Training‐Image Based Geostatistical Inversion Using a Spatial Generative Adversarial Neural Network

Laloy, Eric; Hérault, Romain; Jacques, Diederik; Linde, Niklas

doi:10.1002/2017wr022148

Cited by 325 publications

(285 citation statements)

References 42 publications

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“…1 Introduction and Scope Laloy et al (2018) recently proposed to use generative adversarial networks (GANs), a game changer data generation algorithm (e.g., Goodfellow et al, 2014Goodfellow et al, , 2016, to define a low-dimensional parameterization encoding complex geologic prior models, thereby allowing efficient and accurate geostatistical inversion with Markov chain Monte Carlo (MCMC) methods (Laloy et al, 2018). GANs have permitted impressive advancements for a wide range of applications such as image and texture synthesis, image-to-image translation and super-resolution (Creswell et al, 2017).…”

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confidence: 99%

“…Laloy et al (2018) have shown that inversions based on such parameterizations work well for global probabilistic inference of complex binary 2D and 3D prior subsurface models. However, exploring the GANderived latent space with state-of-the-art MCMC sampling (Vrugt et al, 2009;Laloy and Vrugt, 2012) still necessitates tens of thousands (or more) forward evaluations (Laloy et al, 2018). Such a computational expense can be prohibitive when using computationallydemanding forward solvers encountered in the geosciences.…”

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confidence: 99%

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Gradient-based deterministic inversion of geophysical data with generative adversarial networks: Is it feasible?

Laloy

Linde

Ruffino

et al. 2019

Computers & Geosciences

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Global probabilistic inversion within the latent space learned by a Generative Adversarial Network (GAN) has been recently demonstrated. Compared to inversion on the original model space, using the latent space of a trained GAN can offer the following benefits: (1) the generated model proposals are geostatistically consistent with the prescribed prior training image (TI), and (2) the parameter space is reduced by orders of magnitude compared to the original model space. Nevertheless, exploring the learned latent space by state-of-the-art Markov chain Monte Carlo (MCMC) methods may still require a large computational effort. As an alternative, parameters in this latent space could possibly be optimized with much less computationally expensive gradient-based methods. This study shows that due to the typically highly nonlinear relationship between the latent space and the associated output space of a GAN, gradient-based deterministic inversion may fail even when considering a linear forward physical model. We tested two deterministic inversion approaches: a quasi-Newton gradient descent using the Adam algorithm and a Gauss-Newton (GN) method that makes use of the Jacobian matrix calculated by finite-differencing. For a channelized binary TI and a synthetic linear crosshole ground penetrating radar (GPR) tomography problem involving 576 measurements with low noise, we observe that when allowing for a total of 10,000 iterations only 13% of the gradient descent trials locate a solution that has the required data misfit. The tested GN inversion was unable to recover a solution with the appropriate data misfit. Our results suggest that deterministic inversion performance strongly depends on the inversion approach, starting model, true reference model, number of iterations and noise realization. In contrast, computationally-expensive probabilistic global optimization based on differential evolution always finds an appropriate solution.

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confidence: 99%

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confidence: 99%

Gradient-based deterministic inversion of geophysical data with generative adversarial networks: Is it feasible?

Laloy

Linde

Ruffino

et al. 2019

Computers & Geosciences

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“…The original adversarial autoencoder framework is composed of fully connected layers (Makhzani et al, 2016), making it increasingly difficult to train as the network gets deeper due to a large number of trainable parameters. To resolve this issue, we develop a CAAE framework based on convolutional layers to leverage their sparse-connectivity and parameter-sharing properties as well as robust capability in image-like data processing Laloy et al, 2018;Shen, 2018). GAN (Goodfellow et al, 2014) is a framework that establishes an adversarial game between two networks: a generative network (·) (i.e., generator) that learns the distribution p data (x) over the data, and a discriminative network (·) (i.e., discriminator) that computes the probability that a sample x is sampled from p data (x), rather than generated by the generator.…”

Section: Caae For Parameterization Of Non-gaussian Random Fieldsmentioning

confidence: 99%

Integration of Adversarial Autoencoders With Residual Dense Convolutional Networks for Estimation of Non‐Gaussian Hydraulic Conductivities

Zabaras

Shi

et al. 2020

Water Resources Research

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Inverse modeling for the estimation of non‐Gaussian hydraulic conductivity fields in subsurface flow and solute transport models remains a challenging problem. This is mainly due to the non‐Gaussian property, the nonlinear physics, and the fact that many repeated evaluations of the forward model are often required. In this study, we develop a convolutional adversarial autoencoder (CAAE) to parameterize non‐Gaussian conductivity fields with heterogeneous conductivity within each facies using a low‐dimensional latent representation. In addition, a deep residual dense convolutional network (DRDCN) is proposed for surrogate modeling of forward models with high‐dimensional and highly complex mappings. The two networks are both based on a multilevel residual learning architecture called residual‐in‐residual dense block. The multilevel residual learning strategy and the dense connection structure ease the training of deep networks, enabling us to efficiently build deeper networks that have an essentially increased capacity for approximating mappings of very high complexity. The CAAE and DRDCN networks are incorporated into an iterative ensemble smoother to formulate an inversion framework. The numerical experiments performed using 2‐D and 3‐D solute transport models illustrate the performance of the integrated method. The obtained results indicate that the CAAE is a robust parameterization method for non‐Gaussian conductivity fields with different heterogeneity patterns. The DRDCN is able to obtain accurate approximations of the forward models with high‐dimensional and highly complex mappings using relatively limited training data. The CAAE and DRDCN methods together significantly reduce the computation time required to achieve accurate inversion results.

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“…We generate 10,000 train samples and 1,000 test samples of this input field over a uniform grid of 32 × 32 cells/pixels. The third input field dataset considered is channelized field [74], defined with binary values 0.01 and 1.0. This dataset is taken from this link https://github.com/cicsnd/pde-surrogate and these samples are obtained by cropping 32 × 32 patches from 1250 × 1250 image.…”

Section: Stochastic Boundary Value Problem In 2dmentioning

confidence: 99%

Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks

Karumuri

Tripathy

Bilionis

et al. 2020

Journal of Computational Physics

128

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Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the curse of dimensionality.State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver.Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from traditional squared residual (SR) based loss function for training the ResNet, we introduce a physicsinformed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems.

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Training‐Image Based Geostatistical Inversion Using a Spatial Generative Adversarial Neural Network

Cited by 325 publications

References 42 publications

Gradient-based deterministic inversion of geophysical data with generative adversarial networks: Is it feasible?

Gradient-based deterministic inversion of geophysical data with generative adversarial networks: Is it feasible?

Integration of Adversarial Autoencoders With Residual Dense Convolutional Networks for Estimation of Non‐Gaussian Hydraulic Conductivities

Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks

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