The field of deep learning is witnessing a significant shift towards understanding the underlying topological and geometric structures that govern the behavior of neural networks. Recent research has highlighted the importance of topological invariance, manifold learning, and categorical invariants in advancing our understanding of learning dynamics. The discovery of topological critical points and the role of curl-like components in non-gradient learning dynamics are opening up new avenues for improving the robustness and efficiency of deep learning models. Notably, the application of topological methods to the study of deep learning is enabling the development of more universal and architecture-agnostic theories. Some noteworthy papers in this regard include:

• One paper proves that the training process induces a bi-Lipschitz mapping between neurons and constrains the topology of the neuron distribution during training, revealing a qualitative difference between small and large learning rates.
• Another paper introduces a novel algorithm for detecting invariant manifolds in ReLU-based RNNs, which can be used to characterize multistability and chaos in these systems.
• A third paper proposes a categorical framework for understanding learning dynamics, which reveals that different training runs producing similar test performance often belong to the same homotopy class of optimization paths.