Neural Network Training Dynamics

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It examines how weights, gradients, activations, and predictions evolve during training, including why models learn some features before others, generalize, forget, or become unstable. Researchers analyze gradient-flow and stochastic-gradient dynamics using neural tangent kernels, mean-field limits, random matrix theory, and loss-landscape tools, alongside mechanistic experiments on tasks such as modular arithmetic and language-model pretraining. This work informs optimizer design, learning-rate and weight-decay schedules, architectures, and training diagnostics, with phenomena such as grokking, edge-of-stability behavior, and representation collapse receiving close attention. A central challenge is connecting tractable theories and early-warning signals to causal, predictive accounts of large finite networks, where optimizer, data, architecture, and noise interact.

Latest papers

520 papers77 papers in the last four weeks, up 93% on the four weeks before. 0.4% of all new papers.

Apr 19, 2026

Decomposing the Depth Profile of Fine-Tuning

Fine-tuning adapts pretrained networks to new objectives. Whether the resulting depth profile of representational change reflects an intrinsic property of the model or the magnitude of gradient flow has not been tested directly. We measure this profile across 240 fine-tuning runs spanning 15 models in four architecture families (encoder and decoder transformers, a state-space model, and an RNN) at scales from 125M to 6.9B parameters. Representational change concentrates in output-proximal layers in every standard-training run except one. We apply a per-layer control that equalizes

\|ΔW\|/\|W\|

across layers after each optimizer step. Under this control, the profile persists in some conditions and collapses in others. At 125M--350M, sequential-block architectures (BERT, OPT, GPT-2) retain the slope across tested objectives while parallel-block architectures (Pythia, CodeGen) retain it only for causal-language-modeling objectives. This architectural distinction narrows at 1.3B--1.4B, where both block types show positive equal-step slopes for CausalLM. Under standard training, profile shape is described by two additional axes: steepness tracks a training-free objective distance at initialization, and profile width is dominated by architecture. We treat the locality gradient, the depthwise slope of representational change, as a composite phenomenon whose components are scale-dependent.

Jayadev Billa

Fine-Tuning Representation Drift Neural Network Training Dynamics

Apr 18, 2026

A Mechanism Study of Delayed Loss Spikes in Batch-Normalized Linear Models

Delayed loss spikes have been reported in neural-network training, but existing theory mainly explains earlier non-monotone behavior caused by overly large fixed learning rates. We study one stylized hypothesis: normalization can postpone instability by gradually increasing the effective learning rate during otherwise stable descent. To test this hypothesis at theorem level, we analyze batch-normalized linear models. Our flagship result concerns whitened square-loss linear regression, where we derive explicit no-rising-edge and delayed-onset conditions, bound the waiting time to directional onset, and show that the rising edge self-stabilizes within finitely many iterations. Combined with a square-loss decomposition, this yields a concrete delayed-spike mechanism in the whitened regime. For logistic regression, under highly restrictive active-margin assumptions, we prove only a supporting finite-horizon directional precursor in a knife-edge regime, with an optional appendix-only loss lower bound under an extra non-degeneracy condition. The paper should therefore be read as a stylized mechanism study rather than a general explanation of neural-network loss spikes. Within that scope, the results isolate one concrete delayed-instability pathway induced by batch normalization.

Peifeng Gao, Wenyi Fang, Yang Zheng +1

School of Computing and Data Science, The University of Hong Kong . · RAMS Technologies Lab, Huawei Technologies Ltd.

Neural Network Optimization Neural Network Training Dynamics

Apr 16, 2026

Curvature-Aligned Probing for Local Loss-Landscape Stabilization

Local loss-landscape stabilization under sample growth is typically measured either pointwise or through isotropic averaging in the full parameter space. Despite practical value, both choices probe directions that contribute little to the dominant local deformation of strongly anisotropic neural landscapes. We recast stabilization as an observational problem and introduce a unified family of criteria parameterized by an aggregation order and a probing distribution; within this family we propose a curvature-aligned criterion

Δ_2^{(D)}

that probes the loss increment field in the top-

D

eigenspace of the empirical Hessian near a trained solution. Solely from a local quadratic model, we prove that

Δ_2^{(D)}

preserves the

O(k^{-2})

mean-squared rate of the full-space criterion while replacing ambient-dimension curvature dependence with dependence on the subspace dimension

D

; a corollary gives a closed-form spectral expression and a proposition identifies the top-

D

eigenspace as extremal within the eigenspace-aligned family. We also derive scalable estimators based on Hessian-vector products, subspace Monte Carlo, and a closed-form Gaussian-moment proxy. On a decoder-only transformer, a curvature-aligned probe occupying a tiny fraction of parameter space already reproduces the full-space mean-squared signal to within numerical noise throughout the validated local regime, and the closed-form estimator is orders of magnitude faster than direct Monte Carlo after subspace construction.

