Nonlinear Dynamics and Pattern Formation

Últimos artículos

• Random Quadratic Form with random forcing: Metastable synchronization by noise

  Anna Shalova · 18 de agosto de 2026

  We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to…

• Balancing Centralized Learning and Distributed Self-Organization: A Hybrid Model for Embodied Morphogenesis

  Takehiro Ishikawa · 30 de julio de 2026

  Background: both embodied intelligence and developmental morphogenesis depend on a division of labour between centralized guidance and distributed material dynamics, but the amount of top-down control needed to steer self-organization remains unclear. Methods: we coupled a compact full-resolution co…

• Instability in Complex Oscillator Networks: Limitations and Potentials of Network Measures and Machine Learning

  Christian Nauck, Michael Lindner, Nora Molkenthin, J\"urgen Kurths, Eckehard Sch\"oll, J\"org Raisch, Frank Hellmann · 20 de julio de 2026

  A central question of network science is how functional properties of systems emerge from their structure. For networked dynamical systems, structure is typically captured through network measures. We investigate the relationship between these measures and stability metrics across non-linear and lin…

• Attention as Frustrated Synchronization

  Joshua Nunley · 18 de junio de 2026

  A network of oscillators that synchronizes perfectly computes nothing further, so an attention architecture built from synchronization must locate its computation in structured departures from agreement. We introduce the Frustrated Synchronization Network (FSN), whose token states are phases on a to…

• Transformer Field Theory: A Response-Theoretic Approach to Mechanistic Interpretability

  David N. Olivieri, Antonio F. P\'erez Rodr\'iguez · 12 de junio de 2026

  Mechanistic interpretability often studies Transformer behavior by intervening on internal activations through activation patching, causal tracing, path patching, and steering directions. This paper develops Transformer Field Theory: a response-theoretic framework in which the residual stream of a f…

• Continuous-Depth Field Theory for Transformer Patching and Mechanistic Interpretability

  David N. Olivieri, Antonio F. P\'erez Rodr\'iguez · 26 de mayo de 2026

  Mechanistic interpretability often uses activation patching, causal tracing, path patching, and steering directions to reveal behaviorally meaningful directions in Transformer activation space. This paper develops a field-theoretic framework for organizing and predicting such interventions. Treating…

• The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks

  Mani Rash Ahmadi · 14 de abril de 2026

  We prove that in a coupled Kuramoto oscillator network at stable equilibrium, the physical phase displacement under weak output nudging is the gradient of the loss with respect to natural frequencies, with equality as the nudging strength beta tends to zero. Prior oscillator equilibrium propagation …

• Identifiability and amortized inference limitations in Kuramoto models

  Emma Hannula, Jana de Wiljes, Matthew T. Moores, Heikki Haario, Lassi Roininen · 24 de marzo de 2026

  Bayesian inference is a powerful tool for parameter estimation and uncertainty quantification in dynamical systems. However, for nonlinear oscillator networks such as Kuramoto models, widely used to study synchronization phenomena in physics, biology, and engineering, inference is often computationa…

• Interpreting the Synchronization Gap: The Hidden Mechanism Inside Diffusion Transformers

  Emil Albrychiewicz, Andr\'es Franco Valiente, Li-Ching Chen, Viola Zixin Zhao · 24 de marzo de 2026

  Recent theoretical models of diffusion processes, conceptualized as coupled Ornstein-Uhlenbeck systems, predict a hierarchy of interaction timescales, and consequently, the existence of a synchronization gap between modes that commit at different stages of the reverse process. However, because these…

• Random Quadratic Form on a Sphere: Synchronization by Common Noise

  Maximilian Engel, Anna Shalova · 9 de marzo de 2026

  We introduce the Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While the one-point dynamics of the system is a Brownian motion and thus has no preferred direction, the two-point motion exh…

• Selective Synchronization Attention

  Hasi Hays · 17 de febrero de 2026

  The Transformer architecture has become the foundation of modern deep learning, yet its core self-attention mechanism suffers from quadratic computational complexity and lacks grounding in biological neural computation. We propose Selective Synchronization Attention (SSA), a novel attention mechanis…

• Synchronization on circles and spheres with nonlinear interactions

  Christopher Criscitiello, Quentin Rebjock, Andrew D. McRae, Nicolas Boumal · 28 de enero de 2026

  We consider the dynamics of $n$ points on a sphere in $\mathbb{R}^d$ ($d \geq 2$) which attract each other according to a function $\varphi$ of their inner products. When $\varphi$ is linear ($\varphi(t) = t$), the points converge to a common value (i.e., synchronize) in various connectivity scenari…

• The Mean-Field Dynamics of Transformers

  Philippe Rigollet · 2 de diciembre de 2025

  We develop a mathematical framework that interprets Transformer attention as an interacting particle system and studies its continuum (mean-field) limits. By idealizing attention continuous on the sphere, we connect Transformer dynamics to Wasserstein gradient flows, synchronization models (Kuramoto…

• Associative Memory and Generative Diffusion in the Zero-noise Limit

  Joshua Hess, Quaid Morris · 18 de noviembre de 2025

  This paper shows that generative diffusion processes converge to associative memory systems at vanishing noise levels and characterizes the stability, robustness, memorization, and generation dynamics of both model classes. Morse-Smale dynamical systems are shown to be universal approximators of ass…

• Multistability of Self-Attention Dynamics in Transformers

  Claudio Altafini · 17 de noviembre de 2025

  In machine learning, a self-attention dynamics is a continuous-time multiagent-like model of the attention mechanisms of transformers. In this paper we show that such dynamics is related to a multiagent version of the Oja flow, a dynamical system that computes the principal eigenvector of a matrix c…

• Stuart-Landau Oscillatory Graph Neural Network

  Kaicheng Zhang, David N. Reynolds, Piero Deidda, Francesco Tudisco · 12 de noviembre de 2025

  Oscillatory Graph Neural Networks (OGNNs) are an emerging class of physics-inspired architectures designed to mitigate oversmoothing and vanishing gradient problems in deep GNNs. In this work, we introduce the Complex-Valued Stuart-Landau Graph Neural Network (SLGNN), a novel architecture grounded i…

• Measure-Theoretic Time-Delay Embedding

  Jonah Botvinick-Greenhouse, Maria Oprea, Romit Maulik, Yunan Yang · 7 de noviembre de 2025

  The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicab…