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REVIEW 3 major objections 2 minor 17 references

Von Neumann Networks

T0 review · 3 major / 2 minor · reviewed 2026-05-08 · grok-4.3

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Pith's one-line read Von Neumann Networks let artificial neurons learn specialized roles on a cellular array, producing self-engineered designs that depend only on input and output locations.

desk verdict The paper sketches a cellular self-designing neural net inspired by von Neumann but the emergence claim and universality proof need the missing equations to hold up. read the letter →

arxiv 2605.05780 v1 submitted 2026-05-07 cs.AI cs.CVcs.LG

Shekhar S. Chandra This is my paper
classification cs.AIcs.CVcs.LG
keywords vonneumannnetworkscellularmachinesneuraloperatorsgreen'sfunctionsdiffusionprocessesself-engineeredarchitecturesparameterefficiencycomputationaluniversality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends John von Neumann's model of the brain as an array of cells with finite states governed by diffusion into a modern neural network setting. It defines the Von Neumann neuron as a cell whose role is learned through optimization rather than fixed in advance. The resulting networks have architectures that emerge automatically from the placement of inputs and outputs on this array. These networks are shown to extend neural operators by learning Green's functions through convolutions on a cellular topology with a diffusion signature. Initial tests indicate that simple Von Neumann networks beat equivalent standard networks on basic tasks while using fewer parameters and can handle new task types.

What carries the argument

The Von Neumann neuron: an artificial neuron on a cellular array whose specialized computational role is learned by optimization, with interactions governed by a diffusion process that enables convolution-based learning of Green's functions.

What would settle it

An experiment in which VNN-based MLPs require hand-specified cell roles or explicit architectural choices beyond input/output positions to match their claimed performance, or in which they fail to outperform standard MLPs at matched parameter counts on the reported tasks.

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Extended reading notes

Core claim

Von Neumann Networks are built by allowing each cell in a fixed array to learn one of a finite set of specialized roles through standard deep learning training. The array follows a diffusion process that governs how cell states interact, and the overall network architecture is determined solely by which cells serve as inputs and which as outputs. Mathematically, the construction extends neural operators by learning Green's functions via convolutions defined on the cellular topology that carries the diffusion signature. The networks belong to the larger class of Cellular Machines, which the paper proves are computationally universal.

Load-bearing premise

Specialized roles for cells can be learned effectively through ordinary deep learning optimization on a cellular array with a diffusion signature, and the resulting network structure truly emerges from input and output locations alone without further hand-engineering.

Editorial extensions

If this is right

  • VNN-based multi-layered perceptrons outperform equivalent deep learning networks on basic tasks.
  • The same networks use fewer parameters than their standard counterparts while achieving the gains.
  • VNNs can learn task types that standard networks of similar size do not handle as readily.
  • VNNs form part of Cellular Machines, a class proven to be computationally universal.
  • The framework permits an extension of the classical Von Neumann computer architecture into cellular arrays.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the diffusion signature holds at larger scales, the approach could reduce reliance on manually designed layer types in deep models.
  • The universality result implies VNNs could in principle simulate any computable function, opening routes to more adaptive AI systems.
  • Hardware implementations built directly on cellular arrays might realize the reported efficiency gains in physical form.
  • The self-engineering property suggests testing whether similar cellular rules can discover useful structures for vision or sequential data without human layer choices.
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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

3 major / 2 minor

Summary. The paper proposes Von Neumann Networks (VNNs) as neural architectures built from 'Von Neumann neurons' on a cellular array inspired by von Neumann's work, where each cell learns a specialized role via a diffusion process. It claims VNNs extend neural operators by learning Green's functions through convolutions on a cellular topology with diffusion signature, form part of computationally universal 'Cellular Machines,' yield self-engineered designs whose architecture depends only on input/output locations, and outperform equivalent MLPs in initial experiments while using fewer parameters and enabling new task types.

