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There Will Be a Scientific Theory of Deep Learning

Jamie Simon, Daniel Kunin, Alexander Atanasov, Enric Boix-Adserà, Blake Bordelon, Jeremy Cohen, Nikhil Ghosh, Florentin Guth, Arthur Jacot, Mason Kamb, Dhruva Karkada, Eric J. Michaud, Berkan Ottlik, Joseph Turnbull

arXiv:2604.21691v1stat.MLcs.LG

TL;DR

Deep learning still lacks a unified scientific framework explaining why or how networks learn. The paper makes the case for an emerging scientific theory and argues that it is best understood as a mechanics of the learning process.

  • Problem

    Despite decades of work, deep learning still lacks a unified scientific framework explaining why or how networks learn, and theory plays little role in practice.

    Source: §1 · p. 2

  • Method

    The paper argues for a scientific theory of deep learning, framing it as a mechanics of the learning process.

    Source: §1 · p. 2

  • Results

    The paper concludes that pieces of a scientific theory of deep learning are emerging and will take the form of learning mechanics.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The proposed name, learning mechanics, emphasizes the paper's view of deep-learning theory as a mechanics of the learning process.

    Source: §1 · p. 2

    Learning mechanics for realistic deep learning may break down in small-scale, handcrafted, or otherwise special cases.

    Source: §1.1 · p. 4

Abstract

from arXiv · show

In this paper, we make the case that a scientific theory of deep learning is emerging. By this we mean a theory which characterizes important properties and statistics of the training process, hidden representations, final weights, and performance of neural networks. We pull together major strands of ongoing research in deep learning theory and identify five growing bodies of work that point toward such a theory: (a) solvable idealized settings that provide intuition for learning dynamics in realistic systems; (b) tractable limits that reveal insights into fundamental learning phenomena; (c) simple mathematical laws that capture important macroscopic observables; (d) theories of hyperparameters that disentangle them from the rest of the training process, leaving simpler systems behind; and (e) universal behaviors shared across systems and settings which clarify which phenomena call for explanation. Taken together, these bodies of work share certain broad traits: they are concerned with the dynamics of the training process; they primarily seek to describe coarse aggregate statistics; and they emphasize falsifiable quantitative predictions. We argue that the emerging theory is best thought of as a mechanics of the learning process, and suggest the name learning mechanics. We discuss the relationship between this mechanics perspective and other approaches for building a theory of deep learning, including the statistical and information-theoretic perspectives. In particular, we anticipate a symbiotic relationship between learning mechanics and mechanistic interpretability. We also review and address common arguments that fundamental theory will not be possible or is not important. We conclude with a portrait of important open directions in learning mechanics and advice for beginners. We host further introductory materials, perspectives, and open questions at learningmechanics.pub.

1 Introduction

The paper argues that a scientific theory of deep learning is beginning to emerge as a “learning mechanics” focused on training dynamics and coarse, quantitative predictions. It presents this theory as a route to scientific understanding, practical guidance, and support for AI safety.

1 Introduction

  • Five research strands signal an emerging theory: solvable settings, tractable limits, empirical laws, disentangled hyperparameters, and universal behaviors.The authors characterize the strands as evidence that a scientific theory is emerging.Source: §1 · pp. 2–3

1.1 What’s in a mechanics?

  • Learning mechanics treats training as movement through parameter space shaped by interactions, with gradients mediating forces and loss minima serving as equilibria.The paper says its research strands emphasize training dynamics, coarse aggregate statistics, and average-case predictions over worst-case bounds.Source: §1.1 · p. 3, §1 · p. 3
  • Its seven desiderata are fundamental, mathematical, predictive, comprehensive, intuitive, useful, and humble, with applicability limits made explicit.The theory should describe training, hidden representations, and final weights at a useful level of resolution, while acknowledging that it may fail in small-scale, handcrafted, or special cases.Source: §1.1 · pp. 3–4

1.2 Why learning mechanics matters

  • Learning mechanics could illuminate intelligence and nature, guide real-world AI design, optimization, scaling, and deployment, and help clarify systems for safety, oversight, and control.The paper identifies potential connections to neuroscience and cognitive science, and to mechanistic interpretability.Source: §1.2 · p. 4

1.3 Plan for this paper

  • The paper surveys five lines of evidence, relates learning mechanics to other perspectives and mechanistic interpretability, addresses objections to fundamental theory, and outlines open directions and introductory resources.It is written for a broad audience, including deep learning scientists and practitioners, AI safety and interpretability researchers, and newcomers.Source: §1.3 · pp. 4–5

2 Evidence of an emerging mechanics of learning

The section presents evidence that deep learning’s measurable training process can support a scientific mechanics, using tractable settings and limits to expose regularities. It surveys empirical laws, hyperparameter theories, and universal behaviors that describe training dynamics and learned structures, while noting that explanations and scope remain unresolved.

