Pith. sign in

REVIEW 4 major objections 5 minor 35 references

Cognition, this paper argues, is a Riemannian gradient flow of a single scalar potential, and the split between fast intuition and slow deliberation follows from the geometry of the metric rather than from separate cognitive systems.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Cognition is re-described as Riemannian gradient flow on a manifold; an anisotropic metric makes fast intuitive and slow deliberative timescales emerge from a single equation.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Standard singular perturbation theory with a grand 'geometric cognition' label; the advertised RL experiments are absent and the fast-slow split is built into the assumptions. the 4 major comments →

arxiv 2512.12225 v3 pith:ACK4UXIR submitted 2025-12-13 cs.AI

A Geometric Theory of Cognition for Machine Intelligence

Laha Ale This is my paper
classification cs.AI
keywords Riemannian gradient flowcognitive potentialanisotropic metricfast-slow dynamicsslow manifolddual-process cognitioncognitive manifoldmachine intelligence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes that cognition is one piece of geometry: an internal state η moves on a manifold by the Riemannian gradient flow η̇ = −G(η)⁻¹∇J(η), where J is a scalar 'cognitive potential' that gathers prediction error, complexity, reward, norms, and effort, and G is a learned metric that makes some directions of change cheap and others expensive. The paper's main theorem says that when G is anisotropic — cheap in some coordinates, ε⁻²-expensive in others — the dynamics split by themselves into fast automatic responses and slow deliberative updates, with the slow variables following a reduced gradient flow on an invariant manifold. No modular System 1/System 2 machinery is needed; the timescale separation comes out of the geometry. If the claim is right, a single dynamical law would underwrite intuition, deliberation, memory, and prediction, and the design of artificial agents would become a problem of learning J and G rather than assembling modules.

Core claim

On its own terms, the paper claims that all cognition is a Riemannian gradient flow ˙η = −G(η)⁻¹∇_η J(η), where J is a twice-differentiable 'cognitive potential' that combines prediction accuracy, representational economy, task utility, norms, and effort, and G is a learned positive-definite metric encoding which directions of state change are cheap or costly. The paper proves (Theorem 1) that under an anisotropic metric, trajectories decompose into fast automatic relaxation of the cheap coordinates h and slow deliberative evolution of the expensive coordinates c, with the slow dynamics confined to a locally invariant manifold and governed by the reduced flow ċ = −ε²∇_cJ(h*(c),c) + O(ε³). Th

What carries the argument

The load-bearing object is the anisotropic Riemannian metric G_ε = diag(I_m, ε⁻²I_k) applied to the gradient of a scalar potential J. The inverse metric amplifies cheap (fast) directions and damps expensive (slow) ones, so the flow ẋ = −G_ε⁻¹∇J has h-variables relaxing on an O(1) timescale and c-variables drifting on an O(ε²) timescale; a standard singular-perturbation argument then shows the c-dynamics reduce to a lower-dimensional gradient flow on a slow manifold h = h*(c) + O(ε²). The metric, not any engineered module, is what produces dual-process cognition.

Load-bearing premise

The load-bearing premise is that every cognitive change is the smooth descent of a single scalar potential through a metric: if real cognition contains non-gradient, stochastic, or path-dependent forces, the universal law collapses.

