Functional data analysis using deep neural networks

Functional data analysis is an evolving field focused on analyzing data that reveals insights into curves, surfaces, or entities within a continuous domain. This type of data is typically distinguished by the inherent dependence and smoothness observed within each data curve. Traditional functional data analysis approaches have predominantly relied on linear models, which, while foundational, often fall short in capturing the intricate, nonlinear relationships within the data. This paper seeks to bridge this gap by reviewing the integration of deep neural networks into functional data analysis. Deep neural networks present a transformative approach to navigating these complexities, excelling particularly in high‐dimensional spaces and demonstrating unparalleled flexibility in managing diverse data constructs. This review aims to advance functional data regression, classification, and representation by integrating deep neural networks with functional data analysis, fostering a harmonious and synergistic union between these two fields. The remarkable ability of deep neural networks to adeptly navigate the intricate functional data highlights a wealth of opportunities for ongoing exploration and research across various interdisciplinary areas. This article is categorized under: Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Statistical Learning and Exploratory Methods of the Data Sciences > Deep Learning Statistical Learning and Exploratory Methods of the Data Sciences > Neural Networks Despite two decades of success, traditional methods in functional data analysis predominantly rely on linear models, which are limited in their ability to encapsulate the complex and nonlinear relationships inherent in data. In contrast, deep neural networks (DNNs), known for their proficiency in managing high‐dimensional spaces, offer a significant advancement over traditional techniques. Recent advancements in research highlight the versatility of DNNs across a spectrum of tasks in functional data analysis. This includes functional regression, classification, and representation, which are effectively applied to both one‐dimensional (1D) and two‐dimensional (2D) functional data.