We outline new approaches to incorporate ideas from deep learning into wave-based least-squares imaging. The aim, and main contribution of this work, is the combination of handcrafted constraints with deep convolutional neural networks, as a way to harness their remarkable ease of generating natural images. The mathematical basis underlying our method is the expectation-maximization framework, where data are divided in batches and coupled to additional "latent" unknowns. These unknowns are pairs of elements from the original unknown space (but now coupled to a specific data batch) and network inputs. In this setting, the neural network controls the similarity between these additional parameters, acting as a "center" variable. The resulting problem amounts to a maximum-likelihood estimation of the network parameters when the augmented data model is marginalized over the latent variables. In this work, we tested an inverse problem framework which includes hard constraints and deep priors. Hard constraints are necessary in many problems, such as seismic imaging, where the unknowns must belong to a feasible set in order to ensure the numerical stability of the forward problem. Deep priors, enforced through adherence to the range of a neural network, provide an additional, implicit type of regularization, as demonstrated by recent work [2, Dittmer et al. [3] ], and corroborated by our numerical results. The resulting algorithm can be mathematically interpreted in light of expectation maximization methods. Furthermore, connections to elastic averaging SGD [10] highlight potential computational benefits of a parallel (synchronous or asynchronous) implementation. On a speculative note, we argue that the presented method, which combines stochastic optimization on the dual variable with on-the-fly estimation of the generative model's weights using Langevin dynamics, reaps information on the "posterior" distribution leveraging multiplicity in the data and the fact that the data is acquired over one and the same Earth model. Our preliminary results seem consistent with a behavior to be expected from a "posterior" distribution. b,c) sample "prior" (before training) and "posterior" distribution functions for two points in the model.
We combine hard handcrafted constraints with a deep prior weak constraint to perform seismic imaging and reap information on the "posterior" distribution leveraging multiplicity in the data.
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