What would settle it

Record the internal state of a system governed by the theory — e.g., neural or behavioural trajectories while an agent performs a fixed repeated task — and look for a stable limit cycle: a closed loop the system retraces indefinitely under constant conditions. Riemannian gradient flow strictly decreases the potential J along every non-stationary trajectory and so cannot close on itself; one reproducible closed orbit would settle the question.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Dual-process behaviour becomes a prediction of geometry: any agent whose state follows the gradient-flow law with an anisotropic metric will show fast intuitive responses and slow deliberative updating, so no separate System 1/System 2 modules are needed.
  • A wide family of existing theories (Bayesian inference, predictive coding, reinforcement learning, free-energy style accounts, and dual-process models) can be read as special cases of the same flow with different J and G, giving them a common mathematical substrate.
  • The reduction theorem licenses practical model-order reduction: long-horizon deliberative dynamics can be simulated on the slow manifold with a proven O(ε²) error, making large-scale cognitive agents cheaper to train and evaluate.
  • The framework yields trajectory-level behavioural predictions (response times, learning curves, mode switching, noise and stress susceptibility) that go beyond explaining means and can be tested directly.
  • For machine intelligence, the design recipe becomes to learn J and G from data rather than to assemble memory, prediction, and planning modules; the geometry itself supplies memory-like persistence and multi-timescale behaviour.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • An implication the paper leaves implicit: the unification claim is a statement about mathematical form, not about mechanism; before treating Bayesian, predictive-coding, and reinforcement-learning models as interchangeable, one must show the data they explain can be fit by a single J and G without residual non-gradient components.
  • Editorial extension: because (G, J) is not identifiable from trajectories alone, the theory becomes testable only if the metric is estimated independently — for instance, by perturbing each cognitive variable and measuring its relaxation rate, then using those estimates to predict response times in new conditions.
  • If taken seriously as an architecture, the law suggests a built-in memory: slow variables c integrate evidence over long windows even without a recurrent module; a natural experimental test is to look for slow-manifold persistence in neural population activity during delay periods.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a unified geometric theory of cognition: the cognitive state is a point on a Riemannian manifold, a scalar cognitive potential J integrates predictive, structural, and task-related pressures, and cognition is the gradient flow ˙η = −G(η)⁻¹∇_η J(η). The central claim is that anisotropic structure in the metric generates fast automatic and slow deliberative regimes without modular or hybrid architectures. Theorem 1 asserts a fast–slow decomposition under a block-diagonal anisotropic metric, with a slow manifold and reduced slow dynamics. The paper closes with toy ODE simulations using the potential J(h,c)=1/2(h−c³)²+1/2c² and a brief discussion of implications for cognitive science and AGI.

Significance. If the framework were fully established, it would provide a suggestive mathematical unification of dual-process theories, Bayesian inference, predictive coding, and reinforcement learning under one dynamical principle. The mathematical core of Theorem 1 is a correct conditional application of singular-perturbation theory (Tikhonov/Fenichel), and the toy simulations reproduce the expected O(1) and O(ε²) scalings. Those are genuine strengths. However, the paper's central conceptual novelty is not established: the fast–slow separation is assumed through the metric, not derived from a more fundamental principle, and the advertised empirical validation in the abstract is absent from the body. The contribution is therefore better characterized as a mathematical reformulation with conditional results than as a validated theory of cognition or machine intelligence.

major comments (4)
  1. [Section 3.2, Assumption 1 / Eq. (2)] The advertised 'emergence' of fast–slow cognition is not derived from geometry. Assumption 1 fixes both the coordinate partition and the ε-scaling before any analysis: G_ε = diag(I_m, ε^{-2} I_k) immediately yields Eq. (2), ˙h = −∇_h J, ˙c = −ε²∇_c J, which already contains the separation of time scales. The sentence 'This distinction is not imposed by hand but emerges from the anisotropic metric' (end of §3.1) is contradicted by this construction. Theorem 1 is a correct conditional statement for such metrics, but the central claim that no modular architecture is needed remains unsupported unless the paper shows how a general or learned metric produces this anisotropy.
  2. [Section 4, Step 3 vs Theorem 1(2)] There is an internal inconsistency in the proof of the main theorem. Theorem 1(2) asserts that the slow manifold M_ε lies O(ε²)-close to M_0, but Step 3 of the proof derives h = h*(c) + O(ε), i.e., only O(ε)-closeness. The subsequent expansion of ∇_c J uses only the O(ε) bound, so the stated O(ε²) closeness is not established by the written argument. A standard two-scale expansion can likely repair this, but the proof as written does not prove the claimed bound.
  3. [Abstract vs. Full Text] The abstract (and the arXiv metadata) states that the framework is instantiated in partially observable RL environments, 'consistently outperforms feedforward baselines,' achieves 'robustness comparable to recurrent architectures,' and produces 'highly predictable latent trajectories.' The body contains none of these experiments. The 'Results' section only reports toy ODE simulations with a hand-chosen potential, J(h,c)=1/2(h−c³)²+1/2c². This is a direct mismatch between the central empirical claims and the evidence provided. Either the missing experiments must be included, or the claims must be substantially weakened to what the paper actually demonstrates.
  4. [Section 2, Definitions 1–2] The universal law ˙η = −G(η)⁻¹∇J presupposes that cognitive change is a deterministic gradient flow of a C² potential. No empirical support is offered for this core assumption; the simulations impose the gradient structure by construction. For a 'universal dynamical law,' a toy potential is not sufficient. The paper would need at least one non-tautological behavioral or neural dataset on which the framework is fitted and predicts, or an explicit disclaimer that the contribution is only a mathematical unification of existing models. This issue is particularly salient because the abstract promises machine-learning validations.
minor comments (5)
  1. [Section 1, Figure 1 caption] 'as shown Figure 1' should read 'as shown in Figure 1'; the sentence beginning 'We represent the internal cognitive state' is missing a definite article before 'cognitive state'.
  2. [Section 2.1] There is a typo: 'thecognitive state' in the run-in definition should be 'the cognitive state.' Also, the notation ∇(G)J in §2.2 is not defined with a subscript; consider writing ∇_G J to avoid confusion with the coordinate gradient.
  3. [Figure 2 caption] The caption says 'Numerical verification of Assumption 1,' but the figure verifies the predicted dynamical consequence of that assumption (scaling of speeds), not the assumption itself. Rephrase to 'verification of the predicted scaling under Assumption 1.'
  4. [Section 4, Step 3] 'Fenichel’s theorem' is invoked without a precise statement or reference. Since the proof is a central part of the paper, include the version used or a citation to a standard textbook.
  5. [References] Reference 34 is listed as 'Lecun, Y., Bengio, Y., & Hinton, G.' — the standard capitalization is 'LeCun.' Several journal names and article formats are inconsistent; please unify the reference style.

