Multifidelity Formulations for Triangular Transport

We develop multifidelity methods for constructing triangular transport maps from samples, when high-fidelity data are scarce but lower-fidelity data are more abundant. Using this set of multifidelity data, we approximate a triangular transport map that bijectively maps between a tractable reference density and the high-fidelity target distribution. We introduce two strategies to leverage low-fidelity data: a hierarchical approach that composes maps between adjacent fidelity levels, and a non-hierarchical method that incorporates low-fidelity information through monotonicity-preserving corrections to the map parameterization. Numerical experiments compare these strategies with single-fidelity transport and demonstrate how the proposed multifidelity approaches can improve map estimation from limited high-fidelity data. To illustrate the broader utility of the learned maps, we also deploy them in a downstream amortized simulation-based inference task. This example shows that multifidelity improvements in map estimation can translate to improved conditional sampling and uncertainty quantification when high-fidelity data are scarce.