Generative and variational approaches to scientific machine learning, inverse problems and PDEs

Abstract

This dissertation collects four pieces of research conducted during my doctoral studies. The chapters are largely independent, except that Chapter 3 builds directly on the framework developed in Chapter 2. Chapter 1 develops a theory of conditional optimal transport in separable infinite-dimensional function spaces, motivated by simulation-based Bayesian inference. We study the solvability of block-triangular Monge maps and their Kantorovich relaxations, derive regularity estimates for the posterior-conditioning maps arising in Bayesian inverse problems, and demonstrate the framework on a Darcy-flow benchmark. Chapter 2 develops a kernel-based framework for learning differential equations from data, together with quantitative worst-case error bounds. The state and the unknown governing operator are both represented in reproducing kernel Hilbert spaces, with the differential equation enforced via least-squares collocation; we also discuss connections of this framework to operator learning and PDE solvers. Chapter 3 builds on the kernel collocation framework of Chapter 2 to develop an all-at-once method for jointly identifying sparse ordinary differential equations and reconstructing the underlying state from scarce, partial and noisy observations. The unknown dynamics are parametrized as a sparse linear combination over a fixed function library, and the resulting nonconvex problem is solved by alternating Levenberg–Marquardt steps on a fixed-support nonlinear least squares problem with an outer sparse-regression update. Chapter 4 determines the sharp value of the hot spots ratio in every dimension and characterizes its asymptotic behavior. The upper bound combines a comparison principle, Talenti's rearrangement inequality, and a quantitative form of the Szegő–Weinberger eigenvalue inequality; the lower bound follows from a Neumann-sieve construction of extremizing sequences.