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.
We present a framework that enables estimation of low-dimensional sub-resolution reservoir properties directly from seismic data, without requiring the solution of a high dimensional seismic inverse problem. Our workflow is based on the Bayesian evidential learning approach and exploits learning the direct relation between seismic data and reservoir properties to efficiently estimate reservoir properties. The theoretical framework we develop allows incorporation of non-linear statistical models for seismic estimation problems. Uncertainty quantification is performed with Approximate Bayesian C…
Seismic imaging is an ill-posed inverse problem that is challenged by noisy data and modeling inaccuracies -- due to errors in the background squared-slowness model. Uncertainty quantification is essential for determining how variability in the background models affects seismic imaging. Due to the costs associated with the forward Born modeling operator as well as the high dimensionality of seismic images, quantification of uncertainty is computationally expensive. As such, the main contribution of this work is a survey-specific Fourier neural operator surrogate to velocity continuation that m…
Seismic fragility curves have been introduced as key components of Seismic Probabilistic Risk Assessment studies. They express the probability of failure of mechanical structures conditional to a seismic intensity measure and must take into account the inherent uncertainties in such studies, the so-called epistemic uncertainties (i.e. coming from the uncertainty on the mechanical parameters of the structure) and the aleatory uncertainties (i.e. coming from the randomness of the seismic ground motions). For simulation-based approaches we propose a methodology to build and calibrate a Gaussian p…
Geostatistical seismic inversion is commonly used to infer the spatial distribution of the subsurface petro-elastic properties by perturbing the model parameter space through iterative stochastic sequential simulations/co-simulations. The spatial uncertainty of the inferred petro-elastic properties is represented with the updated a posteriori variance from an ensemble of the simulated realizations. Within this setting, the large-scale geological (metaparameters) used to generate the petro-elastic realizations, such as the spatial correlation model and the global a priori distribution of the pr…
We propose to use techniques from Bayesian inference and deep neural networks to translate uncertainty in seismic imaging to uncertainty in tasks performed on the image, such as horizon tracking. Seismic imaging is an ill-posed inverse problem because of bandwidth and aperture limitations, which is hampered by the presence of noise and linearization errors. Many regularization methods, such as transform-domain sparsity promotion, have been designed to deal with the adverse effects of these errors, however, these methods run the risk of biasing the solution and do not provide information on unc…
In this study, we present a Bayesian hierarchical framework to model fluid-induced seismicity. The framework is based on a non-homogeneous Poisson process (NHPP) with a fluid-induced seismicity rate proportional to the rate of injected fluid. The fluid-induced seismicity rate model depends upon a set of physically meaningful parameters, and has been validated for six fluid-induced case studies. In line with the vision of hierarchical Bayesian modeling, the rate parameters are considered as random variables. We develop both the Bayesian inference and updating rules, which are used to develop a …
Spatially-Heterogeneous Causal Bayesian Networks for Seismic Multi-Hazard Estimation: A Variational Approach with Gaussian Processes and Normalizing Flows
Johns Hopkins University（约翰斯·霍普金斯大学）；Data Science and AI Institute, Johns Hopkins University（约翰斯·霍普金斯大学数据科学与人工智能学院）
This work develops rigorous theoretical basis for the fact that deep Bayesian neural network (BNN) is an effective tool for high-dimensional variable selection with rigorous uncertainty quantification. We develop new Bayesian non-parametric theorems to show that a properly configured deep BNN (1) learns the variable importance effectively in high dimensions, and its learning rate can sometimes "break" the curse of dimensionality. (2) BNN's uncertainty quantification for variable importance is rigorous, in the sense that its 95% credible intervals for variable importance indeed covers the truth…
China University of Petroleum（中国石油大学）；Yangtze University（长江大学）
Seismic inverse modeling is a common method in reservoir prediction and it plays a vital role in the exploration and development of oil and gas. Conventional seismic inversion method is difficult to combine with complicated and abstract knowledge on geological mode and its uncertainty is difficult to be assessed. The paper proposes an inversion modeling method based on GAN consistent with geology, well logs, seismic data. GAN is a the most promising generation model algorithm that extracts spatial structure and abstract features of training images. The trained GAN can reproduce the models with…
Geometry-Aware Bayesian Quantification via Compositional Data Analysis
Istituto di Scienza e Tecnologie dell’Informazione Consiglio Nazionale delle Ricerche（国家研究委员会信息科学与技术研究所）；Artificial Intelligence Center University of Oviedo（奥维耶多大学人工智能中心）
