Artificial intelligence (AI), when interfaced with laboratory automation, can accelerate materials optimization and scientific discovery. For example, it may be used to efficiently map a phase diagram with intelligent sampling along phase boundaries, or in ‘retrosynthesis’ problems where a material with a target structure is desired but its synthetic route is unknown. These AI-driven laboratories are especially promising in polymer physics, where design parameters (e.g. chemical composition, molecular weight, topology, processing) are vast, and properties and functions are intimately tied to design features. However, for AI to operate efficiently in these spaces, they must be ‘encoded’ with domain expertise specific to the problems being tackled. In this talk, we focus on the problem of defining appropriate ‘distance’ metrics to describe differences between functions sampled within a design space. Such functions may be spectroscopic (e.g. UV-Vis absorption, fluorescence, impedance) or scattering profiles (SAXS, SANS) of materials. Traditional ‘distance’ metrics, such as Euclidean and parametric definitions, often fail when important features of the measured functions are subtle and/or when sampling takes place far from the target. We have thus developed a new shape-based similarity metric using Riemannian geometry (Amplitude-Phase Distance) that has been successfully implemented in both retrosynthesis and phase mapping problems. This talk will introduce the audience to the functional data analysis framework by first discussing the definition of the Amplitude-Phase Distance metric. We then demonstrate its implementation in an autonomous batch retrosynthesis problem using spectroscopic signatures in a model system of metallic nanostructures. We then showcase example implementations of the new distance metric in phase-mapping problems involving block-copolymers, polymer blends, and inorganic materials and extraction of design rules from novel material synthesis spaces.