Functional data analysis (FDA) refers to a broad collection of statistical and machine learning methods that deal with the data in the form of random functions. In general, functional data are assumed to lie in a constrained functional space, e.g., images, and smooth curves, rather than the conventional Euclidean space, e.g., scalar vectors. The explosion of massive data and high-performance computational resources brings exciting opportunities as well as new challenges to this field. On one hand, the rich information from modern functional data enables an investigation into the underlying data patterns at an unprecedented scale and resolution. On the other hand, the inherent complex structures and huge data sizes of modern functional data pose additional practical challenges to model building, model training, and model interpretation under various circumstances. This dissertation discusses recent advances in machine learning for analyzing complex structured functional data. Chapter 1 begins with a general introduction to examples of modern functional data and related data analysis challenges.

Chapter 2 introduces a novel machine learning framework, artificial perceptual learning (APL), to tackle the problem of weakly supervised learning in functional remote sensing data. Chapter 3 develops a flexible function-on-scalar regression framework, Wasserstein distributional learning (WDL), to address the challenge of modeling density functional outputs. Chapter 4 concludes the dissertation and discusses future directions.