Neural Embeddings Evolve as Interacting Particles

We consider drawing a parallel between the training dynamics of neural embeddings and the dynamics of physical systems, in that embeddings can be regarded as interacting particles called ``repons''. We investigate this intuitive picture on neural networks performing addition, where each number is associated with a learnable $d$-dimensional embedding vector, interpreted as the spatial coordinates of a repon. We find that the evolution of repons can be successfully modeled by a conservative force (i.e., a force defined from a potential energy) with both attractive and repulsive components, and learn this potential field with a second neural network. The attractive potential is locally quadratic, while the repulsive potential resembles the famous Higgs potential, revealing an intriguing symmetry breaking mechanism. Beyond these preliminary results, this work also proposes a novel paradigm whereby physics ansatzes make it tractable to use auxiliary neural networks to interpret the training dynamics of another neural network.