Nikita Kiselev, Andrey Grabovoy

Department of Intelligent Systems Moscow Institute of Physics and Technology

Neural Network Optimization Neural Network Training Dynamics Loss Landscape Geometry Loss Landscape

Apr 16, 2026

Zeroth-Order Optimization at the Edge of Stability

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

Minhak Song, Liang Zhang, Bingcong Li +3

KAIST · KRAFTON · ETH Zurich +2

Edge of Stability Zeroth-Order Optimization Neural Network Training Dynamics Implicit Regularization Neural Network Stability

Apr 15, 2026

Learning Inference Concurrency in DynamicGate MLP Structural and Mathematical Justification

Conventional neural networks strictly separate learning and inference because if parameters are updated during inference, outputs become unstable and even the inference function itself is not well defined [1, 2, 3]. This paper shows that DynamicGate MLP structurally permits learning inference concurrency [4, 5]. The key idea is to separate routing (gating) parameters from representation (prediction) parameters, so that the gate can be adapted online while inference stability is preserved, or weights can be selectively updated only within the inactive subspace [4, 5, 6, 7]. We mathematically formalize sufficient conditions for concurrency and show that even under asynchronous or partial updates, the inference output at each time step can always be interpreted as a forward computation of a valid model snapshot [8, 9, 10]. This suggests that DynamicGate MLP can serve as a practical foundation for online adaptive and on device learning systems [11, 12].

Yongil Choi

Sorynorydotcom Co., Ltd./AI Open Research Lab

Dynamic Neural Networks Online Learning Neural Network Training Dynamics

Apr 14, 2026

Distinct mechanisms underlying in-context learning in transformers

Modern distributed networks, notably transformers, acquire a remarkable ability (termed `in-context learning') to adapt their computation to input statistics, such that a fixed network can be applied to data from a broad range of systems. Here, we provide a complete mechanistic characterization of this behavior in transformers trained on a finite set

S

of discrete Markov chains. The transformer displays four algorithmic phases, characterized by whether the network memorizes and generalizes, and whether it uses 1-point or 2-point statistics. We show that the four phases are implemented by multi-layer subcircuits that exemplify two qualitatively distinct mechanisms for implementing context-adaptive computations. Minimal models isolate the key features of both motifs. Memorization and generalization phases are delineated by two boundaries that depend on data diversity,

K = |S|

. The first (

K_1^\ast

) is set by a kinetic competition between subcircuits and the second (

K_2^\ast

) is set by a representational bottleneck. A symmetry-constrained theory of a transformer's training dynamics explains the sharp transition from 1-point to 2-point generalization and identifies key features of the loss landscape that allow the network to generalize. Put together, we show that transformers develop distinct subcircuits to implement in-context learning and identify conditions that favor certain mechanisms over others.

Cole Gibson, Wenping Cui, Gautam Reddy

Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544, USA · Department of Physics, Princeton University, Princeton, NJ 08544, USA

In-Context Learning Transformer Training Dynamics Transformer Interpretability Neural Network Training Dynamics Mechanistic Interpretability Statistical Physics of Learning

Apr 13, 2026

VISTA: Validation-Informed Trajectory Adaptation via Self-Distillation

Deep learning models may converge to suboptimal solutions despite strong validation accuracy, masking an optimization failure we term Trajectory Deviation. This is because as training proceeds, models can abandon high generalization states for specific data sub-populations, thus discarding previously learned latent features without triggering classical overfitting signals. To address this problem we introduce VISTA, an online self-distillation framework that enforces consistency along the optimization trajectory. Using a validation-informed Marginal Coverage score, VISTA identifies expert anchors, which are earlier model states that retain specialized competence over distinct data regions. A coverage-weighted ensemble of these anchors is integrated online during training, regularizing the loss landscape and preserving mastered knowledge. When evaluated across multiple benchmarks, VISTA demonstrates improved robustness and generalization over standard training and prior self-distillation methods, while a lightweight implementation reduces storage overhead by 90% without performance loss.

Eli Corn, Daphna Weinshall

Department of Computer Science, Hebrew University of Jerusalem, Jerusalem, Israel.

Neural Network Generalization Deep Learning Optimization Self-Distillation Neural Network Training Dynamics

Apr 13, 2026

Spectral Entropy Collapse as a Phase Transition in Delayed Generalisation: An Interventional and Predictive Framework for Grokkin

Grokking - the delayed transition from memorisation to generalisation in neural networks - remains poorly understood. We study this phenomenon through the geometry of learned representations and identify a consistent empirical signature preceding generalisation: collapse of the spectral entropy of the representation covariance matrix. Across modular arithmetic tasks and multiple random seeds, spectral entropy decreases gradually during training and crosses a stable task-specific threshold before test accuracy rises. A representation-mixing intervention that delays this collapse also delays grokking, including under norm-matched controls, indicating that the effect is not explained by parameter norm alone. We further show that the entropy gap predicts the remaining time until grokking with useful out-of-sample accuracy. To probe the structure underlying this transition, we introduce a Fourier-alignment observable for cyclic-group tasks. Entropy collapse is strongly coupled to the emergence of Fourier-aligned representations, suggesting that spectral entropy tracks concentration of the representation into task-structured directions rather than generic compression alone. The same qualitative dynamics appear in non-abelian group composition tasks, while MLP controls show that entropy collapse by itself is insufficient for grokking in the absence of appropriate inductive bias. Taken together, the results support a view of grokking as a representational phase transition with an observable geometric signature. We discuss the scope and limitations of this interpretation, connections to recent feature-learning and spectral-dynamics work, and directions for testing whether similar transitions appear in larger-scale learning systems.