2 Evidence of an emerging mechanics of learning

  • Deep learning exposes its governing equations and measurable weights, activations, gradients, and losses, but coupled nonlinear, high-dimensional dynamics make complexity—not opacity—the central obstacle.The authors argue that this transparency makes experiments easier to design, replicate, and interrogate, enabling empirical regularities and theoretical predictions to be tested.Source: §2 · p. 5

2.1 Analytically solvable settings exist

  • Deep linear analyses show greedy low-rank learning: task components linked to larger singular values emerge first, a bias also seen in nonlinear networks [Saxe et al., 2014, Kalimeris et al., 2019, Simon et al., 2023b].Small initialization, increased depth, stronger mini-batch noise, and explicit ℓ2 regularization can further strengthen this bias.Source: §2.1 · p. 7
  • NTK-linearized networks make least-squares dynamics and predictions tractable, linking architecture to inductive bias and generalization, but miss feature learning and nonconvex optimization.The NTK framework also captures phenomena such as double descent and scaling laws.Source: §2.1 · pp. 7–8
  • Fully nonlinear toy models increasingly isolate solvable learning mechanisms, but each captures only a slice of nonlinear dynamics and no unified framework has emerged.Examples include feature learning on structured targets, teacher-student dynamics, associative memory, modular arithmetic, and attention models.Source: §2.1 · p. 8

2.2 Insightful limits reveal fundamental behavior

  • At infinite width, initialization scaling separates lazy NTK training, with negligible representation change, from rich training, where features adapt to data; analogous scale sensitivity appears at finite width.In the rich regime, neuron populations can specialize and representations change during training.Source: §2.2 · pp. 9–10
  • Depth, optimization, and joint data–model scaling limits provide further tractable descriptions, with noncommuting limits and finite-size behavior important to connecting them to practice.For residual networks, [depth]^-1 and [depth]^-1/2 layer scaling yield qualitatively different limits; joint data–model limits can capture scaling-law behavior that fixed-data infinite-parameter limits miss.Source: §2.2 · pp. 10–11
  • The Discretization Hypothesis holds that finite-size corrections to limits usually worsen performance while reducing data, time, memory, and compute costs, and is falsifiable if they yield otherwise unattainable benefits.The paper says this unproven hypothesis has implicitly underpinned important work on analytical study of large models.Source: §2.2 · p. 11

2.3 Simple empirical laws capture meaningful macroscopic statistics

  • Within an architectural family, final test loss follows predictable power laws governed by compute, dataset size, and network size [Kaplan et al., 2020, Hestness et al., 2017].The exponents remain unexplained, and no framework robustly predicts them a priori across realistic settings; empirical laws are a fruitful target for experimental theory-building.Source: §2.3 · pp. 12, 14, Figure 3 · p. 12
  • With full-batch gradient descent, loss sharpness rises progressively and then plateaus near 2/η, the edge of stability [Cohen et al., 2021a].The 2/η threshold is the maximum stable sharpness in convex optimization; analyses connect stabilization to third-order curvature and describe dynamics as time-averaged gradient flow plus unstable-direction oscillations, with trajectory predictions matching experiments.Source: §2.3 · pp. 12–13, Figure 4 · p. 13
  • At training’s end, class representations cluster around their means, which form a regular simplex; this geometry is energy-minimizing with cross-entropy and small weight decay [Papyan et al., 2020, Zhu et al., 2021].This provides a condition-specific account of neural collapse.Source: §2.3 · p. 13
  • First-layer weight Gram matrices heuristically align with average gradient outer products, while parameterization symmetries yield gradient-flow conservation laws in linear and nonlinear networks [Radhakrishnan et al., 2024, Saxe et al., 2014, Du et al., 2018, Arora et al., 2019a, Kunin et al., 2021].The first-layer heuristic can accurately predict top eigenvectors, but its theoretical explanation remains partial; symmetry-specific statistics are conserved under gradient flow and weakly broken by SGD in predictable ways.Source: §2.3 · p. 13

2.4 Hyperparameters can be disentangled and understood

  • SDE theory predicts SGD’s linear scaling rule, critical-batch models describe serial-time–compute tradeoffs, and µP enables near-optimal hyperparameter transfer from small proxy models to wide models [Mandt et al., 2017, McCandlish et al., 2018, Yang et al., 2022].For adaptive optimizers, Malladi et al. [2022] argued that learning rate should scale with the square root of batch size. Full-batch trajectories can also be modeled as curvature-penalized gradient flow, simplifying analysis, though extending this account to stochastic and adaptive optimizers remains open [Cohen et al., 2025].Source: §2.4 · pp. 14–16

