@conference {orozco2022SPIEadjoint,
	title = {Adjoint operators enable fast and amortized machine learning based Bayesian uncertainty quantification},
	booktitle = {SPIE Medical Imaging Conference},
	year = {2023},
	note = {(SPIE, San Diego)},
	month = {02},
	abstract = {Machine learning algorithms such as Normalizing Flows, VAEs, and GANs are powerful tools in Bayesian uncertainty quantification (UQ) of inverse problems. Unfortunately, when using these algorithms medical imaging practitioners are faced with the challenging task of manually defining neural networks that can handle complicated inputs such as acoustic data. This task needs to be replicated for different receiver types or configurations since these change the dimensionality of the input. We propose to first transform the data using the adjoint operator {\textemdash}ex: time reversal in photoacoustic imaging (PAI) or back-projection in computer tomography (CT) imaging {\textemdash} then continue posterior inference using the adjoint data as an input now that it has been standardized to the size of the unknown model. This adjoint preprocessing technique has been used in previous works but with minimal discussion on if it is biased. In this work, we prove that conditioning on adjoint data is unbiased for a certain class of inverse problems. We then demonstrate with two medical imaging examples (PAI and CT) that adjoints enable two things: Firstly, adjoints partially undo the physics of the forward operator resulting in faster convergence of an ML based Bayesian UQ technique. Secondly, the algorithm is now robust to changes in the observed data caused by differing transducer subsampling in PAI and number of angles in CT. Our adjoint-based Bayesian inference method results in point estimates that are faster to compute than traditional baselines and show higher SSIM metrics, while also providing validated UQ.},
	keywords = {Amortized Inference, Bayesian inference, deep learning, Inverse problems, machine learning, medical imaging, Normalizing flows, SPIE, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SPIE/2023/orozco2022SPIEadjoint/SPIE_2022_adjoint.html},
	author = {Rafael Orozco and Ali Siahkoohi and Gabrio Rizzuti and Tristan van Leeuwen and Felix J. Herrmann}
}
@unpublished {orozco2023invnet,
	title = {InvertibleNetworks.jl: A Julia package for scalable normalizing flows},
	year = {2023},
	month = {12},
	abstract = {InvertibleNetworks.jl is a Julia package designed for the scalable implementation of normalizing flows, a method for density estimation and sampling in high-dimensional distributions. This package excels in memory efficiency by leveraging the inherent invertibility of normalizing flows, which significantly reduces memory requirements during backpropagation compared to existing normalizing flow packages that rely on automatic differentiation frameworks. InvertibleNetworks.jl has been adapted for diverse applications, including seismic imaging, medical imaging, and CO2 monitoring, demonstrating its effectiveness in learning high-dimensional distributions.},
	keywords = {Bayesian inference, computing, conditional normalizing flows, deep learning, HPC, Inverse problems, memory, Normalizing flows, software, Uncertainty quantification},
	doi = {10.48550/arXiv.2312.13480},
	author = {Rafael Orozco and Philipp A. Witte and Mathias Louboutin and Ali Siahkoohi and Gabrio Rizzuti and Bas Peters and Felix J. Herrmann}
}
@article {louboutin2023lmi,
	title = {Learned multiphysics inversion with differentiable programming and machine learning},
	journal = {The Leading Edge},
	volume = {42},
	year = {2023},
	note = {(The Leading Edge)},
	month = {07},
	pages = {452-516},
	abstract = {We present the Seismic Laboratory for Imaging and Modeling/Monitoring (SLIM) 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, our software is designed to be both readable and scalable. This allows researchers to easily formulate their problems in an abstract fashion while exploiting the latest developments in high-performance computing. We illustrate and demonstrate our design principles and their benefits 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.},
	keywords = {CCS, deep learning, Fourier neural operators, GCS, Imaging, inversion, invertible networks, machine learning, monitoring, multiphysics, Normalizing flows, SLIM, software, time-lapse},
	url = {https://library.seg.org/doi/10.1190/tle42070474.1},
	presentation = {https://slim.gatech.edu/Publications/Public/Journals/TheLeadingEdge/2023/louboutin2023lmi/le_software.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Journals/TheLeadingEdge/2023/louboutin2023lmi/le_software.html},
	author = {Mathias Louboutin and Ziyi Yin and Rafael Orozco and Thomas J. Grady II and Ali Siahkoohi and Gabrio Rizzuti and Philipp A. Witte and Olav M{\o}yner and Gerard J. Gorman and Felix J. Herrmann}
}
@article {siahkoohi2022ravi,
	title = {Reliable amortized variational inference with physics-based latent distribution correction},
	journal = {Geophysics},
	volume = {88},
	number = {3},
	year = {2023},
	note = {(Geophysics)},
	month = {05},
