Modelling early life growth trajectories through representation learning with functional autoencoders

Analysis of individual growth curves for length/height, weight and body mass index (BMI) through infancy and childhood is important for characterising normative skeletal growth and body composition development, and how these patterns vary between individuals. These growth curves can be examined in relation to prenatal and early life factors to shed light on genetic and environment influences on development, and in relation to later outcomes, to investigate how developmental trajectories might influence health across the life course (Molinari, Gasser, Hauspie, & others, 2004). By modelling repeated measurements from the same individuals over time, growth curve analysis can also identify key landmarks on individual trajectories, such as the infant BMI peak and the childhood BMI rebound, providing deeper insight into growth dynamics, their determinants, and health consequences. Collectively the analysis of growth trajectories may ultimately support the development of early screening tools to identify at-risk infants and help inform effective interventions (Cole, 2004; Rolland-Cachera, 2024). Analysing such growth patterns requires appropriate statistical methods for handling repeated measurements over time. Traditional methods, such as mixed effect models, are based on discrete observation collected at specific time points and model how these values change over time through population-level and subject-specific effects (Laird & Ware, 1982). In contrast, Functional Data Analysis (FDA) considers the entire trajectory as the unit of analysis, representing it as a smooth continuous function and allowing the analysis of variation in curve shape across individuals (Ramsay James O, 2005; Wang, Chiou, & Müller). Within this framework, Functional Principal Component Analysis (FPCA) is one of the most widely used techniques. FPCA summarises individual trajectories using a small number of functional principal component (FPC) scores, which capture differences in timing, magnitude, and overall shape (Ramsay James O, 2005; Karuppusami Reka, 2022) . More recently, deep learning methods have been increasingly applied in longitudinal biomedical data (Cascarano Anna, 2023; Swinckels Laura, 2024; Rodriguez-Marin Mauro, 2025). These approaches used either alone or in combination with traditional statistical tools, offer greater flexibility for modelling complex and nonlinear relationships, particularly in high-dimensional data. Consequently, there is growing interest in integrating deep learning with FDA to leverage the strengths of both approaches. This integration enables the modelling of complex longitudinal patterns while preserving the functional nature and interpretability of the data (Wang Shuoyang, 2024; Cao, Wu, Nanshan, Shi, & Wang, 2025). Among these approaches, the Functional Autoencoder (FAE), developed by (Wu, Beaulac, & Cao, 2024), is a neural network model that extends the autoencoder framework to functional data by incorporating concepts from Functional Data Analysis. It integrates basis function representations into the network architecture, allowing the direct modelling of longitudinal trajectories and the reconstruction of smooth growth curves. This approach is designed to capture complex patterns by learning nonlinear, low-dimensional representations of functional trajectories and may complement traditional methods such as FPCA, which represents trajectories as linear combinations of FPC scores. However, the application of FAE to childhood growth trajectories remains unexplored, and its potential for capturing relevant growth features has not yet been fully explored. In this study, we aim to apply the FAE approach to analyse early life growth trajectories using two independent birth cohorts. Within each cohort, the FAE will be trained and tuned using the training data and evaluated using a held-out test dataset. Results will then be compared across cohorts to assess whether the FAE shows consistent performance and identifies similar growth patterns across independent populations. We will also evaluate its ability to reconstruct height, weight and BMI trajectories, and compare its performance with FPCA using the Fast Covariance Estimation (FACE) approach (Xiao, Li, Checkley, & Crainiceanu, 2018). Models will be fitted jointly using data from both sexes and fairness will be assessed by comparing model performance between males and females during model selection. We will also explore relationships between the FAE latent representations, key BMI features such as the BMI peak and BMI rebound, and the FPC scores obtained from FPCA-FACE, to examine which features of the growth trajectories are captured by the latent representations. Finally, we will use regression models to examine the associations between the infant BMI peak and the childhood BMI rebound estimates, derived from the fitted trajectories, and adiposity and blood pressure at 10 years of age.