Temporal Geometry of Deep Networks: Hyperbolic Representations of Training Dynamics for Intrinsic Explainability

This paper investigates how multilayer perceptrons (MLPs) can be represented in non-Euclidean spaces, with emphasis on the Poincaré model of hyperbolic geometry. We aim to capture the geometric evolution of their weighted topology and self-organization over time. Instead of restricting analysis to single checkpoints, we construct temporal parameter-graphs across $T$ snapshots of the optimization process. This reflects the view that neural networks encode information not only in their weights but also in the trajectory traced during training. Drawing on the idea that many complex networks admit embeddings in hidden metric spaces where distances correspond to connection likelihood, we present a geometric and temporal graph-based meta learning framework for obtaining dynamic hyperbolic representations of the underlying neural parameter graphs. Our model embeds temporal parameter-graphs in the Poincaré ball and learns from them while maintaining equivariance to within-snapshot neuron permutations and invariance to permutations of past snapshots. In doing so, it preserves functional equivalence across time and recovers the network’s latent geometry. Experiments on regression and classification tasks with trained MLPs show that hyperbolic temporal representations expose how structure emerges during training, offering intrinsic explanations of self-organisation in a given model training environment.