	abstract = {Bayesian inference for high-dimensional inverse problems is challenged by the computational costs of the forward operator and the selection of an appropriate prior distribution. Amortized variational inference addresses these challenges where a neural network is trained to approximate the posterior distribution over existing pairs of model and data. When fed previously unseen data and normally distributed latent samples as input, the pretrained deep neural network{\textendash}-in our case a conditional normalizing flow{\textendash}-provides posterior samples with virtually no cost. However, the accuracy of this approach relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems because of the highly heterogeneous structure of the Earth. In addition, accurate amortized variational inference requires the observed data to be drawn from the training data distribution. As such, we offer a solution that increases the resilience of amortized variational inference when faced with data distribution shifts, e.g., changes in the forward model or prior distribution. Our method involves a physics-based correction to the conditional normalizing flow latent distribution to provide a more accurate approximation to the posterior distribution for the observed data at hand. To accomplish this, instead of a standard Gaussian latent distribution, we parameterize the latent distribution by a Gaussian distribution with an unknown mean and diagonal covariance. These unknown quantities are then estimated by minimizing the Kullback-Leibler divergence between the corrected and true posterior distributions. While generic and applicable to other inverse problems, by means of a seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in number of source experiments, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty with approximately the same cost as five reverse-time migrations.},
	keywords = {Inverse problems, Normalizing flows, seismic imaging, Uncertainty quantification, Variational Inference},
	doi = {10.1190/geo2022-0472.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2023/siahkoohi2022ravi/paper.html},
	software = {https://github.com/slimgroup/ReliableAVI.jl},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Rafael Orozco and Felix J. Herrmann}
}
@presentation {orozco2022ML4SEISMICaos,
	title = {Adjoint operators as summary functions in amortized Bayesian inference frameworks},
	year = {2022},
	month = {11},
	abstract = {An important concern in seismic inverse problems is the large and varying size of observed data. The large size can cause computational cost concerns and its varying size (such as when changing receiver geometries) implies the need to rerun inference algorithms from scratch for each new observation. Motivated by these two problems, we take inspiration from the statistics literature which commonly relies on summary statistic of observed data. Summary statistic compress the observed data leaving only information needed for inference. In this work, we argue that the adjoint operator provides a natural candidate for a summary function in the context of physics-based inverse problems. We first mathematically show that for certain general assumptions transforming data under the adjoint operator defines a new conditional distribution which preserves the expectations of the original posterior. We validate our hypothesis by evaluating our framework in a learned amortized inference algorithm. Our seismic and medical synthetic experiments show computational gains and increased quality of point estimates using our framework. We discuss statistical metrics that show our learned posterior is well calibrated therefore justifying its use in uncertainty quantification.},
	keywords = {Amortized Inference, Bayesian inference, deep learning, Inverse problems, machine learning, medical imaging, ML4SEISMIC, Normalizing flows, SLIM, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/ML4SEISMIC/2022/orozco2022ML4SEISMICaos/index.html},
	author = {Rafael Orozco and Mathias Louboutin and Ali Siahkoohi and Gabrio Rizzuti and Felix J. Herrmann}
}
@presentation {siahkoohi2022ML4SEISMICluq,
	title = {Low-cost uncertainty quantification for large-scale inverse problems},
	year = {2022},
	month = {11},
	abstract = {Bayesian inference for large-scale inverse problems is challenged by the computationally costly forward operator evaluations during posterior distribution sampling. Recent advances in variational inference and deep learning reduce these costs by pretraining a neural network capable of sampling the posterior distribution for previously unseen observed data. In geophysical applications, however, the accuracy of these methods depends on sufficiently capturing subsurface variability through a training dataset, which is challenging given the heterogeneity of the Earth{\textquoteright}s subsurface and our lack of access to it. Moreover, these methods may be unreliable in the presence of data distribution shifts, e.g., a change in the number of source experiments, noise distribution, or geological features to be imaged. As such, we present a solution that increases the robustness of deep-learning-based Bayesian inference approaches when faced with changes in data distribution. Our proposed method involves a physics-based adaptation to the latent distribution of a conditional normalizing flow that is pretrained to approximate the posterior distribution for previously unseen data. Instead of feeding standard Gaussian latent samples to the conditional normalizing flow, this method parameterizes the latent distribution by a Gaussian distribution with an unknown mean and diagonal covariance, estimated by minimizing the Kullback-Leibler divergence between predicted and true posterior distributions. This method is applicable to a wide range of inverse problems and has the potential to significantly reduce the costs of Bayesian variational inference. By means of a realistic seismic imaging example we demonstrate that the proposed latent distribution adaptation method mitigates the Bayesian inference errors induced by data distribution shifts, including shifts in the forward model and prior distribution.},