2.5 Universal phenomena appear across settings and tasks

  • Different architectures can achieve similar performance on matched vision tasks and near-identical diffusion-model outputs, while shared data statistics such as power laws and multiscale structure help explain cross-domain learning [Liu et al., 2022, Smith et al., 2023, Zhang et al., 2024].The paper also describes locality and geometric adaptivity as inductive biases that yield quantitative predictions for diffusion models [Kadkhodaie et al., 2024, Kamb and Ganguli, 2025, Niedoba et al., 2025].Source: §2.5 · p. 17
  • Representations become similar across initializations, widths, architectures, tasks, and modalities, with similarity increasing alongside model size and performance; the extent of convergence and its dependence on comparison metrics remain unsettled [Raghu et al., 2017, Huh et al., 2024].Explanations include feature-kernel concentration, SGD’s implicit regularization, and shared data structure; the observed range and degree of convergence are not fully known, and depend crucially on the chosen similarity metric.Source: §2.5 · p. 17

3 Relation to other perspectives

The paper presents statistical, information-theoretic, physics, neuroscience, and interpretability approaches as complementary perspectives on deep learning. It argues that studying learning mechanics can help build a science of deep learning and work symbiotically with mechanistic interpretability.

3 Relation to other perspectives

  • The statistical perspective explains learning through implicit bias toward simple, generalizing functions and the ease of optimization in overparameterized networks, but calls for precise accounts of training.The authors argue that these explanations depend on properties of deep learning and natural data, so understanding training mechanics is necessary.Source: §3 · p. 18
  • The information-theoretic perspective views learning as compressing datasets into parameters or representations while retaining predictive information and discarding irrelevant information.Making this account actionable requires examining how architecture, training, and data interact to implement compression and affect its success.Source: §3 · p. 18
  • The physics-of-deep-learning community seeks satisfying average-case theories of neural-network learning and is described as pursuing a mechanics of learning.The paper says clarifying important problems and coordinating effort are challenges for this community.Source: §3 · p. 18
  • Neuroscience-inspired approaches study brain hypotheses and interpretable circuits, and the authors expect learning dynamics to matter for testing related ideas in deep learning.Examples include edge-selective receptive fields in visual cortex and convolutional networks, and systems neuroscience’s resemblance to mechanistic interpretability.Source: §3 · pp. 18–19
  • Developmental interpretability uses singular learning theory to model training as Bayesian, sequential phase transitions mediated by loss-landscape geometry.The authors see it as pursuing a fundamental mechanics of learning and a rigorous foundation for interpretability, with potential for cross-pollination and tool-sharing.Source: §3 · p. 19
  • A fundamental mechanics of learning is presented as the foundation for turning machine learning’s trial-and-error practice into a science.The authors note that systematic organization around benchmarks supported empirical progress, while training and application of large models remain more alchemy than science.Source: §3 · p. 19

3.1 Learning mechanics ⇄mechanistic interpretability

  • Their symbiosis pairs interpretability’s qualitative accounts of trained mechanisms with learning mechanics’ quantitative laws, which can formalize assumptions and explain how mechanisms form.The authors also describe interpretability as offering empirical phenomena for mathematical theory.Source: §3.1 · pp. 19–20
  • Mechanistic interpretability gives learning mechanics concrete modeling targets, including induction heads, Fourier features in algebraic tasks, and feature geometry shaped by data correlations.The authors argue that interpretability’s attention to data structure can help bridge the gap between simplified theoretical data models and observed behavior.Source: §3.1 · p. 20

4 Reasons for skepticism and responses

The paper argues that developing a theory of deep learning is possible and worthwhile despite skepticism. It responds that empirical study, useful local theories, and connections to applied work can make progress valuable before a comprehensive account of large models exists.

4 Reasons for skepticism and responses

  • Recent deep-learning successes have exposed new phenomena and enabled empirical tests, while a larger, more diverse research community and the long timescale of science support optimism.The authors note that model scaling revealed apparent convergence to universal representations and created a tight feedback loop for asking questions and checking answers.Source: §4 · p. 21
  • Although a comprehensive theory of LLMs will take time, local theories of stack components already inform large-model hyperparameter scaling and other practical work.Examples include empirical scaling laws, hyperparameter-scaling prescriptions, neural-tangent-kernel data attribution [Park et al., 2023b], and theoretically motivated optimizers [Gupta et al., 2018, Jordan et al., 2024]; similar theories could address instabilities, dataset selection, attribution, and normalization.Source: §4 · p. 21
  • A rigorous understanding of primitive objects can support applied work on large models even without a theory that explains the whole model.The authors hope learning mechanics will provide tools that adjacent fields such as mechanistic interpretability can apply.Source: §4 · p. 21
  • Understanding deep learning requires multiple levels of study, from learning mechanics as physics through mechanistic interpretability as biology to high-level model psychology.The authors suggest these levels roughly parallel Marr’s levels of computational analysis.Source: §4 · p. 21
  • A theory of learning should be developed alongside a theory of data structure; both belong to the project of learning mechanics.The paper points to its discussion of data structure in Section 2.5 and Open Direction 2.Source: §4 · p. 21
  • Even if AI may understand itself before humans do, theory can have near-term impact, likely progress will involve human scientists working with AI, and human-parseable theory can aid oversight.The authors argue that such a theory gives people a foothold in oversight if AI safety is the goal.Source: §4 · p. 22

