Transport-Based Generative Modeling for Stochastic Dynamics, Uncertainty Quantification, and High-Dimensional Sampling

Abstract

This dissertation develops and analyzes generative modeling methods for scientific computing, where repeated sampling from complex probability distributions is needed for uncertainty quantification, estimation of quantities of interest, and statistical inference. The main contributions are pseudo-reversible normalizing flows and training-free diffusion models, which provide two complementary transport-based approaches for generating samples from target distributions by transforming samples from simple reference distributions.

In the first part of the dissertation, we present a pseudo-reversible normalizing flow method for efficiently generating samples of the state of a stochastic differential equation (SDE) with various initial distributions. The main novelty of our normalizing flow model is that it can learn the conditional distribution of the state, i.e., the distribution of the final state conditional on any initial state, such that the model only needs to be trained once and the trained model can be used to handle various initial distributions. We provide a rigorous convergence analysis of the learned distribution to the target distribution in the Kullback–Leibler divergence metric. Numerical experiments are provided to demonstrate the effectiveness of the proposed normalizing flow model.

Next, we extend this framework to uncertainty quantification for noisy physical models. A conditional pseudo-reversible normalizing flow is constructed directly from input-output data to generate samples from conditional distributions relevant to forward and inverse uncertainty propagation. The method does not require prior knowledge of the deterministic model component or the distribution of additive noise. Theoretical results establish convergence for the forward conditional distribution, and numerical experiments include benchmark examples, high-dimensional uncertainty propagation, and a geologic carbon storage application.

In this dissertation, we also study a training-free diffusion model for high-dimensional sampling. A Gaussian mixture model constructed from available samples yields an analytically tractable score function and avoids propagating score function approximation errors through the reverse process. The resulting error bounds exhibit favorable dimension dependence, scaling as $\mathcal{O}(d)$ in the $\ell_2$ norm and $\mathcal{O}(\log d)$ in the $\ell_\infty$ norm. Importantly, the proposed error estimates are fully numerically verifiable with respect to both time-step size and dimensionality, thereby bridging the gap between theoretical analysis and observed numerical behavior.