Deep Learning of Subsurface Flow via Theory-guided Neural Network (1911.00103v1)

Abstract: Active researches are currently being performed to incorporate the wealth of scientific knowledge into data-driven approaches (e.g., neural networks) in order to improve the latter's effectiveness. In this study, the Theory-guided Neural Network (TgNN) is proposed for deep learning of subsurface flow. In the TgNN, as supervised learning, the neural network is trained with available observations or simulation data while being simultaneously guided by theory (e.g., governing equations, other physical constraints, engineering controls, and expert knowledge) of the underlying problem. The TgNN can achieve higher accuracy than the ordinary Artificial Neural Network (ANN) because the former provides physically feasible predictions and can be more readily generalized beyond the regimes covered with the training data. Furthermore, the TgNN model is proposed for subsurface flow with heterogeneous model parameters. Several numerical cases of two-dimensional transient saturated flow are introduced to test the performance of the TgNN. In the learning process, the loss function contains data mismatch, as well as PDE constraint, engineering control, and expert knowledge. After obtaining the parameters of the neural network by minimizing the loss function, a TgNN model is built that not only fits the data, but also adheres to physical/engineering constraints. Predicting the future response can be easily realized by the TgNN model. In addition, the TgNN model is tested in more complicated scenarios, such as prediction with changed boundary conditions, learning from noisy data or outliers, transfer learning, and engineering controls. Numerical results demonstrate that the TgNN model achieves much better predictability, reliability, and generalizability than ANN models due to the physical/engineering constraints in the former.

• The paper introduces the TgNN framework that integrates governing PDEs, boundary conditions, and expert knowledge to enhance subsurface flow modeling.
• It employs a comprehensive loss function combining data mismatch, PDE residuals, and engineering controls to achieve lower relative L2 error and higher R2 scores than conventional ANN models.
• The study demonstrates that TgNN improves model generalizability and computational efficiency, enabling effective applications in uncertainty quantification and parameter inversion.

Deep Learning of Subsurface Flow via Theory-guided Neural Network

The field of subsurface flow modeling, which is central to numerous geological and engineering applications, frequently encounters the challenge of data scarcity and the complexity of heterogeneity in model parameters. The paper "Deep Learning of Subsurface Flow via Theory-guided Neural Network" by Wang, Zhang, Chang, and Li introduces an innovative approach integrating scientific knowledge with data-driven models to address these challenges. The Theory-guided Neural Network (TgNN) framework is proposed to enhance deep learning models by incorporating domain-specific theories, thereby improving model accuracy, generalizability, and robustness.

Methodological Framework

The TgNN framework leverages deep neural networks (DNNs) enhanced with physical laws, such as governing partial differential equations (PDEs), boundary conditions, and additional constraints from engineering and expert knowledge. This is accomplished through the construction of a comprehensive loss function that includes terms accounting for data mismatch, PDE residuals, boundary and initial conditions, expert knowledge, and engineering controls. The idea is to guide model learning by not solely relying on data but also on the physical validity derived from the theoretical foundations of the problem domain.

Key components of the TgNN model for subsurface flow include:

• Data Term: Captures the fit to available observations or simulation outputs.
• PDE Constraint: Ensures model adherence to fundamental hydrodynamic principles through enforced PDE compliance.
• Boundary and Initial Conditions: Incorporated to respect physical boundaries and initial states of the system.
• Expert Knowledge and Engineering Controls: Introduced as additional penalties in scenarios where response behaviors are governed by operational constraints, such as allowable hydraulic head ranges.

Numerical Experiments

The paper demonstrates the TgNN framework using several numerical experiments designed to validate its capacity to deal with different complexities, such as transient subsurface flow, changing boundary conditions, noisy data, and outliers. A notable example is the application of TgNN to a two-dimensional transient saturated flow in porous media, showcasing enhanced accuracy and stability in predictions compared to traditional artificial neural networks (ANNs).

Analysis of the results indicates the TgNN's superior performance in predicting future responses accurately, even with varied boundary conditions, significant data noise, and outlier contamination. The evaluation metrics include relative L2 error and the R2 score, consistently showing improved outcomes for TgNN across multiple scenarios. For example, in noisy data settings, TgNN preserved accuracy with a relative L2 error much smaller than that observed with conventional ANN models.

Theoretical and Practical Implications

The proposed TgNN framework holds significant implications for advancing modeling techniques in fields requiring physical consistency and reliability where data is limited. It offers a systematic approach to integrating domain knowledge into data-driven models, enhancing their capacity to extrapolate beyond the training dataset's confines.

Furthermore, this framework opens avenues for leveraging TgNN as a surrogate model for tasks like uncertainty quantification and parameter inversion. Considering its capacity to integrate transfer learning, the TgNN can adapt efficiently to new conditions without undergoing complete retraining, significantly reducing the computational load. This computational efficiency, combined with robust predictive capabilities, underscores the potential for TgNN applications in real-world scenarios involving large-scale and heterogeneous environmental systems.

Conclusion

The research provides a compelling framework for integrating theoretical knowledge with neural networks, bringing about reliable and physically reasonable subsurface flow predictions. Future work can extend this framework to encompass broader applications involving intricate system dynamics and widen the application in hybrid modeling approaches. By melding theoretical rigor with machine learning, the TgNN represents a significant step forward in bridging the gap between numerical modeling and practical engineering applications.