# Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks

arXiv: https://arxiv.org/abs/2205.09653 · HTML: https://mlsanity.com/paper/2205.09653/

## Key facts

- Cited by 105 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 121 references that are in the map.
- First posted on arXiv 19 May 2022.

## Builds on (most cited first, 30 of 59)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Deep Information Propagation](https://mlsanity.com/paper/1611.01232/) (2016)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [Learning curves of generic features maps for realistic datasets with a teacher-student model](https://mlsanity.com/paper/2102.08127/) (2021)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Neural Networks as Kernel Learners: The Silent Alignment Effect](https://mlsanity.com/paper/2111.00034/) (2021)

## Built on by (most cited first, 30 of 105)

- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [How Does Critical Batch Size Scale in Pre-training?](https://mlsanity.com/paper/2410.21676/) (2024)
- [Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit](https://mlsanity.com/paper/2309.16620/) (2023)
- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [Grokking as a First Order Phase Transition in Two Layer Networks](https://mlsanity.com/paper/2310.03789/) (2023)
- [Dynamics of Finite Width Kernel and Prediction Fluctuations in Mean Field Neural Networks](https://mlsanity.com/paper/2304.03408/) (2023)
- [Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning](https://mlsanity.com/paper/2406.06158/) (2024)
- [Feature-Learning Networks Are Consistent Across Widths At Realistic Scales](https://mlsanity.com/paper/2305.18411/) (2023)
- [Second-order regression models exhibit progressive sharpening to the edge of stability](https://mlsanity.com/paper/2210.04860/) (2022)
- [Infinite Limits of Multi-head Transformer Dynamics](https://mlsanity.com/paper/2405.15712/) (2024)
- [On the Stepwise Nature of Self-Supervised Learning](https://mlsanity.com/paper/2303.15438/) (2023)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [The Onset of Variance-Limited Behavior for Networks in the Lazy and Rich Regimes](https://mlsanity.com/paper/2212.12147/) (2022)
- [Mechanism of feature learning in deep fully connected networks and kernel machines that recursively learn features](https://mlsanity.com/paper/2212.13881/) (2022)
- [Super Consistency of Neural Network Landscapes and Learning Rate Transfer](https://mlsanity.com/paper/2402.17457/) (2024)
- [Generalization through variance: how noise shapes inductive biases in diffusion models](https://mlsanity.com/paper/2504.12532/) (2025)
- [Universal Sharpness Dynamics in Neural Network Training: Fixed Point Analysis, Edge of Stability, and Route to Chaos](https://mlsanity.com/paper/2311.02076/) (2023)
- [Scaling Collapse Reveals Universal Dynamics in Compute-Optimally Trained Neural Networks](https://mlsanity.com/paper/2507.02119/) (2025)
- [Hyperparameter Transfer Enables Consistent Gains of Matrix-Preconditioned Optimizers Across Scales](https://mlsanity.com/paper/2512.05620/) (2025)
- [Infinite-width limit of deep linear neural networks](https://mlsanity.com/paper/2211.16980/) (2022)
- [Quantitative CLTs in Deep Neural Networks](https://mlsanity.com/paper/2307.06092/) (2023)
- [The Influence of Learning Rule on Representation Dynamics in Wide Neural Networks](https://mlsanity.com/paper/2210.02157/) (2022)
- [Meta-Principled Family of Hyperparameter Scaling Strategies](https://mlsanity.com/paper/2210.04909/) (2022)
- [Phase diagram of early training dynamics in deep neural networks: effect of the learning rate, depth, and width](https://mlsanity.com/paper/2302.12250/) (2023)
- [Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer](https://mlsanity.com/paper/2502.02531/) (2025)
- [Mixed Dynamics In Linear Networks: Unifying the Lazy and Active Regimes](https://mlsanity.com/paper/2405.17580/) (2024)
- [The Hidden Width of Deep ResNets: Tight Error Bounds and Phase Diagram](https://mlsanity.com/paper/2509.10167/) (2025)
- [Theory of Scaling Laws for In-Context Regression: Depth, Width, Context and Time](https://mlsanity.com/paper/2510.01098/) (2025)

## Abstract (arXiv)

We analyze feature learning in infinite-width neural networks trained with gradient flow through a self-consistent dynamical field theory. We construct a collection of deterministic dynamical order parameters which are inner-product kernels for hidden unit activations and gradients in each layer at pairs of time points, providing a reduced description of network activity through training. These kernel order parameters collectively define the hidden layer activation distribution, the evolution of the neural tangent kernel, and consequently output predictions. We show that the field theory derivation recovers the recursive stochastic process of infinite-width feature learning networks obtained from Yang and Hu (2021) with Tensor Programs . For deep linear networks, these kernels satisfy a set of algebraic matrix equations. For nonlinear networks, we provide an alternating sampling procedure to self-consistently solve for the kernel order parameters. We provide comparisons of the self-consistent solution to various approximation schemes including the static NTK approximation, gradient independence assumption, and leading order perturbation theory, showing that each of these approximations can break down in regimes where general self-consistent solutions still provide an accurate description. Lastly, we provide experiments in more realistic settings which demonstrate that the loss and kernel dynamics of CNNs at fixed feature learning strength is preserved across different widths on a CIFAR classification task.