Physica D: Nonlinear Phenomena最新文献_第9页

The complexity dynamics of grokking grokking的复杂性动力学

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-07 DOI: 10.1016/j.physd.2025.134859

Branton DeMoss, Silvia Sapora, Jakob Foerster, Nick Hawes, Ingmar Posner

{"title":"The complexity dynamics of grokking","authors":"Branton DeMoss, Silvia Sapora, Jakob Foerster, Nick Hawes, Ingmar Posner","doi":"10.1016/j.physd.2025.134859","DOIUrl":"10.1016/j.physd.2025.134859","url":null,"abstract":"<div><div>We demonstrate the existence of a complexity phase transition in neural networks by studying the grokking phenomenon, where networks suddenly transition from memorization to generalization long after overfitting their training data. To characterize this phase transition, we introduce a theoretical framework for measuring complexity based on rate–distortion theory and Kolmogorov complexity, which can be understood as principled lossy compression for networks. We find that properly regularized networks exhibit a sharp phase transition: complexity rises during memorization, then falls as the network discovers a simpler underlying pattern that generalizes. In contrast, unregularized networks remain trapped in a high-complexity memorization phase. We establish an explicit connection between our complexity measure and generalization bounds, providing a theoretical foundation for the link between lossy compression and generalization. Our framework achieves compression ratios 30-40<span><math><mo>×</mo></math></span> better than naïve approaches, enabling precise tracking of complexity dynamics. Finally, we introduce a regularization method based on spectral entropy that encourages networks toward low-complexity representations by penalizing their intrinsic dimension.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"482 ","pages":"Article 134859"},"PeriodicalIF":2.9,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144858481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Physics-informed spectral approximation of Koopman operators Koopman算子的物理信息谱近似

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-06 DOI: 10.1016/j.physd.2025.134835

Dimitrios Giannakis , Claire Valva

{"title":"Physics-informed spectral approximation of Koopman operators","authors":"Dimitrios Giannakis , Claire Valva","doi":"10.1016/j.physd.2025.134835","DOIUrl":"10.1016/j.physd.2025.134835","url":null,"abstract":"<div><div>Koopman operators and transfer operators represent nonlinear dynamics in state space through its induced action on linear spaces of observables and measures, respectively. This framework enables the use of linear operator theory for supervised and unsupervised learning of nonlinear dynamical systems, and has received considerable interest in recent years. Here, we propose a data-driven technique for spectral approximation of Koopman operators of continuous-time, measure-preserving ergodic systems that is asymptotically consistent and makes direct use of known equations of motion (physics). Our approach is based on a bounded transformation of the Koopman generator (an operator implementing directional derivatives of observables along the dynamical flow), followed by smoothing by a Markov semigroup of kernel integral operators. This results in a skew-adjoint, compact operator whose eigendecomposition is expressible as a variational generalized eigenvalue problem. We develop Galerkin methods to solve this eigenvalue problem and study their asymptotic consistency in the large-data limit. A key aspect of these methods is that they are physics-informed, in the sense of making direct use of dynamical vector field information through automatic differentiation of kernel functions. Solutions of the eigenvalue problem reconstruct evolution operators that preserve unitarity of the underlying Koopman group while spectrally converging to it in a suitable limit. In addition, the computed eigenfunctions have representatives in a reproducing kernel Hilbert space, enabling out-of-sample evaluation of learned dynamical features. Numerical experiments performed with this method on integrable and chaotic low-dimensional systems demonstrate its efficacy in extracting dynamically coherent observables under complex dynamics.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"482 ","pages":"Article 134835"},"PeriodicalIF":2.9,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144887062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Spontaneous symmetry breaking in continuous waves, dark solitons, and vortices in linearly coupled bimodal systems 连续波中的自发对称性破缺，线性耦合双峰系统中的暗孤子和涡旋

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-05 DOI: 10.1016/j.physd.2025.134854

Hidetsugu Sakaguchi , Boris A. Malomed

{"title":"Spontaneous symmetry breaking in continuous waves, dark solitons, and vortices in linearly coupled bimodal systems","authors":"Hidetsugu Sakaguchi , Boris A. Malomed","doi":"10.1016/j.physd.2025.134854","DOIUrl":"10.1016/j.physd.2025.134854","url":null,"abstract":"<div><div>We introduce a model governing the copropagation of two components which represent circular polarizations of light in the optical fiber with relative strength <span><math><mrow><mi>g</mi><mo>=</mo><mn>2</mn></mrow></math></span> of the nonlinear repulsion between the components, and linear coupling between them. A more general system of coupled Gross–Pitaevskii (GP) equations, with <span><math><mrow><mi>g</mi><mo>≠</mo><mn>2</mn></mrow></math></span> and the linear mixing between the components, is considered too. The latter system is introduced in its one- and two-dimensional (1D and 2D) forms. A new finding is the spontaneous symmetry breaking (SSB) of bimodal CW (continuous-wave) states in the case of <span><math><mrow><mi>g</mi><mo>></mo><mn>1</mn></mrow></math></span> (in the absence of the linear coupling, it corresponds to the immiscibility of the nonlinearly interacting components). The SSB is represented by an exact asymmetric CW solution. An exact solution is also found, in the case of <span><math><mrow><mi>g</mi><mo>=</mo><mn>3</mn></mrow></math></span>, for stable dark solitons (DSs) supported by the asymmetric CW background. For <span><math><mrow><mi>g</mi><mo>≠</mo><mn>3</mn></mrow></math></span>, numerical solutions are produced for stable DSs supported by the same background. Moreover, we identify a parameter domain where the fully miscible (symmetric) CW background maintains stable DSs with the <em>inner SSB</em> (separation between the components) in its core. In 2D, the GP system produces stable vortex states with a shift between the components and broken isotropy. The vortices include ones with the inter-component shift imposed by the asymmetric CW background, and states supported by the symmetric background, in which the intrinsic shift (splitting) is exhibited by vortical cores of the two components.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"481 ","pages":"Article 134854"},"PeriodicalIF":2.9,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144779737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Phase boundary dynamics for ice nucleation and growth processes in fresh and sea water 淡水和海水中冰成核和生长过程的相边界动力学