Circularity Check

2 steps flagged

Theorem 1's fast–slow 'emergence' is put in by hand: h/c labels and the block-diagonal anisotropic metric prescribe the O(ε²) timescale, so the dual-process conclusion reduces to Assumption 1.

specific steps
  1. self definitional [Section 3.1–3.2 (Assumption 1, Eq. (2)) and Theorem 1(4)]
    "Components in h correspond to aspects of cognition that can be updated cheaply or rapidly (such as intuitive, habitual, or automatic adjustments), whereas those in c correspond to processes that are intrinsically costly or slow to change (such as beliefs, goals, or controlled processes). ... This distinction is not imposed by hand but emerges from the anisotropic metric introduced below."

    The h/c split is first labelled fast versus slow, and then Assumption 1 chooses Gε = diag(I_m, ε^{-2} I_k), which directly makes the c-equation ċ = −ε²∇cJ in Eq. (2). Theorem 1(4) then 'derives' that h relaxes on an O(1) scale while c changes on an O(ε²) scale, and claims this needs no dual-process assumptions. The timescale separation is an input, not a consequence: the theorem's advertised emergence is the assumption restated.

  2. fitted input called prediction [Results, Figure 2, Section 'Results']
    "The empirical curves follow the predicted asymptotic behavior: ⟨|˙h|⟩ ∼ O(1) and ⟨|˙c|⟩ ∼ O(ε²), directly confirming the effect of the anisotropic metric on the Riemannian gradient flow. Together, these results provide a direct numerical demonstration that the anisotropic metric indeed generates a fast–slow dynamical structure, validating Assumption 1."

    The 'prediction' is obtained by simulating Eq. (2), whose scalings are algebraic consequences of Gε. There is no independent data source; the simulation enforces the anisotropic metric, and the measured O(1)/O(ε²) speeds are exactly what the equation says. Calling this a verification of the assumption is confirmation by construction.

full rationale

The main advertised result — that dual-process cognition 'emerges' from Riemannian geometry without modular assumptions — is partially circular. The paper defines the h-coordinates as cheap/fast and c-coordinates as costly/slow before introducing the metric, and Assumption 1 then sets the block-diagonal metric to realize precisely that split. Theorem 1's fast relaxation and O(ε²) slow drift therefore follow from the assumed Gε by construction; the standard Fenichel/Tikhonov machinery adds persistence of the slow manifold and reduced dynamics, which is nontrivial, but the central 'no modular or dual-process assumptions' claim is not supported because the h/c partition and anisotropy are the dual-process assumption. The numerical section compounds this by 'verifying' the O(1)/O(ε²) speeds from simulations of the very equations that contain them. There is no self-citation chain or imported uniqueness theorem: the framework is self-contained as a mathematical exercise, and the monotonicity Proposition 1 is a standard identity. Separately, the proof of Theorem 1 has an internal inconsistency (Step 3 states O(ε) closeness; item (2) claims O(ε²)), which is a correctness gap, not a circularity. Because the central 'emergence' claim reduces to an assumption, the circularity score is 6 rather than a lower score.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The framework relies on the free choice of potential J and metric G, plus a hand-set anisotropy ε. The theorem is standard; the cognitive interpretation adds no new entities with independent falsifiable handles.

