In the ATLAS experiment at the Large Hadron Collider, we use a suite of detectors to collect high-dimensional data -- from which we perform complex reconstructions of particle candidates in order to measure properties of the Standard Model, or search for physics beyond it. This thesis presents work on two distinct but thematically-related topics in this realm. The first is a search for displaced vertices and missing transverse energy, conducted using 137 inverse femtobarns of 13 TeV data collected by the ATLAS detector during Run 2, that targets beyond-Standard Model processes that produce long-lived particles. I will explain the analysis method and present its results, giving particular attention to its relevance to and interpretation in the context of a model combining axion physics with supersymmetry. The second topic -- addressing the complex reconstruction tasks inherent to collider physics analyses -- is a Lorentz-equivariant neural network architecture called PELICAN. I will describe the basic network design, and results from top quark identification and momentum reconstruction benchmarks.