Enforcing probabilistic constraints with Data Consistent Inversion and Normalizing Flows

Flow-based models such as Normalizing Flows (NFs) have become a powerful framework for learning and sampling from complex probability distributions. While highly expressive flow architectures can accurately approximate target distributions, they often introduce unnecessary computational cost due to overparameterization, which is particularly acute when only localized distributional changes must be incorporated into an existing stochastic model. In this work, we introduce a Data-Consistent Inversion (DCI)-based framework to rebase input distributions that allows for (1) systematic refinement of NFs to achieve desired distributional properties while avoiding excessive overparameterization, and (2) adapting pre-trained stochastic models. Using the iterative variant of DCI (iDCI), we construct a reweighted sample set from a NF base that is subsequently used to learn an updated base distribution whose push-forward constraints are satisfied by design. This enables the sequential refinement of coarse NF models, allowing them to approach a target distribution to a prescribed tolerance without retraining from scratch. Furthermore, the proposed approach provides a mechanism for adapting pre-trained stochastic models to distribution drift, offering a new perspective on domain adaptation through distribution-consistent updates. Numerical experiments demonstrate that the proposed methodology efficiently improves model fidelity while reducing the burden associated with full retraining, eliminating the need to retain or reuse the original target samples employed during pre-training. Although such samples can be still be incorporated when available, model improvements can be achieved using only the drifted observations and the pre-trained model itself.