free parameters (3)
  • ε (anisotropy parameter) = 0.05, 0.1, 0.2, 0.3 in simulations
    Small parameter controlling the cost ratio in the metric (Assumption 1); chosen by hand for each simulation, not estimated from cognitive data.
  • Toy potential J(h,c)=1/2(h-c³)²+1/2c² = coefficients all 1
    Hand-specified potential used to make the slow manifold h=c³ explicit; arbitrary and not derived from any cognitive task.
  • Decision-task potential J(h,c,t) = not specified
    Hand-constructed to create two basins and a tipping-point switch; details of the landscape deformation are omitted from the manuscript.
axioms (4)
  • standard math Fenichel / Tikhonov singular perturbation theory: normally hyperbolic invariant manifolds persist and yield reduced slow dynamics.
    Invoked in the proof of Theorem 1, Section 4 Step 3, to assert existence of M_ε and the reduced dynamics.
  • domain assumption Cognitive state lies in R^n and evolves deterministically; J ∈ C², G positive-definite.
    Definitions 1–2, Section 2. This is the core modeling premise: cognition is a smooth potential-driven system.
  • domain assumption Assumption 2: unique fast minimizer h*(c), strong convexity (λ_min ≥ α > 0), smooth dependence on c.
    Section 3.3 (J1–J4). Required for exponential stability and normal hyperbolicity; not empirically verified.
  • ad hoc to paper Anisotropic block-diagonal metric G_ε = diag(I_m, ε^{-2} I_k).
    Assumption 1, Section 3.2. This metric choice directly imposes the timescale separation that the paper later 'derives'; it is not learned or justified by data.

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of A Geometric Theory of Cognition for Machine Intelligence." pith.science (2026). https://pith.science/paper/ACK4UXIR

@misc{pith2026251212225,
  author       = {Pith},
  title        = {Pith review of: A Geometric Theory of Cognition for Machine Intelligence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACK4UXIR}},
  note         = {Machine review of arXiv:2512.12225}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Developing artificial agents that unify representation, memory, adaptation, and prediction remains a fundamental challenge in artificial intelligence. Here we introduce a geometric framework in which cognitive computation emerges from Riemannian gradient flow on a learned latent manifold. The learned metric encodes representational constraints and computational preferences, while anisotropies in the geometry naturally generate multiple timescales of behaviour, yielding both rapid reactive responses and slower adaptive dynamics without explicit memory modules or recurrent mechanisms. We instantiate this framework through Riemannian representation and dynamics models and evaluate them in partially observable reinforcement-learning environments. Across observation masking, sensory blackouts, dynamics perturbations, and predictive latent-modelling tasks, the proposed approach consistently outperforms feedforward baselines, achieves robustness comparable to recurrent architectures, and produces highly predictable latent trajectories with low long-horizon rollout error. These results suggest that learned latent geometry can serve simultaneously as a substrate for representation, memory, adaptation, and prediction. More broadly, the framework provides a principled connection between dynamical systems, representation learning, and world-model-based intelligence.

Figures

Figures reproduced from arXiv: 2512.12225 by Laha Ale.