Significance. If the universality proof and emergence claim hold with reproducible code and parameter-free derivations, this would offer a novel bridge between cellular automata theory and operator learning, potentially enabling architectures that adapt topology from I/O placement alone and suggesting hardware extensions of von Neumann architectures to cellular arrays. The parameter-efficiency results, if verified, could impact efficient model design.

major comments (3)
  1. [Abstract, §3] Abstract and §3 (mathematical framework): the assertion that VNN architecture 'is only dependent on the structure and locations of its inputs and outputs on this cellular array' is load-bearing for the self-engineering claim, yet the description of the cellular topology, neighborhood structure, and diffusion kernel appear to require independent specification of grid dimensions and connectivity priors equivalent to those in CNNs or graph networks; this risks reducing the emergence to a reparameterization whose performance gains cannot be attributed solely to the von Neumann model.
  2. [§4] §4 (universality proof for Cellular Machines): the proof that VNNs are part of computationally universal Cellular Machines must be checked for circularity, as the abstract's definition of the framework as 'extension of neural operators' and 'learning of Green's functions' may depend on paper-specific axioms for cell roles and diffusion signature rather than reducing to a standard universal model like Turing machines.
  3. [Experimental section] Experimental section (initial experiments): the claim that VNN-based MLPs outperform equivalent deep learning variants while being more parameter efficient requires details on baseline architectures, hyperparameter matching, and whether post-hoc choices in cellular array setup affect results; without these, the outperformance cannot be isolated from the diffusion convolution design.
minor comments (2)
  1. [Abstract] The abstract introduces 'Von Neumann neuron' and 'Cellular Machines' without prior definitions or citations to von Neumann's original cellular automaton work; add explicit references and a brief recap of the historical model.
  2. [§3] Notation for Green's functions and diffusion convolutions should be clarified with an equation or diagram early in the mathematical framework to avoid ambiguity with standard neural operator formulations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. We address each major comment point by point below, providing clarifications and noting where revisions will be incorporated to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (mathematical framework): the assertion that VNN architecture 'is only dependent on the structure and locations of its inputs and outputs on this cellular array' is load-bearing for the self-engineering claim, yet the description of the cellular topology, neighborhood structure, and diffusion kernel appear to require independent specification of grid dimensions and connectivity priors equivalent to those in CNNs or graph networks; this risks reducing the emergence to a reparameterization whose performance gains cannot be attributed solely to the von Neumann model.

    Authors: We agree that the dimensions of the cellular array and its basic neighborhood structure must be specified as an initial setup, analogous to choosing the input dimensionality or number of layers in a conventional neural network. However, the self-engineering claim holds because, once this uniform topology is fixed, the specialized roles are learned end-to-end via the diffusion process, and the resulting effective architecture (which cells activate and how they interact) depends solely on the positions of the inputs and outputs. No additional hand-crafted connectivity priors are imposed beyond the regular cellular grid. We will revise the abstract and §3 to explicitly separate the initial grid configuration from the emergent, input/output-dependent properties and to contrast this with the fixed kernels in CNNs. revision: partial

  2. Referee: [§4] §4 (universality proof for Cellular Machines): the proof that VNNs are part of computationally universal Cellular Machines must be checked for circularity, as the abstract's definition of the framework as 'extension of neural operators' and 'learning of Green's functions' may depend on paper-specific axioms for cell roles and diffusion signature rather than reducing to a standard universal model like Turing machines.

    Authors: The universality result in §4 is established independently by showing that Cellular Machines—defined through finite learnable cell roles and diffusion-based state propagation on a cellular array—can simulate a universal Turing machine via suitable role configurations, directly extending von Neumann's original cellular automaton construction. The neural operator and Green's function interpretations are downstream applications in the differentiable setting and are not used in the universality argument. To eliminate any perception of circularity, we will expand §4 in the revision with an explicit, step-by-step reduction to a standard Turing machine, grounding all steps in classical cellular automata theory without introducing non-standard axioms. revision: yes

  3. Referee: [Experimental section] Experimental section (initial experiments): the claim that VNN-based MLPs outperform equivalent deep learning variants while being more parameter efficient requires details on baseline architectures, hyperparameter matching, and whether post-hoc choices in cellular array setup affect results; without these, the outperformance cannot be isolated from the diffusion convolution design.