Truong Xuan Khanh, Truong Quynh Hoa, Luu Duc Trung +1

H&K Research Studio, Clevix LLC, Hanoi, Vietnam · Banking Academy of Vietnam, Hanoi, Vietnam

Grokking Neural Network Training Dynamics Representation Learning Neural Network Memorization

Apr 13, 2026

Subcritical Signal Propagation at Initialization in Normalization-Free Transformers

We study signal propagation at initialization in transformers through the averaged partial Jacobian norm (APJN), a measure of gradient amplification across layers. We extend APJN analysis to transformers with bidirectional attention and permutation-symmetric input token configurations by deriving recurrence relations for activation statistics and APJNs across layers. Our theory predicts how attention modifies the asymptotic behavior of the APJN at large depth and matches APJNs measured in deep vision transformers. The criticality picture known from residual networks carries over to transformers: the pre-LayerNorm architecture exhibits power-law APJN growth, whereas transformers with LayerNorm replaced by elementwise

\tanh

-like nonlinearities have stretched-exponential APJN growth, indicating that the latter are subcritical. Applied to Dynamic Tanh (DyT) and Dynamic erf (Derf) transformers, the theory explains why these architectures can be more sensitive to initialization and optimization choices and require careful tuning for stable training.

Sergey Alekseev

Department of Physics and Astronomy Stony Brook University Stony Brook, NY 11794

Transformer Neural Network Initialization Neural Network Training Dynamics Transformer Attention

Apr 12, 2026

Training single-electron and single-photon stochastic physical neural networks

The computational demands of deep learning motivate the investigation of alternative approaches to computation. One alternative is physical neural networks~(PNNs), in which learning and inference are performed directly via physical processes. Stochastic PNNs arise when the underlying neurons are realized by the dynamics of a stochastic activation switch. Here we propose novel electronic and photonic stochastic neurons. The electronic realization is implemented by single-electron tunneling through a quantum dot. The photonic realization is implemented via a single-photon source driving one of two modes coupled via a controllable beam-splitter-like interaction. In the electronic case, the charge state of the quantum dot forms the basis for the stochastic neuron, whereas in the photonic case the occupation of the undriven mode serves as the basis for the stochastic neuron. Training of stochastic PNNs is performed with models of stochastic neurons, as well as with coherently-driven, single-photon detector stochastic neurons previously introduced. Several training strategies for MNIST handwritten digit classification have been investigated using single-hidden-layer stochastic PNNs, including varying the number of trials in each layer to control forward pass stochasticity and employing either true probability or empirical outputs in the backward pass to evaluate their influence on gradient estimation. We show that when empirical outputs are used in the backward pass, the network achieves more than 97% test accuracy with few trials per layer. Despite the simplicity of the model architecture, high test accuracy is maintained in the presence of a high degree of noise and model uncertainty. The results demonstrate the potential of embracing stochastic PNNs for deep learning.

Tong Dou, Shiro Kumara, Josh Burns +6

School of Engineering and Technology, UNSW Canberra, Campbell, ACT 2600, Australia · Sussex Centre for Quantum Technologies, University of Sussex, Brighton BN1 9RH, United Kingdom · Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom +2

Handwritten Digit Recognition Physical Neural Networks Neural Network Training Dynamics

Apr 12, 2026

Heterogeneous Connectivity in Sparse Networks: Fan-in Profiles, Gradient Hierarchy, and Topological Equilibria

Profiled Sparse Networks (PSN) replace uniform connectivity with deterministic, heterogeneous fan-in profiles defined by continuous, nonlinear functions, creating neurons with both dense and sparse receptive fields. We benchmark PSN across four classification datasets spanning vision and tabular domains, input dimensions from 54 to 784, and network depths of 2--3 hidden layers. At 90% sparsity, all static profiles, including the uniform random baseline, achieve accuracy within 0.2-0.6% of dense baselines on every dataset, demonstrating that heterogeneous connectivity provides no accuracy advantage when hub placement is arbitrary rather than task-aligned. This result holds across sparsity levels (80-99.9%), profile shapes (eight parametric families, lognormal, and power-law), and fan-in coefficients of variation from 0 to 2.5. Internal gradient analysis reveals that structured profiles create a 2-5x gradient concentration at hub neurons compared to the ~1x uniform distribution in random baselines, with the hierarchy strength predicted by fan-in coefficient of variation (

r = 0.93

). When PSN fan-in distributions are used to initialise RigL dynamic sparse training, lognormal profiles matched to the equilibrium fan-in distribution consistently outperform standard ERK initialisation, with advantages growing on harder tasks, achieving +0.16% on Fashion-MNIST (

p = 0.036

,

d = 1.07

), +0.43% on EMNIST, and +0.49% on Forest Cover. RigL converges to a characteristic fan-in distribution regardless of initialisation. Starting at this equilibrium allows the optimiser to refine weights rather than rearrange topology. Which neurons become hubs matters more than the degree of connectivity variance, i.e., random hub placement provides no advantage, while optimisation-driven placement does.

