Two-Stage Framework Combining Derivative-Free MCMC Sampling and Transport Maps for Black-Box Bayesian Inverse Problems

This study addresses the challenge of estimating Bayesian posterior distributions, a key task in uncertainty quantification, particularly within settings where the mapping from parameters to observations is a black box. Such models, often representing complex, high-dimensional stochastic systems, can be based on physical equations, surrogate models, or data-based machine learning functions, allowing only forward evaluations. Considering real-world noisy observations, we propose a two-stage framework. First, derivative-free Markov chain Monte Carlo (MCMC) methods sample the posterior. Second, these samples optimize a transport map (using polynomial or neural network bases) to construct a closed-form approximation of the posterior. This functional representation aids in characterizing parameter uncertainty and enables efficient uncertainty propagation. We compare the basis function choices and demonstrate the approach on Bayesian inverse problems for the heat equation and generalized Lotka–Volterra equations.