5 Open directions in learning mechanics

The paper presents a curated agenda of open questions that it expects a theory of learning mechanics could address over the next decade, aiming to sharpen a shared research agenda.

Open Direction 1

  • Can solvable models capture both deep nonlinear parameter dynamics and nonlinear function learning, while illuminating features, depth, optimization, architectures, and modern learning paradigms?The paper notes that existing workhorse models capture these forms of nonlinearity separately, with only a few special cases combining them and no unified framework yet.Source: §5 · p. 22

Open Direction 2

  • What structure in natural data drives learning, and can a predictive theory identify the minimal sufficient statistics that models use across architectures and training stages?The paper also asks whether relevant dataset structure can be described by a model with free parameters found through empirical fit.Source: §5 · p. 22

Open Direction 3

  • Do deep networks minimize a precise notion of functional complexity among low-loss functions, and when is that minimization exact or approximate?Candidate notions include Kolmogorov complexity, circuit complexity, and weight norms; the paper also asks whether mechanistic-interpretability features and circuitry emerge from such minimization.Source: §5 · p. 22

Open Direction 4

  • Can learned features, circuits, and mechanisms be defined mathematically from first principles, and can those definitions formalize assumptions such as linear representability, locality, sparsity, and compositionality?The paper asks how these formal notions connect to the rich-versus-lazy picture of feature learning.Source: §5 · pp. 22–23

Open Direction 5

  • Are finite networks best understood as discretizations of continuum systems, with model size and learning-rate changes reducing discretization error at additional computational cost?The proposed view treats width as a limit over hidden-layer neuron-activity measures, residual-network depth as a neural SDE or ODE discretization, and small-step stochastic optimization as approximately a flow.Source: §5 · p. 23

Open Direction 6

  • How far can systematic analysis and limiting procedures eliminate hyperparameters, and are any hyperparameters irreducible?The paper asks what remains if all hyperparameters can be removed.Source: §5 · p. 23

Open Direction 7

  • Can a theory explain why scaling-law power laws arise and predict their nontrivial exponents from first principles across realistic settings?The paper asks which dataset, architecture, and optimization measurements such predictions require; it notes that no current framework does this robustly across realistic settings.Source: §5 · p. 23

Open Directions 8–9

  • How do curvature dynamics shape learned features and generalization, and what principles explain optimizer performance, stability, and failure?The paper asks why curvature rises without implicit regularization, how architecture and data affect this progressive sharpening, and why adaptive methods such as Adam and Muon outperform SGD on large language models.Source: §5 · p. 23

6 How to get involved in the development of learning mechanics

The authors encourage newcomers to enter learning mechanics from diverse backgrounds, using experiments and simple explanations to build scientific understanding through collaboration and exploration. They recommend learning foundational tools and making use of introductory materials and open research directions.

6 How to get involved in the development of learning mechanics

  • Useful work requires no specific academic background; undergraduate mathematics, familiarity with deep learning, and willingness to learn are the main prerequisites.Experience in other fields can contribute through cross-pollination.Source: §6 · p. 24

6.1 Tenets for getting started

  • Experiments should be frequent and simple, while researchers should prioritize understandable insights over technical complexity.Experiments check assumptions, inform and test theories, and surface phenomena; useful ideas can be more valuable than difficult theorems or solutions to particular problems.Source: §6.1 · p. 24
  • Deep learning theory should prioritize scientific understanding over state-of-the-art performance, without forcing benchmark gains when they do not follow naturally from the science.Fundamental science may eventually improve performance, and such gains can demonstrate understanding when they arise naturally.Source: §6.1 · p. 24
  • Newcomers should seek human mentorship and collaboration, and try several problems before committing deeply to one.The authors also recommend watching paper talks when available, since live presentations convey project nuance.Source: §6.1 · pp. 24–25
  • Researchers should invest in foundational tools, including statistical physics, random matrix theory, and classical optimization, while exploring less-used tools such as wavelets, graphical models, and information theory.The authors point newcomers to introductory material at learningmechanics.pub and encourage tackling the open directions in Section 5.Source: §6.1 · p. 25

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