# Uncertainty quantification and inverse modeling for subsurface flow in 3D heterogeneous formations using a theory-guided convolutional encoder-decoder network

arXiv: https://arxiv.org/abs/2111.08691 · HTML: https://mlsanity.com/paper/2111.08691/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 40 references that are in the map.
- First posted on arXiv 14 Nov 2021.

## Builds on (most cited first, 15 of 15)

- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Deep Learning of Subsurface Flow via Theory-guided Neural Network](https://mlsanity.com/paper/1911.00103/) (2019)
- [A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks](https://mlsanity.com/paper/2105.13136/) (2021)
- [Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification](https://mlsanity.com/paper/1812.09444/) (2018)
- [Efficient Uncertainty Quantification for Dynamic Subsurface Flow with Surrogate by Theory-guided Neural Network](https://mlsanity.com/paper/2004.13560/) (2020)
- [CCSNet: a deep learning modeling suite for CO2 storage](https://mlsanity.com/paper/2104.01795/) (2021)
- [Weak Form Theory-guided Neural Network (TgNN-wf) for Deep Learning of Subsurface Single and Two-phase Flow](https://mlsanity.com/paper/2009.04543/) (2020)
- [Theory-guided Auto-Encoder for Surrogate Construction and Inverse Modeling](https://mlsanity.com/paper/2011.08618/) (2020)
- [Deep Neural Network Approach to Forward-Inverse Problems](https://mlsanity.com/paper/1907.12925/) (2019)

## Built on by (most cited first, 3 of 3)

- [Graph Network Surrogate Model for Subsurface Flow Optimization](https://mlsanity.com/paper/2312.08625/) (2023)
- [Knowledge-Based Convolutional Neural Network for the Simulation and Prediction of Two-Phase Darcy Flows](https://mlsanity.com/paper/2404.03240/) (2024)
- [WellPINN: Accurate Well Representation for Transient Fluid Pressure Diffusion in Subsurface Reservoirs with Physics-Informed Neural Networks](https://mlsanity.com/paper/2507.09330/) (2025)

## Abstract (arXiv)

We build surrogate models for dynamic 3D subsurface single-phase flow problems with multiple vertical producing wells. The surrogate model provides efficient pressure estimation of the entire formation at any timestep given a stochastic permeability field, arbitrary well locations and penetration lengths, and a timestep matrix as inputs. The well production rate or bottom hole pressure can then be determined based on Peaceman's formula. The original surrogate modeling task is transformed into an image-to-image regression problem using a convolutional encoder-decoder neural network architecture. The residual of the governing flow equation in its discretized form is incorporated into the loss function to impose theoretical guidance on the model training process. As a result, the accuracy and generalization ability of the trained surrogate models are significantly improved compared to fully data-driven models. They are also shown to have flexible extrapolation ability to permeability fields with different statistics. The surrogate models are used to conduct uncertainty quantification considering a stochastic permeability field, as well as to infer unknown permeability information based on limited well production data and observation data of formation properties. Results are shown to be in good agreement with traditional numerical simulation tools, but computational efficiency is dramatically improved.