On Compositionality and Emergence in Physical Systems Generativie Modeling

The principle of compositionality plays a pivotal role in both machine learning and physical sciences but remains under-explored, particularly in the context of synthetic data derived from physical energy potentials. This study aims to bridge this gap by examining the compositional nature of synthetic datasets generated using composite energy potentials. By combining established Lennard-Jones and Morse potentials into a composite potential, we generate synthetic datasets using Markov Chain Monte Carlo (MCMC) techniques. These datasets serve as training grounds for machine learning models, specifically Neural Ordinary Differential Equations (ODEs). Our primary focus is to investigate whether the properties of the composite datasets retain the characteristics of their individual components, effectively testing the principle of compositionality. The findings not only shed light on the compositional integrity of synthetic physical datasets but also lay the groundwork for more robust and interpretable machine learning models applied to complex physical systems by using the formalism of Category Theory.