Designing Dynamic Measure Transport for Sampling
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maurais-maurais-phd-aeroastro-2026-thesis.pdf
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Author(s)
Maurais, Aimee E.
Advisor(s)
Marzouk, Youssef M.
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
The task of sampling from a target probability distribution is fundamental to modern computational science and machine learning. Sampling is the essence of Monte Carlo integration, enables uncertainty quantification in Bayesian inference, and underlies generative models that have the ability to synthesize convincing text, images, and far beyond. A powerful, emerging approach to sampling is dynamic measure transport (DMT): the idea is to design an ordinary or stochastic differential equation that evolves samples from a tractable reference distribution (e.g., a Gaussian) to the desired target distribution. DMT is state-of-the-art in generative modeling and underlies techniques such as diffusion models and flow-matching, but DMT pipelines for density-driven sampling tasks, as arising in computational chemistry and Bayesian inference, have seen significantly less development.
In this thesis we develop tractable algorithms for density- and likelihood-driven dynamic measure transport and introduce frameworks to enable effective design and structure exploitation within DMT. In Chapter 1 we introduce a family of closed-form, gradient-free interacting particle systems for density-driven sampling which can even be used in Bayesian inference settings where only samples from a prior and evaluations of the likelihood (rather than access to the full posterior density) are available. We demonstrate that these particle systems are able to effectively sample from distributions with features such as multimodality, anisotropy, and concentration on manifolds which cannot be sampled faithfuly by preexisting interacting particle system algorithms which rely on parametric approximations of the target. Mathematically, our interacting particle systems possess Fisher-Rao gradient flow characterizations and, at the same time, can be interpreted through the lens of optimal transport.
In Chapter 2 we study choices of paths of distributions for density-driven DMT and highlight the fact that certain off-the-shelf paths can be highly unsuitable for DMT when the reference distribution is unimodal and the target is multimodal. The presence of "teleportation behavior" along such paths can lead to catastrophically poor sampling performance, via mode collapse, in practice. We introduce a PDE-constrained optimization framework to enable principled design of paths and, within this framework, introduce computational solution methods grounded in Gaussian process methods for partial differential equations. We demonstrate that our framework is able to correct teleportation behavior and yield paths corresponding to smooth velocity fields which are tractable to approximate and avoid mode-collapse.
Finally, in Chapter 3 we demonstrate how sparse conditional dependence structure can be exploited to enable scalability of DMT-based sampling approaches to high dimensions. We show, under certain regularity assumptions, that sparse conditional dependence in the target and reference distributions implies approximate sparse variable dependence in velocity fields for transport over two commonly used paths of measures and introduce computational approaches for learning velocities with corresponding sparse variable dependence in practice. We demonstrate that imposing this sparse dependence structure on DMT velocity fields can lead to a favorable bias-variance tradeoff when the data or computational resources available for learning the velocity fields are limited and subsequently result in better sampling performance.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Massachusetts Institute of Technology. Center for Computational Science and Engineering
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