Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model.

Then dynamics ofwt is like a physical process - a satellite's motion around the earth (see illustration in Fig.1): according to

Self-consistent Field (SCF) equation is a type of nonlinear eigenvalue problem in which the matrix to be eigen-decomposed is a function of its own eigenvectors. It is of great significance in computational science for its connection to the Schrödinger equation. Traditional fixed-point iteration methods for solving such equations suffer from non-convergence issues. In this work, we present a novel perspective on such SCF equations as a principal component analysis (PCA) for non-stationary time series, in which a distribution and its own top principal components are mutually updated over time, and the equilibrium state of the model corresponds to the solution of the SCF equations. By the new perspective, online PCA techniques are able to engage in so as to enhance the convergence of the model towards the equilibrium state, acting as a new set of tools for converging the SCF equations. With several numerical adaptations, we then develop a new algorithm for converging the SCF equation, and demonstrated its high convergence capacity with experiments on both synthesized and real electronic structure scenarios.

Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks, and use surrogate derivatives or compute gradients with respect to the spiking time to deal with the problem. These approaches either accumulate approximation errors or only propagate information limitedly through existing spikes, and usually require information propagation along time steps with large memory costs and biological implausibility. In this work, we consider feedback spiking neural networks, which are more brain-like, and propose a novel training method that does not rely on the exact reverse of the forward computation.

In this paper, we comprehensively reveal the learning dynamics of normalized neural network using Stochastic Gradient Descent (with momentum) and Weight Decay (WD), named as Spherical Motion Dynamics (SMD). Most related works focus on studying behavior of equilibrium state, i.e. assuming weight norm remains unchanged. However, their discussion on why this equilibrium can be reached is either absent or less convincing. Our work directly explores the cause of equilibrium, as a special state of SMD. Specifically, 1) we introduce the assumptions that can lead to equilibrium state in SMD, and prove equilibrium can be reached in a linear rate regime under given assumptions; 2) we propose ``angular update as a substitute for effective learning rate to depict the state of SMD, and derive the theoretical value of angular update in equilibrium state; 3) we verify our assumptions and theoretical results on various large-scale computer vision tasks including ImageNet and MSCOCO with standard settings. Experiment results show our theoretical findings agree well with empirical observations. We also show that the behavior of angular update in SMD can produce interesting effect to the optimization of neural network in practice.

The orderly behaviors observed in large-scale groups, such as fish schooling and the organized movement of crowds, are both ubiquitous and essential for the survival and stability of these systems. Understanding how such complex collective behaviors emerge from simple local interactions and behavioral adjustments is a significant scientific challenge. Historically, research has predominantly focused on imitation and social learning, where individuals adopt the strategies of more successful peers to refine their behavior. However, in recent years, an alternative learning approach based on self-exploration and introspective learning has garnered increasing attention. In this paradigm, individuals assess their own circumstances and select strategies that best align with their specific conditions. Two examples are coordination and anti-coordination, where individuals align with and diverge from the local majority, respectively. In this study, we analyze networked systems of coordinating and anti-coordinating individuals, exploring the combined effects of system dynamics, network structure, and behavioral patterns. We address several practical questions, including the number of equilibria, their characteristics, the equilibrium time, and the resilience of the system. We find that the number of equilibrium states can be extremely large, even increasing exponentially with minor alterations to the network structure. Moreover, the network structure has a significant impact on the average equilibrium time. Despite the complexity of these findings, we find that variations can be captured by a single, simple network characteristic (the average path length), which we illustrate in both synthetic and empirical networks.