  Predictions for physical systems often rely upon knowledge acquired from
ensembles of entities, e.g., ensembles of cells in biological sciences. For
qualitative and quantitative analysis, these ensembles are simulated with
parametric families of mechanistic models (MM). Two classes of methodologies,
based on Bayesian inference and Population of Models, currently prevail in
parameter estimation for physical systems. However, in Bayesian analysis,
uninformative priors for MM parameters introduce undesirable bias. Here, we
propose how to infer parameters within the framework of stochastic inverse
problems (SIP), also termed data-consistent inversion, wherein the prior
targets only uncertainties that arise due to MM non-invertibility. To
demonstrate, we introduce new methods to solve SIP based on rejection sampling,
Markov chain Monte Carlo, and generative adversarial networks (GANs). In
addition, to overcome limitations of SIP, we reformulate SIP based on
constrained optimization and present a novel GAN to solve the constrained
optimization problem.
