Physics Informed Deep Learning Application In High-Dimensional Groundwater Inverse Modeling Of Hydraulic Tomography
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Guo, Quan
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Abstract
Groundwater flow simulation plays a crucial role in understanding subsurface water dynamics. Achieving accurate groundwater flow simulations necessitates solving the groundwater inverse problem, which involves the estimation of spatially varying hydrogeological parameter fields based on indirect observations. Hydraulic tomography (HT), also known as sequential pumping tests, has demonstrated great potential for aquifer characterization with relatively low cost and simple data collection techniques. By alternatively switching pumping and monitoring wells in a well network, HT can provide larger and more informative data than a traditional single-well pumping test. The enhanced information density reduces the non-uniqueness of potential inverse solutions for the unknown parameter field like hydraulic conductivities. A common and effective approach to solving HT inverse problems is the gradient-based geostatistical approach (GA). The major challenge faced by GA is its high computational cost when the dimension of the target variable is large for estimating a high-resolution parameter field. In addition, GA is not applicable for non-Gaussian fields featuring factures or channels, notable for their discontinuity.
In this thesis, we develop a physics-informed deep learning framework, which consists of physics-informed neural networks (PINNs), neural operators, and generative models. Its primary objective is to tackle inverse problems associated with various hydrogeological parameter random fields by utilizing HT observations.
For smooth Gaussian Random Fields (GRFs) characterized by Gaussian covariance functions, a hydraulic tomography – physics informed neural network (HT-PINN) is developed for inverting two-dimensional large-scale spatially distributed transmissivity. HT-PINN involves a neural network model of transmissivity and a series of neural network models to describe transient or steady-state sequential pumping tests. All the neural network models are jointly trained by minimizing the total loss function including data fitting errors and PDE constraints. Batch training of collocation points is used to limit the number of collocation points per training iteration. As a result, the developed HT-PINN exhibits great scalability and robustness, maintaining consistent data requirements and computational costs in inverting fields with different resolutions ranging from coarse to fine.
For Gaussian Random Fields (GRFs) with exponential covariance functions, which exhibit fine-scale non-smoothness, we have devised a solution by integrating Fourier Neural Operator (FNO) with the Reformulated Geostatistical Approach (RGA). FNO serves as a neural network surrogate forward model, specifically trained to solve groundwater Partial Differential Equations (PDEs) within the context of HT. On the other hand, RGA is utilized to fully harness geostatistical information and prior distributions. A well-trained FNO demonstrates the ability to solve groundwater forward problems efficiently and accurately. Importantly, its auto-differentiation eliminates the need for iterative forward simulations as seen in traditional optimization methods. The combined FNO-RGA model excels in accuracy, efficiency, and adaptability when employed for inverse modeling of non-smooth GRFs.
For non-Gaussian channel fields with discontinuous characteristics around channels and banks, we introduce the HT-INV-NN model, which comprises a Deep Neural Network (DNN) acting as a predictor and a Generative Adversarial Network (GAN) as a decoder. This DNN directly learns the inverse process, mapping hydraulic head measurements to latent variables of random fields. Simultaneously, the GAN can reconstruct high-dimensional non-Gaussian channel fields from the predicted latent variables. HT-INV-NN has undergone rigorous testing in multiple numerical experiments, spanning binary discontinuous or continuous non-Gaussian channel fields. By redefining the inverse problem as a predictive problem, our approach significantly enhances the efficiency of inverse modeling, meanwhile, the model's accuracy is deemed satisfactory.
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Date
2024-02-27
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Dissertation