The field of deep learning continues to evolve with a strong focus on understanding and improving the training dynamics and stability of neural networks. Recent research has made significant strides in analyzing the fixed points of deep neural networks (DNNs), revealing insights into their formation, stability, and applications across various learning paradigms. The exploration of grokking, a phenomenon where models suddenly generalize after prolonged overfitting, has led to the identification of Softmax Collapse (SC) as a barrier to generalization and the development of new activation functions and training algorithms to mitigate this issue. Additionally, the study of emergent weight morphologies in DNNs has provided a deeper understanding of how training can lead to the spontaneous formation of periodic structures within networks, independent of the training data. The convergence properties of dynamic routing algorithms in capsule networks have been rigorously analyzed, offering a mathematical foundation for their optimization. Furthermore, the derivation of effective gradient flow equations has shed light on the interpretability of supervised learning processes, while the convergence analysis of Real-time Recurrent Learning (RTRL) has demonstrated its potential for training models on long data sequences. The impact of batch size on the convergence of stochastic gradient descent with momentum (SGDM) has been theoretically and empirically validated, suggesting that increasing batch sizes can lead to faster convergence. Lastly, the study of gradient descent dynamics in shallow linear networks has revealed a trade-off between convergence speed and implicit regularization, highlighting the benefits of training at the 'Edge of Stability'.nn### Noteworthy Papersn- Fixed Points of Deep Neural Networks: Emergence, Stability, and Applications: Demonstrates the existence and stability of fixed points in DNNs, with applications in image encoding/decoding and restoration.n- Grokking at the Edge of Numerical Stability: Introduces StableMax and $perp$Grad to address Softmax Collapse, enabling grokking without regularization.n- Emergent weight morphologies in deep neural networks: Shows that training DNNs can lead to the emergence of periodic channel structures, impacting network performance.n- The Convergence of Dynamic Routing between Capsules: Provides a mathematical proof of convergence for dynamic routing algorithms in capsule networks.n- Derivation of effective gradient flow equations and dynamical truncation of training data in Deep Learning: Offers insights into the interpretability of supervised learning through gradient flow equations.n- Convergence Analysis of Real-time Recurrent Learning (RTRL) for a class of Recurrent Neural Networks: Proves the convergence of RTRL, highlighting its potential for analyzing long data sequences.n- Increasing Batch Size Improves Convergence of Stochastic Gradient Descent with Momentum: Theoretically and empirically validates that increasing batch sizes can enhance SGDM convergence.n- Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks: Reveals a trade-off between convergence speed and implicit regularization in shallow linear networks.n- Overshoot: Taking advantage of future gradients in momentum-based stochastic optimization: Introduces Overshoot, a novel optimization method that outperforms standard and Nesterov's momentum by leveraging future gradients.