Neural Processes with Normalizing Flows for Wheat Height Estimation

Abstract In this work, we investigate modeling plant traits over time using neural processes, a class of machine learning models that learn distributions over functions. Plant growth is an inherently stochastic process with complex dynamics measured mostly at irregular times throughout the growing seasons. While individual trait trajectories may be simple, their distributions are shaped by complex interactions between genotype, environment, and other factors. In particular, we focus on plant height in wheat, a deceptively simple-looking trait with complex dynamics. To model these trajectory distributions, we evaluate neural processes and in particular extensions using normalizing flows, with different combinations of genotype and environmental covariates. For controlled evaluations, we generate synthetic wheat height trajectories calibrated against Swiss weather station records and the Field Phenotyping Platform 1.0 (FIP1) wheat dataset. To fully evaluate these trajectory distributions, we use signatures, vector representations of sequential data, together with the signature maximum mean discrepancy (Sig-MMD) and its recently introduced censored variant (CSig-MMD). Sig-MMD enables direct pathwise comparison of predicted and simulator trajectory distributions, while CSig-MMD focuses this comparison on the tail, including lodged trajectories. Together, these metrics allow us to assess whether the models capture the full distribution of growth trajectories, including rare outcomes. We compare the neural processes with a lodging-mixture spline baseline and apply them to the FIP1 data.