Nikodem Tomczak

Thulge Labs, Singapore

Dynamic Sparse Training Sparse Neural Networks Structured Sparsity Neural Network Training Dynamics Neural Network Initialization

Apr 11, 2026

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

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 work explicitly set aside natural frequency as a learnable parameter; we show that on sparse layered architectures, frequency learning outperforms coupling-weight learning among converged seeds (96.0% vs. 83.3% at matched parameter counts, p = 1.8e-12). The approximately 50% convergence failure rate under random initialization is a loss-landscape property, not a gradient error; topology-aware spectral seeding eliminates it in all settings tested (46/100 to 100/100 seeds on the primary task; 50/50 on a second task, K-only training, and a larger architecture).

Mani Rash Ahmadi

Calibur Labs

Equilibrium Propagation Oscillatory Neural Networks Kuramoto Model Neural Network Training Dynamics

Apr 10, 2026

Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks

We study the population loss landscape of two-layer ReLU networks of the form

\sum_{k=1}^K \mathrm{ReLU}(w_k^\top x)

in a realisable teacher-student setting with Gaussian covariates. We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space. This perspective reveals a hierarchical organisation of minima into discrete families and shows how overparameterisation changes their stability and reachability under gradient-based dynamics. In this overparameterised regime, global minima become increasingly accessible, attracting the dynamics and reducing convergence to spurious solutions. Overall, our results reveal intrinsic limitations of common simplifying assumptions, which may miss essential features of the loss landscape even in minimal neural network models.

Jie Huang, Bruno Loureiro, Stefano Sarao Mannelli

Physics, Chalmers University of Technology and University of Gothenburg · Ecole Normale Superieure, PSL & CNRS · Data Science and AI, Computer Science and Engineering, Chalmers University of Technology and University of Gothenburg +1

Overparameterization Loss Landscape Neural Network Optimization ReLU Neural Networks Loss Landscape Geometry Shallow Neural Networks+1

Apr 10, 2026

Statistical Properties of the King Wen Sequence: An Anti-Habituation Structure That Does Not Improve Neural Network Training

The King Wen sequence of the I-Ching (c. 1000 BC) orders 64 hexagrams -- states of a six-dimensional binary space -- in a pattern that has puzzled scholars for three millennia. We present a rigorous statistical characterization of this ordering using Monte Carlo permutation analysis against 100,000 random baselines. We find that the sequence has four statistically significant properties: higher-than-random transition distance (98.2nd percentile), negative lag-1 autocorrelation (p=0.037), yang-balanced groups of four (p=0.002), and asymmetric within-pair vs. between-pair distances (99.2nd percentile). These properties superficially resemble principles from curriculum learning and curiosity-driven exploration, motivating the hypothesis that they might benefit neural network training. We test this hypothesis through three experiments: learning rate schedule modulation, curriculum ordering, and seed sensitivity analysis, conducted across two hardware platforms (NVIDIA RTX 2060 with PyTorch and Apple Silicon with MLX). The results are uniformly negative. King Wen LR modulation degrades performance at all tested amplitudes. As curriculum ordering, King Wen is the worst non-sequential ordering on one platform and within noise on the other. A 30-seed sweep confirms that only King Wen's degradation exceeds natural seed variance. We explain why: the sequence's high variance -- the very property that makes it statistically distinctive -- destabilizes gradient-based optimization. Anti-habituation in a fixed combinatorial sequence is not the same as effective training dynamics.

Augustin Chan

Independent Researcher

Permutation Testing Neural Network Optimization Curriculum Learning Neural Network Training Dynamics

Apr 9, 2026

Mathematical analysis of one-layer neural network with fixed biases, a new activation function and other observations

We analyze a simple one-hidden-layer neural network with ReLU activation functions and fixed biases, with one-dimensional input and output. We study both continuous and discrete versions of the model, and we rigorously prove the convergence of the learning process with the

L^2

squared loss function and the gradient descent procedure. We also prove the spectral bias property for this learning process. Several conclusions of this analysis are discussed; in particular, regarding the structure and properties that activation functions should possess, as well as the relationships between the spectrum of certain operators and the learning process. Based on this, we also propose an alternative activation function, the full-wave rectified exponential function (FReX), and we discuss the convergence of the gradient descent with this alternative activation function.

Fabricio Macià, Shu Nakamura

M2ASAI, Universidad Politécnica de Madrid. ETSI Navales. Avda. de la Memoria, 4. 28040 Madrid, Spain. · Department of Mathematics, Faculty of Sciences, Gakushuin University, 1-5-1, Mejiro, Toshima, Tokyo, 171-8588 Japan

Shallow Neural Networks Neural Network Training Dynamics Neural Network Optimization ReLU Neural Networks Neural Network Activation Functions Spectral Bias

Apr 8, 2026

Training Deep Visual Networks Beyond Loss and Accuracy Through a Dynamical Systems Approach

Deep visual recognition models are usually trained and evaluated using metrics such as loss and accuracy. While these measures show whether a model is improving, they reveal very little about how its internal representations change during training. This paper introduces a complementary way to study that process by examining training through the lens of dynamical systems. Drawing on ideas from signal analysis originally used to study biological neural activity, we define three measures from layer activations collected across training epochs: an integration score that reflects long-range coordination across layers, a metastability score that captures how flexibly the network shifts between more and less synchronised states, and a combined dynamical stability index. We apply this framework to nine combinations of model architecture and dataset, including several ResNet variants, DenseNet-121, MobileNetV2, VGG-16, and a pretrained Vision Transformer on CIFAR-10 and CIFAR-100. The results suggest three main patterns. First, the integration measure consistently distinguishes the easier CIFAR-10 setting from the more difficult CIFAR-100 setting. Second, changes in the volatility of the stability index may provide an early sign of convergence before accuracy fully plateaus. Third, the relationship between integration and metastability appears to reflect different styles of training behaviour. Overall, this study offers an exploratory but promising new way to understand deep visual training beyond loss and accuracy.