	keywords = {Inverse problems, ML4SEISMIC, Normalizing flows, seismic imaging, SLIM, Uncertainty quantification, Variational Inference},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/ML4SEISMIC/2022/siahkoohi2022ML4SEISMICluq/siahkoohi2022ML4SEISMICluq_pres.pdf},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Rafael Orozco and Felix J. Herrmann}
}
@conference {siahkoohi2022EAGEweb,
	title = {Wave-equation based inversion with amortized variational Bayesian inference},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2022},
	note = {(EAGE, Madrid)},
	month = {06},
	pages = {Session 2: Velocity model building and imaging (different domains)},
	abstract = {Solving inverse problems involving measurement noise and modeling errors requires regularization in order to avoid data overfit. Geophysical inverse problems, in which the Earth{\textquoteright}s highly heterogeneous structure is unknown, present a challenge in encoding prior knowledge through analytical expressions. Our main contribution is a generative-model-based regularization approach, robust to out-of-distribution data, which exploits the prior knowledge embedded in existing data and model pairs. Utilizing an amortized variational inference objective, a conditional normalizing flow (NF) is pretrained on pairs of low- and high-fidelity migrated images in order to achieve a low-fidelity approximation to the seismic imaging posterior distribution for previously unseen data. The NF is used after pretraining to reparameterize the unknown seismic image in an inversion scheme involving physics-guided data misfit and a Gaussian prior on the NF latent variable. Solving this optimization problem with respect to the latent variable enables us to leverage the benefits of data-driven conditional priors whilst being informed by physics and data. The numerical experiments demonstrate that the proposed inversion scheme produces seismic images with limited artifacts when dealing with noisy and out-of-distribution data.},
	keywords = {conditional priors, EAGE, Normalizing flows, seismic imaging},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2022/siahkoohi2022EAGEweb/abstract.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2022/siahkoohi2022EAGEweb/Mon-Herrmann.pdf},
	software = {https://github.com/slimgroup/ConditionalNFs4Imaging.jl},
	author = {Ali Siahkoohi and Rafael Orozco and Gabrio Rizzuti and Felix J. Herrmann}
}
@conference {siahkoohi2021EarthMLfar,
	title = {Fast and reliability-aware seismic imaging with conditional normalizing flows},
	booktitle = {Intelligent illumination of the Earth},
	year = {2021},
	note = {(KAUST, virtual)},
	month = {09},
	abstract = {The posterior probability distribution provides a comprehensive description of the solution in ill-posed inverse problems. Sampling from the posterior distribution in the context of seismic imaging is challenged by the high-dimensionality of the unknown and the expensive-to-evaluate forward operator. These challenges limit the applicability of Markov Chain sampling methods due to the costs associated with the forward operator. Moreover, explicitly choosing a prior distribution that captures the true heterogeneity exhibited by the Earth{\textquoteright}s subsurface further complicates casting seismic imaging into a Bayesian framework. To handle this situation and to assess uncertainty, we propose a data-driven variational inference approach based on conditional normalizing flows (NFs). The proposed scheme leverages existing data, which are in the form of low- and high-fidelity migrated image pairs, to train a conditional NF capable of characterizing the posterior distribution. After training, the NF can be used to sample from the posterior distribution associated with a previously unseen seismic survey, which is in some sense close, e.g., data from a neighboring survey area. 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 band-limited 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 image{\textquoteright}s reliability.},
	keywords = {Normalizing flows, seismic imaging, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/KAUST/2021/siahkoohi2021EarthMLfar/siahkoohi2021EarthMLfar.pdf},
	author = {Ali Siahkoohi and Rafael Orozco and Gabrio Rizzuti and Philipp A. Witte and Mathias Louboutin and Felix J. Herrmann}
}
@conference {witte2021JULIACONmedlj,
	title = {InvertibleNetworks.jl - Memory efficient deep learning in Julia},
	booktitle = {JuliaCon},
	year = {2021},
	note = {(JuliaCon, virtual)},
	month = {07},
	abstract = {We present InvertibleNetworks.jl, an open-source package for invertible neural networks and normalizing flows using memory-efficient backpropagation. InvertibleNetworks.jl uses manually implement gradients to take advantage of the invertibility of building blocks, which allows for scaling to large-scale problem sizes. We present the architecture and features of the library and demonstrate its application to a variety of problems ranging from loop unrolling to uncertainty quantification.},
