Mean-Field Theory

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Mean-field theory replaces a large collection of interacting units with dynamics for their shared distribution, making systems with many neurons, agents, or particles easier to analyze. In machine learning, it describes infinite-width neural-network training and dynamic mean-field behavior; related approaches use state evolution for approximate message passing and McKean–Vlasov equations for interacting particles, control, and games. Researchers derive these limits with gradient-flow, kinetic, and cavity methods, then test how well finite systems follow them using propagation-of-chaos bounds, particle simulations, and learning benchmarks. A central challenge is quantifying when the simplified population dynamics remain accurate for finite systems, especially with non-exchangeable interactions, common noise, and transitions between lazy and feature-learning regimes.

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34 papers1 paper in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Fluctuations of Nonlinear Observables in Mean Field Neural Network Training
Oct 7, 2026Arnaud Descours, Geoffrey LacourNeural Network Training Dynamics Uncertainty Quantification
UCBL · MaIAGE
Attention Mean Fields Predict Average Representation Dynamics and Reveal Context-Specific Computation
Sep 14, 2026Micah Adler, John W. Byers, Mark CrovellaAttention Head Analysis In-Context Learning
Department of Computer Science, Boston University · Faculty of Computing & Data Sciences, Boston University
Sparse Prototype Code Underlies Classification and Prediction Across Modalities
Aug 16, 2026Yehonatan Avidan, Daniel D. Lee, Haim SompolinskyNeural Representation Geometry Mean-Field Theory
The Racah Institute of Physics, Hebrew University · Edmond and Lily Safra Center for Brain Sciences, Hebrew University · Department of Electrical and Computer Engineering, Cornell Tech, Cornell University +1
Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model
Aug 14, 2026Taegyun KimPhase Transitions Mean-Field Theory
A cylindrical neural approximation theorem for conditional laws of McKean-Vlasov equations with common noise
Aug 8, 2026Nacira Agram, Reda Hmioui, Jan RemsMean-Field Theory Neural Network Approximation Theory
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden. · Digital Futures, Sweden. · Grenoble INP – Ensimag, Université Grenoble Alpes, Grenoble, France. +1
Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees
Jun 12, 2026Pekka Malo, Lauri Viitasaari, Patrik Nummi +3Evolutionary Optimization Optimization Convergence Analysis
Aalto University, Department of Information and Service Management, Finland. · Aalto University, Department of Information and Communications Engineering, Finland. · Indian Institute of Management Ahmedabad, Operations and Decision Sciences, India. +1
Discovering and decoding latent mean-field structure with variational autoencoders
Jun 7, 2026Marco Biroli, Max Welling, Vincenzo VitelliVariational Autoencoders Mean-Field Theory
Department of Physics and the James Franck Institute, University of Chicago, Chicago, IL, USA · CuspAI, Cambridge, England · AMLab, University of Amsterdam, Amsterdam, Netherlands +1
Beyond Linear and Overcomplete Regimes: A Mean-Field Analysis of Bottleneck Autoencoders
Jun 5, 2026Santanu Das, Ramyak Bilas, Pascal Esser +1Autoencoders Mean-Field Theory
STCS department Tata Institute Of Fundamental Research Mumbai, India-400005 · Department of Mathematics Indiana University Bloomington Indiana, USA-47405 · Department of Mathematics Ludwig-Maximilians-Universität München Germany, München-80799 +1
Nonlocal Mean Field Schrödinger Bridge with Learned Interactions
Jun 2, 2026Daisuke Inoue, Dante Kalise, Mathieu LaurièreNeural Surrogate Modeling Schrödinger Bridges
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom. · Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning, NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, Shanghai 200126, People’s Republic of China.
The Crowded Embedding Space: A Mean-Field Mechanism for Emergent Marginalization in Retrieval-Augmented Agents
Jun 1, 2026Shwan Ashrafi, Dan RothFokker–Planck Equation Algorithmic Fairness
Oracle AI
Mean-Field Diffuser: Scaling Offline MARL to Thousands of Agents
May 28, 2026Wenhao Li, Xiangfeng Wang, Bo JinMulti-Agent System Optimization Mean Field Games
Tongji University · East China Normal University
Kernel-based potential mean-field games with unbiased random Fourier $U$ -statistics
May 28, 2026Yumiharu NakanoMulti-Agent Coordination Mean Field Games
Department of Mathematical and Computing Science, Institute of Science Tokyo
The incremental voter model: mean-field analysis and convergence to equilibrium
May 27, 2026Fei Cao, Xiaoqian GongOpinion Dynamics Computational Social Science
Amherst College - Department of Mathematics, Amherst, MA 01002, USA
Thinned Mean Field Langevin Dynamics
May 27, 2026Zonghao Chen, Heishiro Kanagawa, François-Xavier Briol +2Langevin Dynamics Mean-Field Theory
University College London · 2Newcastle University · 3The Alan Turing Institute +1
Feature Learning in Wide Neural Networks under $μ$ P: Identifiability and Sparse-Dictionary Decomposition of the Mean-Field Limit
May 23, 2026Akmal XodarevRepresentation Learning Parameter Identifiability
Independent Researcher · Tashkent, Uzbekistan
Move on Muon : A Hamiltonian probability gradient flow perspective of Muon optimizer
May 22, 2026Aratrika Mustafi, Soumya Mukherjee, Bharath K. SriperumbudurMomentum Methods Muon Optimizer
Department of Statistics, Pennsylvania State University
Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks
May 21, 2026Margalit Glasgow, Joan BrunaShallow Neural Networks Mean-Field Theory
Massachusetts Institute of Technology · Courant Institute School of Mathematics, Computing and Data Science, New York University
Dropout Universality: Scaling Laws and Optimal Scheduling at the Edge-of-Chaos
May 20, 2026Lucas Fernandez SarmientoMean-Field Theory
Carnegie Mellon University (CMU), Pittsburgh, Pennsylvania, USA.
Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime
May 11, 2026Albert Alcalde, Leon Bungert, Konstantin Riedl +1Transformer Inference Transformer
Department of Mathematics Friedrich-Alexander-University Erlangen-Nürnberg · Institute of Mathematics CAIDAS University of Würzburg · Mathematical Institute University of Oxford Reuben College +1
Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow
May 10, 2026Lénaïc Chizat, Maria Colombo, Roberto Colombo +1Dynamical Systems Optimization Convergence Analysis
Kinetic theory for Transformers and the lost-in-the-middle phenomenon
May 9, 2026Mitia Duerinckx, Borjan Geshkovski, Stefano RossiLost-in-the-Middle Effect Self-Attention
Learnability and Competition in High-Dimensional Multi-Component ICA
May 8, 2026Eser Ilke Genc, Samet Demir, Zafer DoganIndependent Component Analysis Unsupervised Representation Learning
MLIP Research Group, KUIS AI Center, Koç University · Department of EEE, Koç University
Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers
May 8, 2026Noboru Isobe, Daisuke Inoue, Masaaki ImaizumiTransformer Transformer Training Dynamics
RIKEN Center for Advanced Intelligence Project · Imperial College London · The University of Tokyo +1
Population-Aware Imitation Learning in Mean-field Games with Common Noise
May 5, 2026Grégoire Lambrecht, Mathieu LaurièreMean Field Games Adversarial Imitation Learning
Center for Data Science, New York University and NYU Shanghai, New York, United States of America · Shanghai Center for Data Science; NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai; NYU Shanghai, Shanghai, People’s Republic of China.
Continuous-time q-learning for mean-field control with common noise, part-I: Theoretical foundations
Apr 30, 2026Zhenjie Ren, Xiaoli Wei, Xiang Yu +1Stochastic Optimal Control Q-Learning
LaMME, Universit´e ´Evry Paris-Saclay, ´Evry, France · Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong · Department of Industrial Engineering and Operations Research, Columbia University, New York, USA
MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data
Apr 1, 2026Liyao Lyu, Xinyue Yu, Hayden SchaefferEmergent Behavior in Multi-Agent Systems Mean-Field Theory
Department of Mathematics University of California, Los Angeles Los Angeles, CA 90024, USA
Rigorous Asymptotics for First-Order Algorithms Through the Dynamical Cavity Method
Mar 15, 2026Yatin Dandi, David Gamarnik, Francisco Pernice +1Approximate Message Passing Dynamical Mean-Field Theory
Statistical Physics of Computation Laboratory, École polytechnique fédérale de Lausanne (EPFL) CH-1015 Lausanne. · Sloan School of Management, Operations Research Center and Institute of Data, Systems and Society (IDSS), MIT. · CSAIL and LIDS, MIT.
A Gaussian Comparison Theorem for Training Dynamics in Machine Learning
Mar 10, 2026Ashkan PanahiDynamical Mean-Field Theory Neural Network Training Dynamics
Department of Computer Science and Engineering, Chalmers University, Gothenburg, Sweden
Gradient Flow Through Diagram Expansions: Learning Regimes and Explicit Solutions
Feb 4, 2026Dmitry Yarotsky, Eugene Golikov, Yaroslav GusevTensor Decomposition Mean-Field Theory
Applied AI Institute, Moscow, Russia · Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.
Local exponential stability of mean-field Langevin descent-ascent and associated particle system
Feb 2, 2026Geuntaek Seo, Minseop Shin, Pierre Monmarché +1Nash Equilibrium Lyapunov Stability