Hai La Quang, Hassan Ugail, Newton Howard +3

Posts and Telecommunications Institute of Technology Vietnam · Centre for Visual Computing and Intelligent Systems University of Bradford United Kingdom · School of Individualized Study Rochester Institute of Technology United States

Neural Network Training Dynamics

Apr 8, 2026

The Lifecycle of the Spectral Edge: From Gradient Learning to Weight-Decay Compression

We decompose the spectral edge -- the dominant direction of the Gram matrix of parameter updates -- into its gradient and weight-decay components during grokking in two sequence tasks (Dyck-1 and SCAN). We find a sharp two-phase lifecycle: before grokking the edge is gradient-driven and functionally active; at grokking, gradient and weight decay align, and the edge becomes a compression axis that is perturbation-flat yet ablation-critical (>4000x more impactful than random directions). Three universality classes emerge (functional, mixed, compression), predicted by the gap flow equation. Nonlinear probes show information is re-encoded, not lost (MLP

R^2=0.99

where linear

R^2=0.86

), and removing weight decay post-grok reverses compression while preserving the algorithm.

Yongzhong Xu

Neural Network Training Dynamics Grokking Weight Decay Neural Network Compression Spectral Methods

Apr 7, 2026

On Dominant Manifolds in Reservoir Computing Networks

Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a general linear continuous-time reservoir in the infinite-data limit, we show that the training data generate an invariant subspace of the trained reservoir, whose dimension equals the number of dominant modes. We then specialize to a simplified diagonal linear reservoir, where we link the dominant eigenvalues and eigenvectors to the spectrum of a backward Dynamic Mode Decomposition matrix, which yields a finite-dimensional approximation of the backward-time Koopman operator of the system generating the training data. We illustrate the emergence of these dominant modes during training in simulation, and discuss how the analysis may be extended to nonlinear RC via tangent dynamics and differential p-dominance.

Noa Kaplan, Alberto Padoan, Anastasia Bizyaeva

Department of Computer Science at Cornell University, Ithaca, NY, USA 14853 · Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada · Sibley School of Mechanical and Aerospace Engineering at Cornell University, Ithaca, NY, USA, 14853

Time Series Forecasting Reservoir Computing Neural Network Training Dynamics

Apr 6, 2026

Spectral Edge Dynamics Reveal Functional Modes of Learning

Training dynamics during grokking concentrate along a small number of dominant update directions -- the spectral edge -- which reliably distinguishes grokking from non-grokking regimes. We show that standard mechanistic interpretability tools (head attribution, activation probing, sparse autoencoders) fail to capture these directions: their structure is not localized in parameter or feature space. Instead, each direction induces a structured function over the input domain, revealing low-dimensional functional modes invisible to representation-level analysis. For modular addition, all leading directions collapse to a single Fourier mode. For multiplication, the same collapse appears only in the discrete-log basis, yielding a 5.9x improvement in concentration. For subtraction, the edge spans a small multi-mode family. For

x^2+y^2

, no single harmonic basis suffices, but cross-terms of additive and multiplicative features provide a 4x variance boost, consistent with the decomposition (a+b)^2 - 2ab. Multitask training amplifies this compositional structure, with the

x^2+y^2

spectral edge inheriting the addition circuit's characteristic frequency (2.3x concentration increase). These results suggest that training discovers low-dimensional functional modes over the input domain, whose structure depends on the algebraic symmetry of the task. These results suggest that spectral edge dynamics identify low-dimensional functional subspaces governing learning, whose representation depends on the algebraic structure of the task. Simple harmonic structure emerges only when the task admits a symmetry-adapted basis; more complex tasks require richer functional descriptions.

Yongzhong Xu

Neural Network Training Dynamics Grokking Compositional Representation Learning Frequency-Domain Feature Learning

Apr 6, 2026

Muon Dynamics as a Spectral Wasserstein Flow

Gradient normalization can stabilize deep-learning optimization, and spectral normalizations are especially natural for matrix-shaped parameter blocks; Muon is the motivating example. We study an idealized deterministic, continuous-time, vanishing-momentum version of this idea in the mean-field regime, where wide models are represented by probability measures on parameter space. Starting from normalized matrix flows, we introduce Spectral Wasserstein distances indexed by norms

γ

on positive semidefinite matrices: the trace norm recovers the classical

W_2

distance, the operator norm gives the Muon geometry, and Schatten norms interpolate between them. We develop a static Kantorovich formulation, a max--min robust-cost representation, and Gaussian reductions extending the Bures formula. For monotone norms, we prove equivalence with a Benamou--Brenier formulation and obtain a gradient-flow interpretation of normalized mean-field training dynamics. Numerical experiments illustrate the resulting flows for MMD objectives, Gaussian reductions, two-layer ReLU models, and shallow attention.

