# Deep Learning of Subsurface Flow via Theory-guided Neural Network

arXiv: https://arxiv.org/abs/1911.00103 · HTML: https://mlsanity.com/paper/1911.00103/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2019.

## Built on by (most cited first, 30 of 32)

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- [Finite Volume Neural Network: Modeling Subsurface Contaminant Transport](https://mlsanity.com/paper/2104.06010/) (2021)
- [Integration of knowledge and data in machine learning](https://mlsanity.com/paper/2202.10337/) (2022)
- [Use of Multifidelity Training Data and Transfer Learning for Efficient Construction of Subsurface Flow Surrogate Models](https://mlsanity.com/paper/2204.11138/) (2022)
- [A Gradient-based Deep Neural Network Model for Simulating Multiphase Flow in Porous Media](https://mlsanity.com/paper/2105.02652/) (2021)
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- [Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations](https://mlsanity.com/paper/2106.01078/) (2021)
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- [Time Series Predictions in Unmonitored Sites: A Survey of Machine Learning Techniques in Water Resources](https://mlsanity.com/paper/2308.09766/) (2023)
- [Surrogate and inverse modeling for two-phase flow in porous media via theory-guided convolutional neural network](https://mlsanity.com/paper/2110.10080/) (2021)
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- [Differentiable modeling to unify machine learning and physical models and advance Geosciences](https://mlsanity.com/paper/2301.04027/) (2023)
- [Uncertainty quantification and inverse modeling for subsurface flow in 3D heterogeneous formations using a theory-guided convolutional encoder-decoder network](https://mlsanity.com/paper/2111.08691/) (2021)
- [Depression Detection Using Digital Traces on Social Media: A Knowledge-aware Deep Learning Approach](https://mlsanity.com/paper/2303.05389/) (2023)
- [TgDLF2.0: Theory-guided deep-learning for electrical load forecasting via Transformer and transfer learning](https://mlsanity.com/paper/2210.02448/) (2022)
- [Domain-guided Machine Learning for Remotely Sensed In-Season Crop Growth Estimation](https://mlsanity.com/paper/2106.13323/) (2021)
- [Transfer Learning Approaches for Knowledge Discovery in Grid-based Geo-Spatiotemporal Data](https://mlsanity.com/paper/2110.00841/) (2021)
- [Filtered Partial Differential Equations: a robust surrogate constraint in physics-informed deep learning framework](https://mlsanity.com/paper/2311.03776/) (2023)
- [A novel Trunk Branch-net PINN for flow and heat transfer prediction in porous medium](https://mlsanity.com/paper/2501.16362/) (2025)
- [Identification of Physical Processes and Unknown Parameters of 3D Groundwater Contaminant Problems via Theory-guided U-net](https://mlsanity.com/paper/2205.00134/) (2022)
- [Uncertainty quantification of two-phase flow in porous media via coupled-TgNN surrogate model](https://mlsanity.com/paper/2205.14301/) (2022)
- [Phy-Taylor: Physics-Model-Based Deep Neural Networks](https://mlsanity.com/paper/2209.13511/) (2022)
- [Worth of knowledge in deep learning](https://mlsanity.com/paper/2307.00712/) (2023)
- [Optimization of geological carbon storage operations with multimodal latent dynamic model and deep reinforcement learning](https://mlsanity.com/paper/2406.04575/) (2024)

## Abstract (arXiv)

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.