Bayesian Inference for Non-linear forward model by using a VAE-based neural network structure

In this paper, a Variational Autoencoder (VAE) based framework is introduced to solve parameter estimation problems for non-linear forward models. In particular, we focus on applications in the field of medical imaging where many thousands of model-based inference analyses might be required to populate a single parametric map. We adopt the concept from Variational Bayes (VB) of using an approximate representation of the posterior, and the concept from the VAE of using the latent space representation to encode the parameters of a forward model. Our work develops the idea of mapping between time-series data and latent parameters using a neural network in variational way. A loss function that differs from the classic VAE formulation and a new sampling strategy are proposed to enable uncertainty estimation as part of the forward model inference. The VAE-based structure is evaluated using simulation experiments on a simple example and two perfusion MRI forward models. Compared with analytical VB (aVB) and Markov Chain Monte Carlo (MCMC), our VAE-based model achieves comparable accuracy, and hundredfold improvement in computational time (100ms/image). We believe this VAE-like framework can be generalized to imaging modularities with higher complexity and thus benefit clinical adoption where otherwise long processing time associated with conventional inference methods is prohibitive.