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-05 DOI: 10.1016/j.physd.2025.134855

Bernd Kutschan , Silke Thoms , Andrea Thom , Raghav Pathak , Tim Ricken

引用次数: 0

Machine learning-based estimation of correlation dimension from short chaotic time series 基于机器学习的短混沌时间序列相关维数估计

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-05 DOI: 10.1016/j.physd.2025.134869

Shuang Zhou , Shiyu Wang , Herbert Ho-Ching Iu , Yingqian Zhang , Hao Zhang

{"title":"Machine learning-based estimation of correlation dimension from short chaotic time series","authors":"Shuang Zhou , Shiyu Wang , Herbert Ho-Ching Iu , Yingqian Zhang , Hao Zhang","doi":"10.1016/j.physd.2025.134869","DOIUrl":"10.1016/j.physd.2025.134869","url":null,"abstract":"<div><div>Fractal dimension is an important tool for describing complex systems, while correlation dimension is a type of fractal dimension of time series. It can give a more accurate understanding of the characteristics of the system and plays an important role in practical applications. To address the challenge of accurately calculating the correlation dimension of short chaotic data, this paper explores a machine learning-based approach. This method leverages the universal approximation theory of neural networks to compare the Long Short-Term Memory (LSTM) network, the Transformer architecture, and the Backpropagation (BP) network in deep learning, and then selects the optimal approach. Ultimately, the method integrates the backpropagation (BP) neural network with a genetic programming (GP) algorithm for the calculation of the correlation dimension, thereby constructing a novel model. This method aims to expand the volume of chaotic short data and thereby better utilize data information. Initially, a BP neural network with ten hidden layer neurons is constructed, combined with the calculation of correlation dimension to form a new method. Subsequently, simulations are conducted on various commonly encountered chaotic systems, including Lorenz, Hénon, Chen, Logistic systems, solar activity time series and daily female births time series. And their mean squared error (MSE) values are recorded after data augmentation. Finally, the correlation dimensions of these expanded data sets are recalculated and compared with their original values. The findings reveal a remarkable proximity to the theoretical values, thereby validating the method's efficacy in augmenting chaotic short-term sequences. This approach not only enhances the utilization of original chaotic time series data but also paves the way for extracting deeper insights and greater value from short chaotic data.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"482 ","pages":"Article 134869"},"PeriodicalIF":2.9,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

N-breather solutions of the Camassa–Holm equation on oscillatory backgrounds 振荡背景下Camassa-Holm方程的n呼吸解

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-05 DOI: 10.1016/j.physd.2025.134847

Ruomeng Li , Xianguo Geng , Abdul-Majid Wazwaz , Manxue Liu

引用次数: 0

Learning dissipation and instability fields from chaotic dynamics 从混沌动力学中学习耗散和不稳定场

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-05 DOI: 10.1016/j.physd.2025.134865

Ludovico T. Giorgini , Andre N. Souza , Domenico Lippolis , Predrag Cvitanović , Peter Schmid

引用次数: 0

Theory of phase reduction from hypergraphs to simplicial complexes: A general route to higher-order Kuramoto models 从超图到简单复合体的相位缩减理论：通往高阶Kuramoto模型的一般途径

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-05 DOI: 10.1016/j.physd.2025.134858

