framework employs multi-layer neural networks to approximate unknown solution variables, while optimizing a complex multi-component loss function informed by the governing PDEs, initial, and boundary conditions. A notable innovation described is the use of a sequential training methodology which incorporates stress-splitting techniques. This approach contrasts with prior methods by showing improved stability and convergence for various benchmark problems, including Mandel's consolidation problem and Barry-Mercer's injection-production problem.

Numerical and Theoretical Implications

Numerical results presented in the paper support the advantages of the stress-split sequential training approach, where the systems exhibit increased stability, avoiding the common pitfall of trivial convergence or deviation during optimization. The experiments demonstrate high fidelity when compared against traditional solutions. Furthermore, by applying PINNs to both single-phase and two-phase flow conditions, the authors have expanded the applicability of neural network-based simulations in poromechanics, which traditionally struggle with high computational demands and sensitivity to hyperparameter settings.

Theoretically, the dimensionless representation of governing equations enhances the training process of neural networks by allowing them to converge more efficiently and accurately. The insights gained from testing both stress-split and strain-split approaches can impact future algorithmic developments, hinting at broader applicability across other domains governed by coupled PDEs.

Future Directions

The work establishes several pathways for ongoing research. Firstly, a deep dive into adaptive strategies for both spatial and temporal sampling could potentially further optimize the PINN framework. Secondly, leveraging more advanced network architectures or hybrid models that incorporate domain-specific knowledge may enhance the robustness of the methodology. Finally, exploring diverse applications beyond the tested benchmark problems, such as irregular geometries or variable material properties, could expand the usability of the presented approach.

Overall, this paper contributes a significant step towards more efficient and robust computational models for poromechanics, paving the way for broader integration of machine learning methods in the simulation of physical systems subject to complex boundary and initial conditions. The proposed methods notably shift the needle towards PINNs being a more viable alternative in tackling such intricate multiphysics problems.