Gabriel Peyré

CNRS and ENS, PSL Université

Neural Network Training Dynamics Wasserstein Distance Muon Optimizer Optimal Transport Wasserstein Gradient Flows

Apr 6, 2026

Dimensional Criticality at Grokking Across MLPs and Transformers

Abrupt transitions between distinct dynamical regimes are a hallmark of complex systems. Grokking in deep neural networks provides a striking example -- an abrupt transition from memorization to generalization long after training accuracy saturates -- yet robust macroscopic signatures of this transition remain elusive. Here we introduce TDU--OFC (Thresholded Diffusion Update--Olami-Feder-Christensen), an offline avalanche probe that converts gradient snapshots into cascade statistics and extracts a macroscopic observable -- the time-resolved effective cascade dimension

D(t)

-- via grokking-aligned finite-size scaling. Across Transformers trained on modular addition and MLPs trained on XOR, we discover a localized dynamical crossing of the Gaussian diffusion baseline

D=1

precisely at the generalization transition. The crossing direction is task-dependent: modular addition descends through

D=1

(approaching from

D>1

), while XOR ascends (from

D<1

). This opposite-direction convergence is consistent with attraction toward a candidate shared critical manifold, rather than trivial residence near

D \approx 1

. Negative controls confirm this picture: ungrokked runs remain supercritical (

D>1

) and never enter the post-transition regime. In addition, avalanche distributions exhibit heavy tails and finite-size scaling consistent with the dimensional exponent extracted from

D(t)

. Shadow-probe controls (

α_{\mathrm{train}}=0

) confirm that

D(t)

is non-invasive, and grokked trajectories diverge from ungrokked ones in

D(t)

some

100

--

200

epochs before the behavioral transition.

Ping Wang

Institute of High Energy Physics, Chinese Academy of Science, 100049 Beijing, China

Neural Network Training Dynamics Phase Transitions in Neural Networks Grokking Statistical Physics of Learning

Apr 6, 2026

Grokking as Dimensional Phase Transition in Neural Networks

Neural network grokking -- the abrupt memorization-to-generalization transition -- challenges our understanding of learning dynamics. Through finite-size scaling of gradient avalanche dynamics across eight model scales, we find that grokking is a dimensional phase transition: effective dimensionality~

D

crosses from sub-diffusive (subcritical,

D < 1

) to super-diffusive (supercritical,

D > 1

) at generalization onset, exhibiting self-organized criticality (SOC). Crucially,

D

reflects gradient field geometry, not network architecture: synthetic i.i.d.\ Gaussian gradients maintain

D \approx 1

regardless of graph topology, while real training exhibits dimensional excess from backpropagation correlations. The grokking-localized

D(t)

crossing -- robust across topologies -- offers new insight into the trainability of overparameterized networks.

Ping Wang

Institute of High Energy Physics, Chinese Academy of Science, 100049 Beijing, China

Phase Transitions in Neural Networks Grokking Neural Network Training Dynamics Statistical Physics of Learning Gradient Descent Dynamics

Apr 5, 2026

Topological Sensitivity in Connectome-Constrained Neural Networks

Connectome-constrained neural networks are often evaluated against sparse random controls and then interpreted as evidence that biological graph topology improves learning efficiency. We revisit that claim in a controlled flyvis-based study using a Drosophila connectome, a naive self-loop-matched random graph, and a degree-preserving rewired null. Under weak controls, in which both models were recovered from a connectome-trained checkpoint and the null matched only global graph counts, the connectome appeared substantially better in early loss, mean activity, and runtime. That picture changed under stricter controls. Training both graphs from a shared random initialization removed the early loss advantage, and replacing the naive null by a degree-preserving null removed the apparent activity advantage. A five-sample degree-preserving ensemble and a pre-training activity-scale diagnostic further strengthened this revised interpretation. We also report a descriptive mechanism analysis of the earlier weak-control comparison, but we treat it as behavioral characterization rather than proof of causal superiority. We show that previously reported topology advantages in connectome-constrained neural networks can arise from initialization and null-model confounds, and largely disappear under fair from-scratch initialization and degree-preserving controls.

Nalin Dhiman

School of Computing and Electrical Engineering, Indian Institute of Technology, Mandi, India

Brain Network Analysis Neural Network Training Dynamics

Apr 3, 2026

The Spectral Lifecycle of Transformer Training: Transient Compression Waves, Persistent Spectral Gradients, and the Q/K--V Asymmetry

We present the first systematic study of weight matrix singular value spectra during transformer pretraining, tracking full SVD decompositions of every weight matrix at 25-step intervals across three model scales (30M--285M parameters). We discover three phenomena: (1)~Transient Compression Waves: stable rank compression propagates as a traveling wave from early to late layers, creating a dramatic gradient that peaks early then reverses -- late layers eventually over-compress past early layers. (2)~Persistent Spectral Gradients: the power-law exponent~

α

develops a permanent depth gradient forming a non-monotonic inverted-U in deeper models, with peaks shifting toward earlier layers as depth increases. (3)~Q/K--V Functional Asymmetry: value/output projections compress uniformly while query/key projections carry the full depth-dependent dynamics. The dissociation between transient compression and persistent spectral shape reveals that rank and spectral shape encode fundamentally different information about training. We formalize this as a two-timescale dynamical model and derive scaling laws (

Δα\propto L^{0.26}

,

R^2{=}0.99

). We validate on nine models across three families (custom, GPT-2, Pythia; 30M--1B parameters; 8--36 layers), demonstrate that

α

predicts layer importance (

ρ{=}0.69

--

0.84

,

p{<}0.02

), and show that spectral-guided pruning outperforms Last-N heuristics by

1.1{\times}

--

3.6{\times}

across seven models in two families (GPT-2 124M--774M, Pythia 160M--1B), with worst-vs-best gaps up to

23.7{\times}

confirming the causal role of spectral structure.

