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Inverse problems and, in particular, inferring unknown or latent parametersfrom data are ubiquitous in engineering simulations. A predominant viewpoint inidentifying unknown parameters is Bayesian inference where both priorinformation about the parameters and the information from the observations vialikelihood evaluations are incorporated into the inference process. In thispaper, we adopt a similar viewpoint with a slightly different numericalprocedure from standard inference approaches to provide insight about thelocalized behavior of unknown underlying parameters. We present a variationalinference approach which mainly incorporates the observation data in apoint-wise manner, i.e. we invert a limited number of observation dataleveraging the gradient information of the forward map with respect toparameters, and find true individual samples of the latent parameters when theforward map is noise-free and one-to-one. For statistical calculations (as theultimate goal in simulations), a large number of samples are generated from atrained neural network which serves as a transport map from the prior toposterior latent parameters. Our neural network machinery, developed as part ofthe inference framework and referred to as Neural Net Kernels (NNK), is basedon hierarchical (deep) kernels which provide greater flexibility for trainingcompared to standard neural networks. We showcase the effectiveness of ourinference procedure in identifying bimodal and irregular distributions comparedto a number of approaches including Markov Chain Monte Carlo samplingapproaches and a Bayesian neural network approach.

computer methods in applied mechanics and engineering

Variational Inference for Nonlinear Inverse Problems via Neural Net  Kernels: Comparison to Bayesian Neural Networks, Application to Topology  Optimization