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Inverse problems, i.e., estimating parameters of physical models fromexperimental data, are ubiquitous in science and engineering. The Bayesianformulation is the gold standard because it alleviates ill-posedness issues andquantifies epistemic uncertainty. Since analytical posteriors are not typicallyavailable, one resorts to Markov chain Monte Carlo sampling or approximatevariational inference. However, inference needs to be rerun from scratch foreach new set of data. This drawback limits the applicability of the Bayesianformulation to real-time settings, e.g., health monitoring of engineeredsystems, and medical diagnosis. The objective of this paper is to develop amethodology that enables real-time inference by learning the Bayesian inversemap, i.e., the map from data to posteriors. Our approach is as follows. Weparameterize the posterior distribution as a function of data. This workoutlines two distinct approaches to do this. The first method involvesparameterizing the posterior using an amortized full-rank Gaussian guide,implemented through neural networks. The second method utilizes a ConditionalNormalizing Flow guide, employing conditional invertible neural networks forcases where the target posterior is arbitrarily complex. In both approaches, welearn the network parameters by amortized variational inference which involvesmaximizing the expectation of evidence lower bound over all possible datasetscompatible with the model. We demonstrate our approach by solving a set ofbenchmark problems from science and engineering. Our results show that theposterior estimates of our approach are in agreement with the correspondingground truth obtained by Markov chain Monte Carlo. Once trained, our approachprovides the posterior distribution for a given observation just at the cost ofa forward pass of the neural network.

Learning to solve Bayesian inverse problems: An amortized variational inference approach using Gaussian and Flow guides