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Abstract Due to their uncertainty quantification, Bayesian solutions to inverse problems are the framework of choice in applications that are risk averse. These benefits come at the cost of computations that are in general, intractable. New advances in machine learning and variational inference (VI) have lowered this computational barrier by leveraging data-driven learning. Two VI paradigms have emerged that represent different tradeoffs: amortized and non-amortized. Amortized VI can produce fast results but due to generalizing to many observed datasets it produces suboptimal inference results. Non-amortized VI is slower at inference but finds better posterior approximations since it is specialized towards a single observed dataset. Current amortized VI techniques run into a sub-optimality wall that cannot be improved without more expressive neural networks or extra training data. We present a solution that enables iterative improvement of amortized posteriors that uses the same networks architectures and training data. The benefits of our method requires extra computations but these remain frugal since they are based on physics-hybrid methods and summary statistics. Importantly, these computations remain mostly offline thus our method maintains cheap and reusable online evaluation while bridging the optimality gap between these two paradigms. We denote our proposed methodASPIRE-Amortized posteriors withSummaries that arePhysics-based andIterativelyREfined. We first validate our method on a stylized problem with a known posterior then demonstrate its practical use on a high-dimensional and nonlinear transcranial medical imaging problem with ultrasound. Compared with the baseline and previous methods in the literature, ASPIRE stands out as an computationally efficient and high-fidelity method for posterior inference. 

Normalizing flows is a density estimation method that provides efficient exact likelihood estimation and sampling (Dinh et al., 2014) from high-dimensional distributions. This method depends on the use of the change of variables formula, which requires an invertible transform. Thus normalizing flow architectures are built to be invertible by design (Dinh et al., 2014). In theory, the invertibility of architectures constrains the expressiveness, but the use of coupling layers allows normalizing flows to exploit the power of arbitrary neural networks, which do not need to be invertible, (Dinh et al., 2016) and layer invertibility means that, if properly implemented, many layers can be stacked to increase expressiveness without creating a training memory bottleneck. The package we present, InvertibleNetworks.jl, is a pure Julia (Bezanson et al., 2017) imple- mentation of normalizing flows. We have implemented many relevant neural network layers, including GLOW 1x1 invertible convolutions (Kingma & Dhariwal, 2018), affine/additive coupling layers (Dinh et al., 2014), Haar wavelet multiscale transforms (Haar, 1909), and Hierarchical invertible neural transport (HINT) (Kruse et al., 2021), among others. These modular layers can be easily composed and modified to create different types of normalizing flows. As starting points, we have implemented RealNVP, GLOW, HINT, Hyperbolic networks (Lensink et al., 2022) and their conditional counterparts for users to quickly implement their individual applications. 

Modern-day reservoir management and monitoring of geologic carbon storage increasingly call for costly time-lapse seismic data collection. We demonstrate how techniques from graph theory can be used to optimize acquisition geometries for low-cost sparse 4D seismic data. Based on midpoint-offset-domain connectivity arguments, our algorithm automatically produces sparse nonreplicated time-lapse acquisition geometries that favor wavefield recovery. 

We present the Seismic Laboratory for Imaging and Modeling/Monitoring open-source software framework for computational geophysics and, more generally, inverse problems involving the wave equation (e.g., seismic and medical ultrasound), regularization with learned priors, and learned neural surrogates for multiphase flow simulations. By integrating multiple layers of abstraction, the software is designed to be both readable and scalable, allowing researchers to easily formulate problems in an abstract fashion while exploiting the latest developments in high-performance computing. The design principles and their benefits are illustrated and demonstrated by means of building a scalable prototype for permeability inversion from time-lapse crosswell seismic data, which, aside from coupling of wave physics and multiphase flow, involves machine learning.