Oct 7, 2026J/K move · Enter open · S save

Fluctuations of Nonlinear Observables in Mean Field Neural Network Training

Arnaud Descours, Geoffrey Lacour

UCBL · MaIAGE

Neural Network Training Dynamics Uncertainty Quantification Mean-Field Theory Shallow Neural Networks

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Mean field limits describe the training dynamics of wide neural networks through the evolution of the empirical distribution of their parameters. Although functional central limit theorems characterize the asymptotic fluctuations of this distribution, quantities of practical interest are typically nonlinear observables of the parameter distribution rather than the distribution itself. In this work, we show how these mean field fluctuations propagate to finite dimensional nonlinear observables for shallow neural networks trained by stochastic gradient descent. Working in the weighted Sobolev space in which the limiting fluctuation process is constructed, we apply a functional Delta method under ordinary Fr{é}chet differentiability, without requiring Lions derivatives with respect to the measure variable. We obtain a central limit theorem for the observables and, under a suitable representation of their differentials, an explicit covariance formula inherited from the underlying mean field fluctuation theory. We also study whether prescribed quantities of interest can be recovered from the selected observations. Under a constant rank assumption, we prove that a quantity of interest factors locally through the observation functional if and only if, throughout a neighborhood, the kernel of the differential of the observation is contained in that of the quantity of interest. Thus, a differential condition expressed directly in the ambient Sobolev space yields an exact nonlinear local factorization. These results provide a framework both for quantifying finite-width uncertainty on observable, statistically or physically meaningful quantities and for assessing whether the chosen observations contain the information required to identify them.