Harvard University（哈佛大学）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Bayesian Neural Networks (BNNs) place priors over the parameters in a neural network. Inference in BNNs, however, is difficult; all inference methods for BNNs are approximate. In this work, we empirically compare the quality of predictive uncertainty estimates for 10 common inference methods on both regression and classification tasks. Our experiments demonstrate that commonly used metrics (e.g. test log-likelihood) can be misleading. Our experiments also indicate that inference innovations designed to capture structure in the posterior do not necessarily produce high quality posterior approxi…
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for un…
We present an application of deep generative models in the context of partial-differential equation (PDE) constrained inverse problems. We combine a generative adversarial network (GAN) representing an a priori model that creates subsurface geological structures and their petrophysical properties, with the numerical solution of the PDE governing the propagation of acoustic waves within the earth's interior. We perform Bayesian inversion using an approximate Metropolis-adjusted Langevin algorithm (MALA) to sample from the posterior given seismic observations. Gradients with respect to the model…
What do data tell us about physics-and what don't they tell us? There has been a surge of interest in using machine learning models to discover governing physical laws such as differential equations from data, but current methods lack uncertainty quantification to communicate their credibility. This work addresses this shortcoming from a Bayesian perspective. We introduce a novel model comprising "leaf" modules that learn to represent distinct experiments' spatiotemporal functional data as neural networks and a single "root" module that expresses a nonparametric distribution over their governi…
Despite numerous studies of deep autoencoders (AEs) for unsupervised anomaly detection, AEs still lack a way to express uncertainty in their predictions, crucial for ensuring safe and trustworthy machine learning systems in high-stake applications. Therefore, in this work, the formulation of Bayesian autoencoders (BAEs) is adopted to quantify the total anomaly uncertainty, comprising epistemic and aleatoric uncertainties. To evaluate the quality of uncertainty, we consider the task of classifying anomalies with the additional option of rejecting predictions of high uncertainty. In addition, we…
University of Pennsylvania（宾夕法尼亚大学）；Purdue University（普渡大学）；University of Michigan（密歇根大学）
Deep neural network (DNN) models have achieved state-of-the-art predictive accuracy in a wide range of supervised learning applications. However, accurately quantifying the uncertainty in DNN predictions remains a challenging task. For continuous outcome variables, an even more difficult problem is to estimate the predictive density function, which not only provides a natural quantification of the predictive uncertainty, but also fully captures the random variation in the outcome. In this work, we propose the Bayesian Deep Noise Neural Network (B-DeepNoise), which generalizes standard Bayesian…
King Abdulllah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
Uncertainty quantification is crucial to inverse problems, as it could provide decision-makers with valuable information about the inversion results. For example, seismic inversion is a notoriously ill-posed inverse problem due to the band-limited and noisy nature of seismic data. It is therefore of paramount importance to quantify the uncertainties associated to the inversion process to ease the subsequent interpretation and decision making processes. Within this framework of reference, sampling from a target posterior provides a fundamental approach to quantifying the uncertainty in seismic …
Deep neural networks have outperformed existing machine learning models in various molecular applications. In practical applications, it is still difficult to make confident decisions because of the uncertainty in predictions arisen from insufficient quality and quantity of training data. Here, we show that Bayesian neural networks are useful to quantify the uncertainty of molecular property prediction with three numerical experiments. In particular, it enables us to decompose the predictive variance into the model- and data-driven uncertainties, which helps to elucidate the source of errors. …
KAIST（韩国科学技术院）；Seoul National University（首尔国立大学）；KAIST Institute for Artificial Intelligence（韩国科学技术院人工智能研究所）
In chemistry, deep neural network models have been increasingly utilized in a variety of applications such as molecular property predictions, novel molecule designs, and planning chemical reactions. Despite the rapid increase in the use of state-of-the-art models and algorithms, deep neural network models often produce poor predictions in real applications because model performance is highly dependent on the quality of training data. In the field of molecular analysis, data are mostly obtained from either complicated chemical experiments or approximate mathematical equations, and then quality …
Center for Pattern Recognition and Data Analytics (PRaDA)（模式识别与数据分析中心（PRaDA））；Deakin University（迪肯大学）
In this paper we consider the problem of finding stable maxima of expensive (to evaluate) functions. We are motivated by the optimisation of physical and industrial processes where, for some input ranges, small and unavoidable variations in inputs lead to unacceptably large variation in outputs. Our approach uses multiple gradient Gaussian Process models to estimate the probability that worst-case output variation for specified input perturbation exceeded the desired maxima, and these probabilities are then used to (a) guide the optimisation process toward solutions satisfying our stability cr…