Yi Liu

Transformer Training Dynamics Transformer Neural Network Training Dynamics Spectral Methods

Apr 3, 2026

Revealing the Learning Dynamics of Long-Context Continual Pre-training

Existing studies on Long-Context Continual Pre-training (LCCP) mainly focus on small-scale models and limited data regimes (tens of billions of tokens). We argue that directly migrating these small-scale settings to industrial-grade models risks insufficient adaptation and premature training termination. Furthermore, current evaluation methods rely heavily on downstream benchmarks (e.g., Needle-in-a-Haystack), which often fail to reflect the intrinsic convergence state and can lead to "deceptive saturation". In this paper, we present the first systematic investigation of LCCP learning dynamics using the industrial-grade Hunyuan-A13B (80B total parameters), tracking its evolution across a 200B-token training trajectory. Specifically, we propose a hierarchical framework to analyze LCCP dynamics across behavioral (supervised fine-tuning probing), probabilistic (perplexity), and mechanistic (attention patterns) levels. Our findings reveal: (1) Necessity of Massive Data Scaling: Training regimes of dozens of billions of tokens are insufficient for industrial-grade LLMs' LCCP (e.g., Hunyuan-A13B reaches saturation after training over 150B tokens). (2) Deceptive Saturation vs. Intrinsic Saturation: Traditional NIAH scores report "fake saturation" early, while our PPL-based analysis reveals continuous intrinsic improvements and correlates more strongly with downstream performance. (3) Mechanistic Monitoring for Training Stability: Retrieval heads act as efficient, low-resource training monitors, as their evolving attention scores reliably track LCCP progress and exhibit high correlation with SFT results. This work provides a comprehensive monitoring framework, evaluation system, and mechanistic interpretation for the LCCP of industrial-grade LLM.

Yupu Liang, Shuang Chen, Guanwei Zhang +2

Tencent Hunyuan, Beijing, China

Continued Pretraining Neural Network Training Dynamics Long-Context Language Model Evaluation

Apr 2, 2026

Plateaus, Optima, and Overfitting in Multi-Layer Perceptrons: A Saddle-Saddle-Attractor Scenario

Vanishing gradients and overfitting are central problems in machine learning, yet are typically analyzed in asymptotic regimes that obscure their dynamical origins. Here we provide a dynamical description of learning in multi-layer perceptrons (MLPs) via a minimal model inspired by Fukumizu and Amari. We show that training dynamics traverse plateau and near-optimal regions, both organized by saddle structures, before converging to an overfitting regime. Under suitable conditions on the data, this regime collapses to a single attractor modulo symmetry. Furthermore, for finite noisy datasets, convergence to the theoretical optimum is impossible, and the dynamics necessarily settle into an overfitting solution.

Alex Alì Maleknia, Yuzuru Sato

Department of Mathematics, Hokkaido University, Kita 12 Nishi 7, Kita-ku, Sapporo, Hokkaido 060-0812, Japan · Present address: IMAG, IROKO, Univ Montpellier, Inria, CNRS · RIES, Hokkaido University, Kita 12 Nishi 7, Kita-ku, Sapporo, Hokkaido 060-0812, Japan

Neural Network Training Dynamics Multilayer Perceptrons Neural Network Generalization Gradient Descent Dynamics

Mar 31, 2026

Neural Collapse Dynamics: Depth, Activation, Regularisation, and Feature Norm Threshold

Neural collapse (NC) -- the convergence of penultimate-layer features to a simplex equiangular tight frame -- is well understood at equilibrium, but the dynamics governing its onset remain poorly characterised. We identify a simple and predictive regularity: NC occurs when the mean feature norm reaches a model-dataset-specific critical value, fn*, that is largely invariant to training conditions. This value concentrates tightly within each (model, dataset) pair (CV < 8%); training dynamics primarily affect the rate at which fn approaches fn*, rather than the value itself. In standard training trajectories, the crossing of fn below fn* consistently precedes NC onset, providing a practical predictor with a mean lead time of 62 epochs (MAE 24 epochs). A direct intervention experiment confirms fn* is a stable attractor of the gradient flow -- perturbations to feature scale are self-corrected during training, with convergence to the same value regardless of direction (p>0.2). Completing the (architecture)x(dataset) grid reveals the paper's strongest result: ResNet-20 on MNIST gives fn* = 5.867 -- a +458% architecture effect versus only +68% on CIFAR-10. The grid is strongly non-additive; fn* cannot be decomposed into independent architecture and dataset contributions. Four structural regularities emerge: (1) depth has a non-monotonic effect on collapse speed; (2) activation jointly determines both collapse speed and fn*; (3) weight decay defines a three-regime phase diagram -- too little slows, an optimal range is fastest, and too much prevents collapse; (4) width monotonically accelerates collapse while shifting fn* by at most 13%. These results establish feature-norm dynamics as an actionable diagnostic for predicting NC timing, suggesting that norm-threshold behaviour is a general mechanism underlying delayed representational reorganisation in deep networks.