Iván León , Riccardo Muolo , Shigefumi Hata , Hiroya Nakao

{"title":"Theory of phase reduction from hypergraphs to simplicial complexes: A general route to higher-order Kuramoto models","authors":"Iván León , Riccardo Muolo , Shigefumi Hata , Hiroya Nakao","doi":"10.1016/j.physd.2025.134858","DOIUrl":"10.1016/j.physd.2025.134858","url":null,"abstract":"<div><div>Kuramoto-type models are paradigmatic models in the study of coupled oscillators, since they provide a simple explanation of the collective synchronization transition. Their universality comes from the fact that they are derived through phase reduction. Recent evidence highlighting the importance of higher-order (many-body) interactions in the description of real-world systems has led to extensions of Kuramoto-type models to include such frameworks. However, most of these extensions were obtained by adding higher-order terms instead of performing systematic phase reduction, leaving the questions of which higher-order couplings naturally emerge in the phase model. In this paper, we fill this gap by presenting a theory of phase reduction for coupled oscillators on hypergraphs, i.e., the most general case of higher-order interactions. We show that, although higher-order connection topology is preserved in the phase reduced model, the interaction topology generally changes, due to the fact that the hypergraph generally turns into a simplicial complex in the reduced Kuramoto-type model. Furthermore, we show that, when the oscillators have certain symmetries, even couplings are irrelevant for the dynamics at the first order. The power and ductility of the phase reduction approach are illustrated by applying it to a population of Stuart–Landau oscillators with an all-to-all configuration and with a ring-like hypergraph topology; in both cases, the analysis of the phase model can provide deeper insights and analytical results.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"482 ","pages":"Article 134858"},"PeriodicalIF":2.9,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Modelling and nonclassical symmetry analysis of a complex porous media flow in a dilating channel 膨胀通道中复杂多孔介质流动的建模与非经典对称性分析

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-08-04 DOI: 10.1016/j.physd.2025.134834

Sougata Mandal, Sukhendu Ghosh

{"title":"Modelling and nonclassical symmetry analysis of a complex porous media flow in a dilating channel","authors":"Sougata Mandal, Sukhendu Ghosh","doi":"10.1016/j.physd.2025.134834","DOIUrl":"10.1016/j.physd.2025.134834","url":null,"abstract":"<div><div>This investigation focuses on the symmetry analysis and explicit solution of a fluid flow inside a channel filled with a porous material. The walls of the channel are weakly permeable and dilating vertically. There is an inflow through the pores of the walls that develops flow within the channel. The configuration pertains to the fluid flow, exhibiting either injection or suction across the porous walls at an absolute velocity while experiencing uniform expansion or contraction. The entire flow dynamics is modelled by the Darcy–Brinkman equations. The classical and nonclassical symmetry analysis based on the invariance principle, brings a fourth-order nonlinear ordinary differential equation. The analytical solution is obtained by a double perturbation method, and the comparison is carried out with the numerical solution calculated using the shooting method. A stronger wall expansion pushes the mean-flow through the channel and strengthens the flow rate. The velocity profiles are much fuller for relatively larger Darcy numbers, and behave like a Hartmann flow for the smaller Darcy numbers. Notably, a flow reversal phenomenon is noticed for a suction flow with strong wall dilation rates. Moreover, this study explores a set of conservation laws for the governing model.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"481 ","pages":"Article 134834"},"PeriodicalIF":2.9,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144767073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Influence of extreme events modeled by Lévy flight on Atlantic meridional overturning circulation stability lsamvy飞行模拟的极端事件对大西洋经向翻转环流稳定性的影响

IF 2.9 3区数学

Physica D: Nonlinear Phenomena Pub Date : 2025-07-30 DOI: 10.1016/j.physd.2025.134849

Daniel Tesfay , Larissa Serdukova , Yayun Zheng , Pingyuan Wei , Jinqiao Duan , Jürgen Kurths

{"title":"Influence of extreme events modeled by Lévy flight on Atlantic meridional overturning circulation stability","authors":"Daniel Tesfay , Larissa Serdukova , Yayun Zheng , Pingyuan Wei , Jinqiao Duan , Jürgen Kurths","doi":"10.1016/j.physd.2025.134849","DOIUrl":"10.1016/j.physd.2025.134849","url":null,"abstract":"<div><div>How will extreme events due to human activities and climate change affect the Atlantic meridional overturning circulation (AMOC) is a key concern in climate predictions. The stability of the thermohaline circulation with respect to extreme events, such as fresh-water oscillations, is examined using a conceptual stochastic Stommel two-compartment model. The extreme fluctuations are modeled by symmetric <span><math><mi>α</mi></math></span>-stable Lévy motions whose pathways are cádlág functions with at most a countable number of jumps. The mean first passage time, escape probability and stochastic basin of attraction are used to perform the stability analysis of <em>on (off)</em> equilibrium states. Our aim is to elucidate the interplay between fresh-water forcing, <span><math><mi>α</mi></math></span>, and the stability of metastable AMOC states. Our results argue that for a model with weak fresh-water forcing strength, the greatest threat to the stability of the <em>on</em>-state represents noise with low jumps and higher frequency (modeled by small values of <span><math><mi>α</mi></math></span>) that can be seen as fresh-water inputs from glacier melting due to ocean warming caused by increased greenhouse gas emissions. On the other hand, the <em>off</em>-state stability is more vulnerable to agitations with moderate jumps (modeled by moderate values of <span><math><mi>α</mi></math></span>) and frequencies that can be interpreted as a possible scenario of recovery of the Atlantic thermohaline circulation. Under the repercussion of stochastic noise, <em>on</em> to <em>off</em> transitions are more expected in the model with the strong fresh-water influx. Moreover, transitions from one metastable state to another are equiprobable when the fresh-water input induces a symmetric potential well.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"481 ","pages":"Article 134849"},"PeriodicalIF":2.9,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144779950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0