Figure 1
Figure 1. Figure 1: Geometric representation of the cognitive state and its dynamics. The cognitive state is modelled as a point on a differentiable manifold whose geometry captures relationships among perceptual, memory, motor, and reasoning variables. The Riemannian metric encodes representational constraints and computational costs, shaping the local curvature of cognitive space. A cognitive potential integrates predictive… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical verification of Assumption 1 (anisotropic metric). (A) Phase portrait of trajectories for multiple values of ε under the gradient flow h˙ = −∇hJ(h,c), ˙c = −ε 2∇cJ(h,c). Trajectories rapidly relax toward the slow manifold h = c 3 (black dashed) in the h-direction before drifting slowly in the c-direction. Decreasing ε sharpens this separation of timescales. (B) Scaling of the average speeds ⟨|h˙|… view at source ↗
Figure 3
Figure 3. Figure 3: Numerical illustration of Assumption 2 (Regularity and Stability) in a cognitive fast–slow system. (A) Trajectory of the full cognitive gradient flow under the anisotropic metric, shown together with the slow manifold h ∗ (c) = c 3 (black dashed), which represents the stable equilibrium of the fast, automatic cognitive coordinates for each deliberative state c. From the initial condition (black), the syste… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical validation of the reduced slow dynamics. (A) Comparison of the slow coordinate obtained from the full fast–slow system cε (t) and from the reduced dynamics cred(t) defined on the slow manifold h ∗ (c) = c 3 , shown here for ε = 0.2. After an initial transient during which the fast coordinate rapidly equilibrates, the slow variable of the full system evolves closely to the reduced trajectory, illu… view at source ↗
Figure 5
Figure 5. Figure 5: Geometric structure of the cognitive potential and validation of the reduced dynamics. (A) Energy landscape J(h,c) for the two-dimensional illustrative system, shown as a smooth surface over the fast coordinate h and slow coordinate c. The black dashed curve marks the slow manifold h ∗ (c) = c 3 , which corresponds to the unique minimizer of J with respect to h for each fixed c. This curve forms a stable v… view at source ↗
Figure 6
Figure 6. Figure 6: Cognitive gradient flow illustrating fast automatic responses and slow deliberative dynamics. (A) Early decision landscape J(h,c,t) showing two approximately symmetric basins in the deliberative coordinate c, each corresponding to a stable decision mode. (B) As external evidence gradually shifts, the energy landscape deforms and one basin deepens, biasing the system toward an alternative decision. (C) Traj… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 7 canonical work pages

  1. [1]

    & Thomson, M

    Gornet, J. & Thomson, M. Automated construction of cognitive maps with visual predictive coding.Nat. Mach. Intell.6, 820–833, DOI: 10.1038/s42256-024-00863-1 (2024)

    work page doi:10.1038/s42256-024-00863-1 2024

  2. [2]

    A., Patai, E

    Epstein, R. A., Patai, E. Z., Julian, J. B. & Spiers, H. J. The cognitive map in humans: spatial navigation and beyond.Nat. Neurosci.20, 1504–1513, DOI: 10.1038/nn.4656 (2017)

    doi:10.1038/nn.4656 2017

  3. [3]

    E., Liu, Y ., Vehar, N., Behrens, T

    Wimmer, G. E., Liu, Y ., Vehar, N., Behrens, T. E. J. & Dolan, R. J. Episodic memory retrieval success is associated with rapid replay of episode content.Nat. Neurosci.23, 1025–1033, DOI: 10.1038/s41593-020-0649-z (2020)

    work page doi:10.1038/s41593-020-0649-z 2020

  4. [4]

    & Zampieri, S

    Betteti, S., Baggio, G., Bullo, F. & Zampieri, S. Input-driven dynamics for robust memory retrieval in hopfield networks. Sci. Adv.11, eadu6991, DOI: 10.1126/sciadv.adu6991 (2025)

    work page doi:10.1126/sciadv.adu6991 2025

  5. [5]

    Mnih, V .et al.Human-level control through deep reinforcement learning.Nature518, 529–533, DOI: 10.1038/nature14236 (2015)

    doi:10.1038/nature14236 2015

  6. [6]

    & Lillicrap, T

    Hafner, D., Pasukonis, J., Ba, J. & Lillicrap, T. Mastering diverse control tasks through world models.Nature640, 647–653, DOI: 10.1038/s41586-025-08744-2 (2025)

    doi:10.1038/s41586-025-08744-2 2025

  7. [7]

    T., Mieczkowski, E., Fang, C., Tenenbaum, J

    Pramod, R. T., Mieczkowski, E., Fang, C., Tenenbaum, J. & Kanwisher, N. Decoding predicted future states from the brain’s ‘physics engine’.J. Vis.24, 1258, DOI: 10.1167/jov.24.10.1258 (2024)

    work page doi:10.1167/jov.24.10.1258 2024

  8. [8]

    M.et al.Building machines that learn and think with people.Nat

    Collins, K. M.et al.Building machines that learn and think with people.Nat. Hum. Behav.8, 1851–1863, DOI: 10.1038/s41562-024-01991-9 (2024)

    doi:10.1038/s41562-024-01991-9 2024

  9. [9]

    L., Tang, N., Xu, H

    Gao, T., Baker, C. L., Tang, N., Xu, H. & Tenenbaum, J. B. The Cognitive Architecture of Perceived Animacy: Intention, Attention, and Memory.Cogn. Sci.43, 1–31, DOI: 10.1111/cogs.12775 (2019)

    work page doi:10.1111/cogs.12775 2019

  10. [10]