We introduce a class of acquisition functions for sample selection that leads to faster convergence in applications related to Bayesian experimental design and uncertainty quantification. The approach follows the paradigm of active learning, whereby existing samples of a black-box function are utilized to optimize the next most informative sample. The proposed method aims to take advantage of the fact that some input directions of the black-box function have a larger impact on the output than others, which is important especially for systems exhibiting rare and extreme events. The acquisition …
This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation. The holding times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning …
We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder-decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since normal neural networks are data intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein's method is scaled to deep convolutional networks to perform approximate Baye…
Model misspecification constitutes a major obstacle to reliable inference in many inverse problems. Inverse problems in seismology, for example, are particularly affected by misspecification of wave propagation velocities. In this paper, we focus on a specific seismic inverse problem - full-waveform moment tensor inversion - and develop a Bayesian framework that seeks robustness to velocity misspecification. A novel element of our framework is the use of transport-Lagrangian (TL) distances between observed and model predicted waveforms to specify a loss function, and the use of this loss to de…
Two data-dependent information metrics are developed to quantify the information of the prior and likelihood functions within a parametric Bayesian model, one of which is closely related to the reference priors from Berger, Bernardo, and Sun, and information measure introduced by Lindley. A combination of theoretical, empirical, and computational support provides evidence that these information-theoretic metrics may be useful diagnostic tools when performing a Bayesian analysis.
Paderborn University（帕德博恩大学）；University of Munich (LMU)（慕尼黑大学）
The idea to distinguish and quantify two important types of uncertainty, often referred to as aleatoric and epistemic, has received increasing attention in machine learning research in the last couple of years. In this paper, we consider ensemble-based approaches to uncertainty quantification. Distinguishing between different types of uncertainty-aware learning algorithms, we specifically focus on Bayesian methods and approaches based on so-called credal sets, which naturally suggest themselves from an ensemble learning point of view. For both approaches, we address the question of how to quan…
We apply a linear Bayesian model to seismic tomography, a high-dimensional inverse problem in geophysics. The objective is to estimate the three-dimensional structure of the earth's interior from data measured at its surface. Since this typically involves estimating thousands of unknowns or more, it has always been treated as a linear(ized) optimization problem. Here we present a Bayesian hierarchical model to estimate the joint distribution of earth structural and earthquake source parameters. An ellipsoidal spatial prior allows to accommodate the layered nature of the earth's mantle. With ou…
We propose the first Bayesian encoder for metric learning. Rather than relying on neural amortization as done in prior works, we learn a distribution over the network weights with the Laplace Approximation. We actualize this by first proving that the contrastive loss is a valid log-posterior. We then propose three methods that ensure a positive definite Hessian. Lastly, we present a novel decomposition of the Generalized Gauss-Newton approximation. Empirically, we show that our Laplacian Metric Learner (LAM) estimates well-calibrated uncertainties, reliably detects out-of-distribution examples…
This work affords new insights into Bayesian CART in the context of structured wavelet shrinkage. The main thrust is to develop a formal inferential framework for Bayesian tree-based regression. We reframe Bayesian CART as a g-type prior which departs from the typical wavelet product priors by harnessing correlation induced by the tree topology. The practically used Bayesian CART priors are shown to attain adaptive near rate-minimax posterior concentration in the supremum norm in regression models. For the fundamental goal of uncertainty quantification, we construct adaptive confidence bands f…
Deep neural networks offer numerous potential applications across geoscience, for example, one could argue that they are the state-of-the-art method for predicting faults in seismic datasets. In quantitative reservoir characterization workflows, it is common to incorporate the uncertainty of predictions thus such subsurface models should provide calibrated probabilities and the associated uncertainties in their predictions. It has been shown that popular Deep Learning-based models are often miscalibrated, and due to their deterministic nature, provide no means to interpret the uncertainty of t…
We begin by introducing the main ideas of the paper under discussion, and we give a brief description of the method proposed. Next, we discuss an alternative approach based on B-spline expansion, and lastly we make some comments on the method's convergence rate.