    Authors: We acknowledge that the current experimental section is preliminary and lacks sufficient detail on baselines and controls. In the revised manuscript we will fully specify the baseline MLP architectures (layer widths, depths, activations), describe the hyperparameter search and matching procedure, and add ablation studies that vary cellular array size and diffusion parameters. These additions will allow the performance and parameter-efficiency gains to be isolated from setup choices and will clarify the contribution of the learned roles and diffusion mechanism. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in derivation chain

full rationale

The paper constructs a mathematical framework explicitly linking VNNs to extensions of neural operators and Green's function learning on a cellular array with diffusion signature, while asserting a self-engineered architecture dependent on I/O locations and proving universality for the introduced Cellular Machines system. No equations, fitted parameters renamed as predictions, or self-citations appear in the provided text that reduce any central claim to its own inputs by construction. The universality statement is presented as a proof within the work rather than imported from prior self-work, and experiments are described as initial validation rather than tautological outputs. The derivation remains self-contained with independent content in the framework construction, universality proof, and empirical results.

Assumptions & free parameters 0 free parameters · 2 assumptions · 2 invented entities

The central claims rest on the existence of a learnable diffusion process on a cellular array that produces specialized roles, the equivalence to neural operators via Green's functions, and the definition of Cellular Machines as a universal class. No explicit free parameters are named in the abstract, but the learning of roles implies optimization parameters. The universality proof likely relies on standard computability axioms plus domain assumptions about the cellular topology.

assumptions (2)
  • domain assumption Cellular arrays with finite states can model diffusion processes that support universal computation when roles are assigned appropriately.
    Invoked when claiming VNNs are part of computationally universal Cellular Machines.
  • domain assumption Neural operators can be extended to cellular topologies via convolutions that learn Green's functions.
    Stated as the mathematical foundation for VNNs.
invented entities (2)
  • Von Neumann neuron
    purpose: A neuron with specialized, learnable roles on a cellular array.
    New postulated neuron type whose behavior is defined by the paper's framework.
  • Cellular Machines
    purpose: A general computational system containing VNNs that is claimed to be Turing-universal.
    New class introduced to contain the proposed networks and support the universality proof.

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Cite this review

Pith. "Pith review of Von Neumann Networks." pith.science (2026). https://pith.science/paper/2605.05780

@misc{pith2026260505780,
  author       = {Pith},
  title        = {Pith review of: Von Neumann Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2605.05780}},
  note         = {Machine review of arXiv:2605.05780}
}
read the original abstract

In the mid-twentieth century, mathematician and polymath John von Neumann created a computational system on an array of cells as a simple model of the human brain, where each cell had one of a finite set of roles or states that he predicted would be modelled by a diffusion process. In this work, we show that such a system, when developed in a modern deep learning setting, enables the construction of an artificial neuron having specialized roles that can be learnt. We refer to this neuron as the Von Neumann neuron, and the resulting neural network from such neurons result in a self-engineered design whose architecture is only dependent on the structure and locations of its inputs and outputs on this cellular array. The mathematical framework for these Von Neumann Networks (VNNs) is also constructed and shows that they are based on the extension of neural operators and the learning of Green's functions with convolutions on a cellular topology having a diffusion signature. We also prove that these VNNs are part of a more general computational system called Cellular Machines that are computationally universal. Initial experiments show that VNN based multi-layered perceptrons outperform their equivalent deep learning variant on basic tasks, while being more parameter efficient and are capable of learning new types of tasks. This includes the ability to solve for and construct an extension of the Von Neumann (hardware) architecture common to all modern computers to cells and suggests new opportunities that could be explored.

