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Register a new userLearning by example: fast reliability-aware seismic imaging with normalizing flows
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Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data from neighboring surveys. To arrive at this result, we train the NF on pairs of low- and high-fidelity migrated images. In our numerical example, we obtain high-fidelity images from the Parihaka dataset and low-fidelity images are derived from these images through the process of demigration, followed by adding noise and migration. During inference, given shot records from a new neighboring seismic survey, we first compute the reverse-time migration image. Next, by feeding this low-fidelity migrated image to the NF we gain access to samples from the posterior distribution virtually for free. We use these samples to compute a high-fidelity image including a first assessment of the images reliability. To our knowledge, this is the first attempt to train a conditional network on what we know from neighboring images to improve the current image and assess its reliability.
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In supervised learning, it is known that overparameterized neural networks with one hidden layer provably and efficiently learn and generalize, when trained using stochastic gradient descent with sufficiently small learning rate and suitable initialization. In contrast, the benefit of overparameterization in unsupervised learning is not well understood. Normalizing flows (NFs) constitute an important class of models in unsupervised learning for sampling and density estimation. In this paper, we theoretically and empirically analyze these models when the underlying neural network is one-hidden-layer overparameterized network. Our main contributions are two-fold: (1) On the one hand, we provide theoretical and empirical evidence that for a class of NFs containing most of the existing NF models, overparametrization hurts training. (2) On the other hand, we prove that unconstrained NFs, a recently introduced model, can efficiently learn any reasonable data distribution under minimal assumptions when the underlying network is overparametrized.
To optimally monitor earthquake-generating processes, seismologists have sought to lower detection sensitivities ever since instrumental seismic networks were started about a century ago. Recently, it has become possible to search continuous waveform archives for replicas of previously recorded events (template matching), which has led to at least an order of magnitude increase in the number of detected earthquakes and greatly sharpened our view of geological structures. Earthquake catalogs produced in this fashion, however, are heavily biased in that they are completely blind to events for which no templates are available, such as in previously quiet regions or for very large magnitude events. Here we show that with deep learning we can overcome such biases without sacrificing detection sensitivity. We trained a convolutional neural network (ConvNet) on the vast hand-labeled data archives of the Southern California Seismic Network to detect seismic body wave phases. We show that the ConvNet is extremely sensitive and robust in detecting phases, even when masked by high background noise, and when the ConvNet is applied to new data that is not represented in the training set (in particular, very large magnitude events). This generalized phase detection (GPD) framework will significantly improve earthquake monitoring and catalogs, which form the underlying basis for a wide range of basic and applied seismological research.
Normalizing flows, which learn a distribution by transforming the data to samples from a Gaussian base distribution, have proven powerful density approximations. But their expressive power is limited by this choice of the base distribution. We, therefore, propose to generalize the base distribution to a more elaborate copula distribution to capture the properties of the target distribution more accurately. In a first empirical analysis, we demonstrate that this replacement can dramatically improve the vanilla normalizing flows in terms of flexibility, stability, and effectivity for heavy-tailed data. Our results suggest that the improvements are related to an increased local Lipschitz-stability of the learned flow.
Incorporating prior knowledge on model unknowns of interest is essential when dealing with ill-posed inverse problems due to the nonuniqueness of the solution and data noise. Unfortunately, it is not trivial to fully describe our priors in a convenient and analytical way. Parameterizing the unknowns with a convolutional neural network (CNN), and assuming an uninformative Gaussian prior on its weights, leads to a variational prior on the output space that favors natural images and excludes noisy artifacts, as long as overfitting is prevented. This is the so-called deep-prior approach. In seismic imaging, however, evaluating the forward operator is computationally expensive, and training a randomly initialized CNN becomes infeasible. We propose, instead, a weak version of deep priors, which consists of relaxing the requirement that reflectivity models must lie in the network range, and letting the unknowns deviate from the network output according to a Gaussian distribution. Finally, we jointly solve for the reflectivity model and CNN weights. The chief advantage of this approach is that the updates for the CNN weights do not involve the modeling operator, and become relatively cheap. Our synthetic numerical experiments demonstrate that the weak deep prior is more robust with respect to noise than conventional least-squares imaging approaches, with roughly twice the computational cost of reverse-time migration, which is the affordable computational budget in large-scale imaging problems.
We present a novel integrator based on normalizing flows which can be used to improve the unweighting efficiency of Monte-Carlo event generators for collider physics simulations. In contrast to machine learning approaches based on surrogate models, our method generates the correct result even if the underlying neural networks are not optimally trained. We exemplify the new strategy using the example of Drell-Yan type processes at the LHC, both at leading and partially at next-to-leading order QCD.
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