    M., Ullman, T

    Lake, B. M., Ullman, T. D., Tenenbaum, J. B. & Gershman, S. J. Building machines that learn and think like people.Behav. Brain Sci.40, e253, DOI: 10.1017/S0140525X16001837 (2017)

    doi:10.1017/s0140525x16001837 2017

  11. [11]

    M., Salakhutdinov, R

    Lake, B. M., Salakhutdinov, R. & Tenenbaum, J. B. Human-level concept learning through probabilistic program induction. Science350, 1332–1338, DOI: 10.1126/science.aab3050 (2015). https://www.science.org/doi/pdf/10.1126/science.aab3050

    doi:10.1126/science.aab3050 2015

  12. [12]

    H., Vempala, S

    Papadimitriou, C. H., Vempala, S. S., Mitropolsky, D., Collins, M. & Maass, W. Brain computation by assemblies of neurons.Proc. Natl. Acad. Sci.117, 14464–14472, DOI: 10.1073/pnas.2001893117 (2020)

    doi:10.1073/pnas.2001893117 2020

  13. [13]

    Kar, K., Kubilius, J., Schmidt, K., Issa, E. B. & DiCarlo, J. J. Evidence that recurrent circuits are critical to the ventral stream’s execution of core object recognition behavior.Nat. Neurosci.22, 974–983, DOI: 10.1038/s41593-019-0392-5 (2019)

    doi:10.1038/s41593-019-0392-5 2019

  14. [14]

    Garcez, A. d. & Lamb, L. C. Neurosymbolic ai: the 3rd wave.Artif. Intell. Rev.56, 12387–12406, DOI: 10.1007/ s10462-023-10448-w (2023)

    2023

  15. [15]

    & Wang, J

    Wang, Z., Zhang, W., Liu, N. & Wang, J. Scalable rule-based representation learning for interpretable classification. In Proceedings of the 35th International Conference on Neural Information Processing Systems, NIPS ’21 (Curran Associates Inc., Red Hook, NY , USA, 2021)

    2021

  16. [16]

    17.Kahneman, D.Thinking, Fast and Slow(Farrar, Straus and Giroux, New York: New York, 2011)

    Dohare, S.et al.Loss of plasticity in deep continual learning.Nature632, 768–774, DOI: 10.1038/s41586-024-07711-7 (2024). 17.Kahneman, D.Thinking, Fast and Slow(Farrar, Straus and Giroux, New York: New York, 2011). 18.Riemann, B. On the Hypotheses Which Lie at the Bases of Geometry.Nature8, 14–17, DOI: 10.1038/008014a0 (1873)

    doi:10.1038/s41586-024-07711-7 2024

  17. [19]

    A mathematical framework of intelligence and consciousness based on riemannian geometry (2024)

    Lu, M. A mathematical framework of intelligence and consciousness based on riemannian geometry (2024). 2407.11024

    Pith/arXiv arXiv 2024

  18. [20]

    K.et al.A principal odor map unifies diverse tasks in olfactory perception.Science381, 999–1006, DOI: 10.1126/science.ade4401 (2023)

    Lee, B. K.et al.A principal odor map unifies diverse tasks in olfactory perception.Science381, 999–1006, DOI: 10.1126/science.ade4401 (2023)

    doi:10.1126/science.ade4401 2023

  19. [21]

    2409.06509

    Muttenthaler, L.et al.Aligning machine and human visual representations across abstraction levels.Nature647, 349–355, DOI: 10.1038/s41586-025-09631-6 (2025). 2409.06509. 13/14

    Pith/arXiv arXiv 2025

  20. [22]

    L., Chater, N

    Griffiths, T. L., Chater, N. & Tenenbaum, J. B.Bayesian Models of Cognition: Reverse Engineering the Mind(MIT Press, Boston, MA, 2024)

    2024

  21. [23]

    Learning riemannian metrics

    Lebanon, G. Learning riemannian metrics. InProceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence, UAI’03, 362–369 (Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2002)

    2002

  22. [24]

    & DURRLEMAN, S

    Gruffaz, S., Poulet, P.-E., Maheux, E., Jedynak, B. & DURRLEMAN, S. Learning Riemannian metric for disease progression modeling. In Ranzato, M., Beygelzimer, A., Dauphin, Y ., Liang, P. S. & Vaughan, J. W. (eds.)Advances in Neural Information Processing Systems, vol. 34, 23780–23792 (Curran Associates, Inc., 2021). 25.Robbins, H. & Monro, S. A stochastic ...