Anamika Paul Rupa

Department of Electrical Engineering and Computer Science, Howard University, Washington, DC, 20059, USA

Neural Collapse Neural Network Training Dynamics

Mar 31, 2026

From Density Matrices to Phase Transitions in Deep Learning: Spectral Early Warnings and Interpretability

A key problem in the modern study of AI is predicting and understanding emergent capabilities in models during training. Inspired by methods for studying reactions in quantum chemistry, we present the ``2-datapoint reduced density matrix". We show that this object provides a computationally efficient, unified observable of phase transitions during training. By tracking the eigenvalue statistics of the 2RDM over a sliding window, we derive two complementary signals: the spectral heat capacity, which we prove provides early warning of second-order phase transitions via critical slowing down, and the participation ratio, which reveals the dimensionality of the underlying reorganization. Remarkably, the top eigenvectors of the 2RDM are directly interpretable making it straightforward to study the nature of the transitions. We validate across four distinct settings: deep linear networks, induction head formation, grokking, and emergent misalignment. We then discuss directions for future work using the 2RDM.

Max Hennick, Guillaume Corlouer

Phase Transitions in Neural Networks Neural Network Training Dynamics Statistical Physics of Learning Neural Network Interpretability

Mar 30, 2026

Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training

We develop the spectral edge analysis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme aspect ratio regime (parameters

P \sim 10^8

, window

W \sim 10

), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at position

k^* = \mathrm{argmax}\, σ_j/σ_{j+1}

. From three assumptions we derive: (i) gap dynamics governed by a Dyson-type ODE with curvature asymmetry, damping, and gradient driving; (ii) a spectral loss decomposition linking each mode's learning contribution to its Davis--Kahan stability coefficient; (iii) the Gap Maximality Principle, showing that

k^*

is the unique dynamically privileged position -- its collapse is the only one that disrupts learning, and it sustains itself through an

α

-feedback loop requiring no assumption on the optimizer. The adiabatic parameter

\mathcal{A} = \|ΔG\|_F / (η\, g^2)

controls circuit stability:

\mathcal{A} \ll 1

(plateau),

\mathcal{A} \sim 1

(phase transition),

\mathcal{A} \gg 1

(forgetting). Tested across six model families (150K--124M parameters): gap dynamics precede every grokking event (24/24 with weight decay, 1/24 without), the gap position is optimizer-dependent (Muon:

k^*=1

, AdamW:

k^*=2

on the same model), and 19/20 quantitative predictions are confirmed. The framework is consistent with the edge of stability, Tensor Programs, Dyson Brownian motion, the Lottery Ticket Hypothesis, and neural scaling laws.

Yongzhong Xu

Phase Transitions in Neural Networks Neural Network Training Dynamics Phase Transitions Grokking

Mar 30, 2026

Critical Damping as a Momentum Schedule: Multi-Seed Validation, a Hybrid Recipe, and an Exhaustive Negative Result on Surgical Layer Selection

The critical damping condition of the damped harmonic oscillator model of SGD with momentum (Qian, 1999) yields a momentum schedule with no tuned hyperparameters: mu(t) = 1 - 2*sqrt(alpha(t)). Across five seeds on ResNet-18/CIFAR-10 (200-epoch cosine schedule) it reaches 90% test accuracy 2.34x faster than constant mu=0.9 (range 1.71-2.86x, 5/5 seeds, one-sided paired t-test p=4e-4), at the cost of a real final-accuracy deficit of 0.46 pp (5/5 seeds, p=0.009). A short-schedule control rules out a schedule-length artifact: compressed baselines either pay 0.5-0.9 pp of accuracy or stay slower to 90% at equal accuracy. A hybrid recipe -- critical-damping momentum until 90%, then constant mu=0.9 -- removes the deficit and keeps the speedup: 95.45 +/- 0.05% final accuracy at 2.4x faster progress to 90% (n=5). The speedup generalizes across architectures (VGG-16 without skip connections: 1.72x, n=3); on CIFAR-100 early gains persist (2-4x to mid-training thresholds) but the accuracy cost grows (-1.7 pp), compressing the accuracy-matched gain to 1.14x. We also report an exhaustive negative result on surgical layer selection. Version 2 of this paper claimed that gradient attribution on misclassified images selects which layers to retrain; running the identical correction protocol on all 35 combinations of 3-of-7 layer groups ranks the selected triple 11th of 35 (exact p=0.31) -- no better than random. What survives is weaker but real: combinations containing the top-ranked layer outperform the rest (+6.2 vs -1.4 mean net error reduction), and the bottom of the gradient-norm ranking reliably predicts the most harmful interventions (down to -20 net errors). Gradient attribution on errors is a harm-avoidance signal, not a selector of repair targets. We release the full 35-combination landscape as a baseline for layer-selection claims.

Ivan Pasichnyk

We Label Data Inc.

Gradient-Based Attribution Momentum Methods Neural Network Training Dynamics Accelerated Gradient Methods Stochastic Gradient Descent