	keywords = {deep learning, Invertible network, Julia, Normalizing flows, segmentation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/JuliaCon/2021/witte2021JULIACONmedlj/witte2021JULIACONmedlj.pdf},
	url2 = {https://www.youtube.com/watch?v=9M-zEGHY4i4},
	author = {Philipp A. Witte and Mathias Louboutin and Ali Siahkoohi and Gabrio Rizzuti and Bas Peters and Felix J. Herrmann}
}
@conference {herrmann2021Delftlwi,
	title = {Learned wave-based imaging - variational inference at scale},
	booktitle = {Delft},
	year = {2021},
	note = {(Delft, virtual)},
	month = {06},
	abstract = {High dimensionality, complex physics, and lack of access to the ground truth rank medical ultrasound and seismic imaging amongst the most challenging problems in the computational imaging sciences. If these challenges were not bad enough, modern applications of computational imaging increasingly call for the assessment of uncertainty on the image itself and on subsequent tasks. During this talk, I will show how recent developments in Normalizing Flows, a new type of invertible neural networks, can be used to cast wave-based imaging into a scalable Bayesian framework. Contrary to conventional methods, where sample images are drawn from the posterior distribution during inversion, our approach trains Normalizing Flows capable of generating samples from the posterior. Aside from greatly reducing the computational cost, this approach gives us access to the image itself (via Maximum a posteriori or mean estimation) and its multidimensional statistical distribution including its pointwise variance.},
	keywords = {medical imaging, Normalizing flows, seismic imaging, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/Delft/2021/herrmann2021Delftlwi/herrmann2021Delftlwi.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/Delft/2021/herrmann2021Delftlwi/herrmann2021Delftlwi.mp4},
	author = {Felix J. Herrmann and Ali Siahkoohi and Rafael Orozco and Gabrio Rizzuti and Philipp A. Witte and Mathias Louboutin}
}
@presentation {siahkoohi2021ML4SEISMICmcn,
	title = {Multifidelity conditional normalizing flows for physics-guided Bayesian inference},
	year = {2021},
	month = {11},
	abstract = {We introduce a scalable Bayesian inference approach that combines techniques from deep learning with a physic-based variational inference formulation. Bayesian inference for ill-posed inverse problems is challenged by the high-dimensionality of the unknown, computationally expensive forward operator, and choosing a prior distribution that accurately encodes prior knowledge on the unknown. To handle this situation and to assess uncertainty, we propose to approximate the posterior distribution using a pretrained conditional normalizing flow, which is trained on existing low- and high-fidelity estimations of the unknown. To further improve the accuracy of this approximation, we use transfer learning and finetune this normalizing flow by minimizing the Kullback-Leibler divergence between the predicted and the desired high-fidelity posterior density. This amounts to minimizing a physic-based variational inference objective with respect to the network weights, which we believe might scale better than Bayesian inference with Markov Chain sampling methods. We apply the proposed Bayesian inference approach to seismic imaging where we use quasi-real data obtained from the Parihaka dataset.},
	keywords = {deep learning, ML4SEISMIC, Normalizing flows, seismic imaging, SLIM, Variational Inference},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/ML4SEISMIC/2021/siahkoohi2021ML4SEISMICmcn/Mon-11-45-Siahkoohi.pdf},
	author = {Ali Siahkoohi and Rafael Orozco and Gabrio Rizzuti and Philipp A. Witte and Mathias Louboutin and Felix J. Herrmann}
}
@conference {siahkoohi2021AABIpto,
	title = {Preconditioned training of normalizing flows for variational inference in inverse problems},
	booktitle = {3rd Symposium on Advances in Approximate Bayesian Inference},
	year = {2021},
	note = {AABI},
	month = {01},
	abstract = {Obtaining samples from the posterior distribution of inverse problems with expensive forward operators is challenging especially when the unknowns involve the strongly heterogeneous Earth. To meet these challenges, we propose a preconditioning scheme involving a conditional normalizing flow (NF) capable of sampling from a low-fidelity posterior distribution directly. This conditional NF is used to speed up the training of the high-fidelity objective involving minimization of the Kullback-Leibler divergence between the predicted and the desired high-fidelity posterior density for indirect measurements at hand. To minimize costs associated with the forward operator, we initialize the high-fidelity NF with the weights of the pretrained low-fidelity NF, which is trained beforehand on available model and data pairs. Our numerical experiments, including a 2D toy and a seismic compressed sensing example, demonstrate that thanks to the preconditioning considerable speed-ups are achievable compared to training NFs from scratch.},
	keywords = {Inverse problems, Normalizing flows, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/AABI/2021/siahkoohi2021AABIpto/siahkoohi2020ABIfab.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/AABI/2021/siahkoohi2021AABIpto/siahkoohi2021AABIpto_pres.pdf},
	url2 = {https://openreview.net/forum?id=P9m1sMaNQ8T},
	software = {https://github.com/slimgroup/Software.siahkoohi2021AABIpto},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Mathias Louboutin and Philipp A. Witte and Felix J. Herrmann}
}