    Pith/arXiv arXiv 2021

  23. [27]

    InProceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, 3988–3996 (Curran Associates Inc., Red Hook, NY , USA, 2016)

    Andrychowicz, M.et al.Learning to learn by gradient descent by gradient descent. InProceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, 3988–3996 (Curran Associates Inc., Red Hook, NY , USA, 2016)

    2016

  24. [28]

    & Rao, R

    Khalvati, K., Kiani, R. & Rao, R. P. N. Bayesian inference with incomplete knowledge explains perceptual confidence and its deviations from accuracy.Nat. Commun.12, 5704, DOI: 10.1038/s41467-021-25419-4 (2021)

    work page doi:10.1038/s41467-021-25419-4 2021

  25. [29]

    Rao, R. P. N. & Ballard, D. H. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects.Nat. Neurosci.2, 79–87, DOI: 10.1038/4580 (1999)

    doi:10.1038/4580 1999

  26. [30]

    Sutton, R. S. & Barto, A. G.Reinforcement Learning: An Introduction(A Bradford Book, Cambridge, MA, USA, 2018)

    2018

  27. [31]

    & Sun, J

    He, K., Zhang, X., Ren, S. & Sun, J. Deep residual learning for image recognition.Proc. IEEE Comput. Soc. Conf. on Comput. Vis. Pattern Recognit.2016-Decem, 770–778, DOI: 10.1109/CVPR.2016.90 (2016). 1512.03385

    Pith/arXiv arXiv 2016

  28. [32]

    The free-energy principle: a unified brain theory?Nat

    Friston, K. The free-energy principle: a unified brain theory?Nat. Rev. Neurosci.11, 127–138, DOI: 10.1038/nrn2787 (2010)

    doi:10.1038/nrn2787 2010

  29. [33]

    & Rand, D

    Bear, A. & Rand, D. G. Intuition, deliberation, and the evolution of cooperation.Proc. Natl. Acad. Sci.113, 936–941, DOI: 10.1073/pnas.1517780113 (2016). 34.Lecun, Y ., Bengio, Y . & Hinton, G. Deep learning.Nature521, 436–444, DOI: 10.1038/nature14539 (2015)

    work page doi:10.1073/pnas.1517780113 2016

  30. [35]

    Silver, D.et al.Mastering the game of Go with deep neural networks and tree search.Nature529, DOI: 10.1038/nature16961 (2016)

    doi:10.1038/nature16961 2016

  31. [36]

    Silver, D.et al.Mastering the game of Go without human knowledge.Nature550, 354–359, DOI: 10.1038/nature24270 (2017)

    doi:10.1038/nature24270 2017

  32. [37]

    J., Freeman, B

    Wu, J., Yildirim, I., Lim, J. J., Freeman, B. & Tenenbaum, J. Galileo: Perceiving physical object properties by integrating a physics engine with deep learning. In Cortes, C., Lawrence, N., Lee, D., Sugiyama, M. & Garnett, R. (eds.)Advances in Neural Information Processing Systems, vol. 28 (Curran Associates, Inc., 2015). 38.LeCun, Y . A path towards auto...

    2015

  33. [39]

    2506.09985

    Assran, M.et al.V-jepa 2: Self-supervised video models enable understanding, prediction and planning (2025). 2506.09985

    Pith/arXiv arXiv 2025

  34. [40]

    & Bengio, Y

    Bengio, E., Jain, M., Korablyov, M., Precup, D. & Bengio, Y . Flow network based generative models for non-iterative diverse candidate generation. In Ranzato, M., Beygelzimer, A., Dauphin, Y ., Liang, P. & Vaughan, J. W. (eds.)Advances in Neural Information Processing Systems, vol. 34, 27381–27394 (Curran Associates, Inc., 2021)

    2021

  35. [41]

    R., Son, S

    Silva, A., Killian, T., Jimenez, I. R., Son, S. H. & Gombolay, M. Optimization Methods for Interpretable Differentiable Decision Trees in Reinforcement Learning.Proc. Mach. Learn. Res.108, 1855–1865 (2020). 1903.09338. 14/14

    Pith/arXiv arXiv 2020

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.