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Register a new userA deep-learning based Bayesian approach to seismic imaging and uncertainty quantification
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Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is via formulating the posterior distribution. Unfortunately, it is often not possible to formulate a prior distribution that precisely encodes our prior knowledge about the unknown. Furthermore, adherence to handcrafted priors may greatly bias the outcome of the Bayesian analysis. To address this issue, we propose to use the functional form of a randomly initialized convolutional neural network as an implicit structured prior, which is shown to promote natural images and excludes images with unnatural noise. In order to incorporate the model uncertainty into the final estimate, we sample the posterior distribution using stochastic gradient Langevin dynamics and perform Bayesian model averaging on the obtained samples. Our synthetic numerical experiment verifies that deep priors combined with Bayesian model averaging are able to partially circumvent imaging artifacts and reduce the risk of overfitting in the presence of extreme noise. Finally, we present pointwise variance of the estimates as a measure of uncertainty, which coincides with regions that are more difficult to image.
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In inverse problems, uncertainty quantification (UQ) deals with a probabilistic description of the solution nonuniqueness and data noise sensitivity. Setting seismic imaging into a Bayesian framework allows for a principled way of studying uncertainty by solving for the model posterior distribution. Imaging, however, typically constitutes only the first stage of a sequential workflow, and UQ becomes even more relevant when applied to subsequent tasks that are highly sensitive to the inversion outcome. In this paper, we focus on how UQ trickles down to horizon tracking for the determination of stratigraphic models and investigate its sensitivity with respect to the imaging result. As such, the main contribution of this work consists in a data-guided approach to horizon tracking uncertainty analysis. This work is fundamentally based on a special reparameterization of reflectivity, known as deep prior. Feasible models are restricted to the output of a convolutional neural network with a fixed input, while weights and biases are Gaussian random variables. Given a deep prior model, the network parameters are sampled from the posterior distribution via a Markov chain Monte Carlo method, from which the conditional mean and point-wise standard deviation of the inferred reflectivities are approximated. For each sample of the posterior distribution, a reflectivity is generated, and the horizons are tracked automatically. In this way, uncertainty on model parameters naturally translates to horizon tracking. As part of the validation for the proposed approach, we verified that the estimated confidence intervals for the horizon tracking coincide with geologically complex regions, such as faults.
Deep Learning methods are known to suffer from calibration issues: they typically produce over-confident estimates. These problems are exacerbated in the low data regime. Although the calibration of probabilistic models is well studied, calibrating extremely over-parametrized models in the low-data regime presents unique challenges. We show that deep-ensembles do not necessarily lead to improved calibration properties. In fact, we show that standard ensembling methods, when used in conjunction with modern techniques such as mixup regularization, can lead to less calibrated models. In this text, we examine the interplay between three of the most simple and commonly used approaches to leverage deep learning when data is scarce: data-augmentation, ensembling, and post-processing calibration methods. We demonstrate that, although standard ensembling techniques certainly help to boost accuracy, the calibration of deep-ensembles relies on subtle trade-offs. Our main finding is that calibration methods such as temperature scaling need to be slightly tweaked when used with deep-ensembles and, crucially, need to be executed after the averaging process. Our simulations indicate that, in the low data regime, this simple strategy can halve the Expected Calibration Error (ECE) on a range of benchmark classification problems when compared to standard deep-ensembles.
Conventional uncertainty quantification methods usually lacks the capability of dealing with high-dimensional problems due to the curse of dimensionality. This paper presents a semi-supervised learning framework for dimension reduction and reliability analysis. An autoencoder is first adopted for mapping the high-dimensional space into a low-dimensional latent space, which contains a distinguishable failure surface. Then a deep feedforward neural network (DFN) is utilized to learn the mapping relationship and reconstruct the latent space, while the Gaussian process (GP) modeling technique is used to build the surrogate model of the transformed limit state function. During the training process of the DFN, the discrepancy between the actual and reconstructed latent space is minimized through semi-supervised learning for ensuring the accuracy. Both labeled and unlabeled samples are utilized for defining the loss function of the DFN. Evolutionary algorithm is adopted to train the DFN, then the Monte Carlo simulation method is used for uncertainty quantification and reliability analysis based on the proposed framework. The effectiveness is demonstrated through a mathematical example.
In inverse problems, we often have access to data consisting of paired samples $(x,y)sim p_{X,Y}(x,y)$ where $y$ are partial observations of a physical system, and $x$ represents the unknowns of the problem. Under these circumstances, we can employ supervised training to learn a solution $x$ and its uncertainty from the observations $y$. We refer to this problem as the supervised case. However, the data $ysim p_{Y}(y)$ collected at one point could be distributed differently than observations $ysim p_{Y}(y)$, relevant for a current set of problems. In the context of Bayesian inference, we propose a two-step scheme, which makes use of normalizing flows and joint data to train a conditional generator $q_{theta}(x|y)$ to approximate the target posterior density $p_{X|Y}(x|y)$. Additionally, this preliminary phase provides a density function $q_{theta}(x|y)$, which can be recast as a prior for the unsupervised problem, e.g.~when only the observations $ysim p_{Y}(y)$, a likelihood model $y|x$, and a prior on $x$ are known. We then train another invertible generator with output density $q_{phi}(x|y)$ specifically for $y$, allowing us to sample from the posterior $p_{X|Y}(x|y)$. We present some synthetic results that demonstrate considerable training speedup when reusing the pretrained network $q_{theta}(x|y)$ as a warm start or preconditioning for approximating $p_{X|Y}(x|y)$, instead of learning from scratch. This training modality can be interpreted as an instance of transfer learning. This result is particularly relevant for large-scale inverse problems that employ expensive numerical simulations.
We propose a novel Bayesian neural network architecture that can learn invariances from data alone by inferring a posterior distribution over different weight-sharing schemes. We show that our model outperforms other non-invariant architectures, when trained on datasets that contain specific invariances. The same holds true when no data augmentation is performed.
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