Department of Biostatistics, University of Oslo（奥斯陆大学生物统计系）；Helsinki Institute for Information Technology HIIT, Department of Computer Science, Aalto University（赫尔辛基信息技术研究所，阿尔托大学计算机科学系）
The computational efficiency of approximate Bayesian computation (ABC) has been improved by using surrogate models such as Gaussian processes (GP). In one such promising framework the discrepancy between the simulated and observed data is modelled with a GP which is further used to form a model-based estimator for the intractable posterior. In this article we improve this approach in several ways. We develop batch-sequential Bayesian experimental design strategies to parallellise the expensive simulations. In earlier work only sequential strategies have been used. Current surrogate-based ABC m…
University of Tübingen（图宾根大学）；University of Genova（热那亚大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；Massachusetts Institute of Technology（麻省理工学院）；Max-Planck-Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit uncertainty quantification. This is arguably even more of a problem for this kind of tasks than elsewhere in machine learning, because the dynamical systems typically described by PDEs often exhibit subtle, multiscale structure that makes errors hard to spot by humans. In this work, we first provide a mathematically detailed Bayesian formulation of the ''shallow'' (li…
We propose a new, two-step empirical Bayes-type of approach for neural networks. We show in context of the nonparametric regression model that the procedure (up to a logarithmic factor) provides optimal recovery of the underlying functional parameter of interest and provides Bayesian credible sets with frequentist coverage guarantees. The approach requires fitting the neural network only once, hence it is substantially faster than Bootstrapping type approaches. We demonstrate the applicability of our method over synthetic data, observing good estimation properties and reliable uncertainty quan…
University of Chicago（芝加哥大学）；Kavli Institute for Cosmological Physics（卡夫利宇宙物理研究所）；Fermi National Accelerator Laboratory（费米国家加速器实验室）
Measuring the structural parameters (size, total brightness, light concentration, etc.) of galaxies is a significant first step towards a quantitative description of different galaxy populations. In this work, we demonstrate that a Bayesian Neural Network (BNN) can be used for the inference, with uncertainty quantification, of such morphological parameters from simulated low-surface-brightness galaxy images. Compared to traditional profile-fitting methods, we show that the uncertainties obtained using BNNs are comparable in magnitude, well-calibrated, and the point estimates of the parameters …
Optical flow refers to the visual motion observed between two consecutive images. Since the degree of freedom is typically much larger than the constraints imposed by the image observations, the straightforward formulation of optical flow as an inverse problem is ill-posed. Standard approaches to determine optical flow rely on formulating and solving an optimization problem that contains both a data fidelity term and a regularization term, the latter effectively resolves the otherwise ill-posedness of the inverse problem. In this work, we depart from the deterministic formalism, and instead tr…
We consider the problem of uncertainty quantification for an unknown low-rank matrix $\mathbf{X}$, given a partial and noisy observation of its entries. This quantification of uncertainty is essential for many real-world problems, including image processing, satellite imaging, and seismology, providing a principled framework for validating scientific conclusions and guiding decision-making. However, existing literature has mainly focused on the completion (i.e., point estimation) of the matrix $\mathbf{X}$, with little work on investigating its uncertainty. To this end, we propose in this work…
Due to the importance of uncertainty quantification (UQ), Bayesian approach to inverse problems has recently gained popularity in applied mathematics, physics, and engineering. However, traditional Bayesian inference methods based on Markov Chain Monte Carlo (MCMC) tend to be computationally intensive and inefficient for such high dimensional problems. To address this issue, several methods based on surrogate models have been proposed to speed up the inference process. More specifically, the calibration-emulation-sampling (CES) scheme has been proven to be successful in large dimensional UQ pr…
Concordia University（康考迪亚大学）；University of Wisconsin（威斯康星大学）
Quantitative ultrasound (QUS) allows estimating the intrinsic tissue properties. Speckle statistics are the QUS parameters that describe the first order statistics of ultrasound (US) envelope data. The parameters of Homodyned K-distribution (HK-distribution) are the speckle statistics that can model the envelope data in diverse scattering conditions. However, they require a large amount of data to be estimated reliably. Consequently, finding out the intrinsic uncertainty of the estimated parameters can help us to have a better understanding of the estimated parameters. In this paper, we propos…