Figures

Figures reproduced from arXiv: 2605.05780 by the authors.

Figure 1
Figure 1. Inspired by the works of John von Neumann, we show that with a suitable state-based view at source ↗
Figure 2
Figure 2. The artificial neuron (a) represents the typical deep learning definition, where inputs view at source ↗
Figure 3
Figure 3. The resulting VNNs final states (thus the architecture) for CLP (left) and two different ALU models. The inputs/outputs and controls in the case of the ALU models are shown in the top left insets with the bottom right showing the final weights of the models view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The resulting VNN tape based model (purely 1D Chua field C with 1D Green’s functions and 1D convolutions) for XOR arithmetic of two 8-bit numbers. The 16 “instructions” learnt as 1D Green’s functions for the model are shown on the right. one needs is a tape (1D) struct…
Figure 5
Figure 5. Figure 5: (a) represents the typical deep learning
Figure 6
Figure 6. Figure 6: The resulting Green’s functions learned for the box ALU model from Figure 3 (middle).
Figure 7
Figure 7. Figure 7: The resulting Green’s functions learned for the tape-like
Figure 8
Figure 8. Figure 8: The resulting Green’s functions learned for the CLP model from Figure 3 (left).
Figure 9
Figure 9. Figure 9: An illustration of a Turing machine with its tape and read/write head. In theory, the tape

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Arthurs and J

    Scikit-learn: Machine learning in Python. Journal of Machine Learning Research 12, 2825–2830. Petersen, F., Borgelt, C., Kuehne, H., Deussen, O., 2022. Deep differentiable logic gate networks, in: Proceedings of the 36th International Conference on Neural Information Processing Systems, Curran Associates Inc., Red Hook, NY , USA. pp. 2006–2018. Pérez, J.,...

    work page doi:10.1002/j.1538- 2022

  2. [2]

    S is the set of all possible states,

    work page

  3. [3]

    Ai is the input alphabet without the blank symbol,

    work page

  4. [4]

    AT is the tape alphabet which includes the blank symbol,

    work page

  5. [5]

    δ : S ˆ AT Ñ S ˆ AT ˆ tL, Ru is the transition function,

    work page

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    s0 P S is the initial state,

    work page

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    sa P S is the accept state,

    work page

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    The Turing machine would function as the following:

    sr P S is the reject state with sr ‰ sa. The Turing machine would function as the following:

    work page

Show all 17 references
  1. [9]

    We assume that the tape is infinite in both directions

  2. [10]

    The rest of the tape is considered empty or blank

    Begin with the input placed in the middle of the tape with the head in the left most cell of the input. The rest of the tape is considered empty or blank

  3. [11]

    The input alphabet Ai does not have the blank symbol, so it is safe to assume that the end of the input is marked with a blank symbol

  4. [12]

    The computation begins by using the tape alphabet AT and following the transition function until either the accept or reject state is reached

  5. [13]

    Otherwise, the machine continues forever without halting

    When either of sa or sr is reached, the machine halts. Otherwise, the machine continues forever without halting. D.3 Equivalences Firstly, if we add higher dimensional tape, such as two dimensional ( 2D) or three dimensional (3D) tape, to increase the versatility of our memory...

    1970

  6. [14]

    Underpopulation: A live cell that has < 2 live neighbouring cells will die

  7. [15]

    Survival: A live cell that has 2-3 live neighbouring cells will remain alive

  8. [16]

    Overpopulation: A live cell with more than 3 live neighbours will die

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    The game begins with an initial state of the universe with a pattern of live cells

    Reproduction: A dead cell with exactly 3 live neighbours will become alive. The game begins with an initial state of the universe with a pattern of live cells. The universe is evolved by applying the above rules to each cell of the universe to determine the next iteration of t...

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