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A Neural Network Training Method Based on Distributed PID Control

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a symmetric differential-equation neural network can be trained without backpropagation, using forward and reverse signal integrals plus distributed PID control, with faster training and improved accuracy on MNIST.

desk verdict A clear, honest description of an unvalidated training heuristic for the author's symmetric differential-equation network; the PID framing is new, but the central accuracy claim is not supported by the evidence presented. read the letter →

arxiv 2411.14468 v1 pith:HB63FYP6 submitted 2024-11-18 cs.LG cs.AIcs.NE

Jiang Kun This is my paper · ORCID
classification cs.LGcs.AIcs.NE MSC 68T07
keywords WuxingneuralnetworksymmetricdifferentialequationsdistributedPIDcontrolbackpropagation-freetrainingreversiblesystemssignalpropagationMNIST
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

This paper aims to show that the Wuxing neural network — a multilayer architecture built from symmetric differential equations rather than artificial neurons — can be trained without backpropagation. The training rule replaces chain-rule gradients with differential-equation signal propagation: each neuron integrates the forward signal and the corresponding reverse error signal over time, and updates a parameter according to the product of those integrals. Because that rule alone leaves the network's three parameter sets redundantly coupled, the paper adds a distributed Proportional-Integral-Derivative strategy, assigning integral, differential, and proportional roles to K1, K2, and K3 respectively. On MNIST, the combined rule is reported to train faster and reach higher accuracy than adjusting a single parameter set. A sympathetic reading is that the paper establishes a proof-of-concept that a reversible, locally controlled dynamical system can learn a digit-classification task from raw pixels.

What carries the argument

The load-bearing object is the Wuxing neural network, a system of five fully symmetric differential equations (Eq. 2.2) whose symmetry makes signal propagation reversible: the inverse system (Eq. 3.1) preserves the causal links between elements, which is what lets a neuron pair forward and backward signals. Training uses the fixed point B0 = (K1 − K2)/K3 to define the zero state, and the correlation variable G1 (Eq. 3.6) — the product of the time integrals of the forward output D(t) and the reverse output D̂(t) — as the raw parameter-adjustment signal. G1 is bounded by an arctangent function (Eq. 3.7) and applied multiplicatively and exponentially (Eq. 3.8/4.10). The distributed PID control maps the three parameter sets to three control modes: integral accumulation for K1, differential input-node gating for K2, and proportional fixed-point adjustment for K3.

What would settle it

Take the same update rule and replace the forward/reverse integrals with randomized values of the same magnitude; if MNIST accuracy still rises, the correlation signal is not the cause of learning. Alternatively, compute the true gradient of the classification loss with respect to K3 on a small subnetwork and compare its sign with the sign of the updates produced by exp(−atan(G1_k3 · kt)/kt); frequent sign disagreement would show the proxy is not tracking descent.

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Extended reading notes

Core claim

The paper's central claim is that the update K3_new = K3_old · exp(−atan(G1_k3 · kt)/kt), where G1_k3 = ∫₀ᵀ D(t)dt · ∫₀ᵀ D̂(t)dt is the product of forward and reverse signal integrals, gives a working parameter-adjustment law for the symmetric differential-equation network. The same exponential-atan structure is used for K1 with an integral-mode accumulation and for K2 with a differential-mode gating by input nodes. The author argues that this preserves the point-to-point causality of backpropagation — reversibility replaces the chain rule — and that the distributed PID framing resolves the redundancy problem that arises when K1 and K3 are adjusted at the same time. The empirical claim is that on MNIST the PID combination improves both speed and accuracy relative to the earlier single-parameter training.

Load-bearing premise

The method works only if the product of the time-integrated forward signal and the time-integrated reverse error signal tells a neuron the right direction and size for updating a parameter; the paper assumes this correlation acts like a useful gradient signal without deriving it from the equations.

Editorial extensions

If this is right

  • If the central claim is right, a backpropagation-free rule based only on local forward/reverse signal products can train a continuous-time differential-equation network on a real image task.
  • The reversibility of the network is what does the work that the chain rule does in backpropagation, so any architecture that is constructed to be symmetric and reversible can in principle use the same training scheme.
  • The PID decomposition provides a concrete answer to the parameter-redundancy problem: the three parameter sets are not interchangeable, and assigning them integral, differential, and proportional roles is what stabilises combined training.
  • On MNIST, the paper's reported accuracy curves show that combined integral-plus-proportional control reaches higher accuracy than adjusting K3 alone, giving a baseline for future biologically inspired training methods.
  • The paper's observation that the differential mode is unstable matches known PID behaviour, so standard PID tuning ideas become available for neural-network training.

Reading between the lines

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

  • Beyond the paper: if the G1 correlation really behaves like a signed gradient proxy, the same update law could be applied to any dynamical system whose forward and reverse propagators are inverses, not just five-element Wuxing systems.
  • Beyond the paper: the time-integral form of G1 suggests that training depends on the settling window T; a testable extension would be to anneal T during training and check whether accuracy and stability improve.
  • Beyond the paper: the assignment of one control mode per parameter set is a design choice, and alternative assignments (for example, swapping the roles of K1 and K3) could be probed on the same MNIST setup to map the control landscape.
  • Beyond the paper: because the method is local by construction, it could be combined with neuromorphic or continuous-time hardware where global gradient signals are unavailable, but the paper does not demonstrate this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper proposes a training method for the Wuxing neural network, a network architecture based on symmetric differential equations introduced in the author's prior preprint. The proposed method replaces backpropagation with forward and reverse signal propagation through the network, and then applies a distributed Proportional-Integral-Derivative (PID) control scheme to update three parameter sets K1, K2, K3. The update rules are based on a correlation variable G1 defined from time integrals of forward and reverse deviation signals, transformed by an arctangent function and applied as multiplicative exponential factors. The author reports experiments on MNIST, claiming faster training and improved accuracy. The paper also motivates the method as biologically plausible because each neuron adjusts parameters using only local forward and reverse signals.

Significance. If the central claim were substantiated, the paper would offer a backpropagation-free, locally controlled training rule for a continuous-time symmetric network, which could be of interest to researchers in biologically plausible learning and control-based optimization. The manuscript does make its update equations explicit and acknowledges some limitations, such as the instability of the differential control component. However, the central claim is not currently supported: the correlation-based update lacks a theoretical foundation, and the experimental validation is minimal, with no baseline, no error bars, and no clear test set performance. The paper also depends on unverified properties of the Wuxing network from a prior preprint, and the full distributed PID method described in the abstract is not actually tested.

major comments (4)
  1. [Section 3.2 and Section 4.3 (Eqs. 3.6-3.8 and 4.8-4.10)] The update rule for K3 is K3_new = K3_old * exp(-G2_k3), where G2_k3 = atan(G1_k3 * kt)/kt and G1_k3 is the product of the time integrals of the forward and reverse deviation signals. This rule is asserted without any derivation from an objective function; there is no proof that the sign or magnitude of G1 is correlated with a decrease in classification error, and no convergence or stability analysis is provided. Because every parameter update in the paper uses this same correlation heuristic, this missing justification is load-bearing for the entire training method.
  2. [Section 4.1 and Figures 3 and 5] The experiments compare only variants of the same heuristic (adjusting K3 alone vs. K1 and K3, or different combinations of the PID strategies). There is no baseline such as standard backpropagation, no random-update or ablated correlation control, no error bars, no indication of the number of independent runs, and no clear statement of whether reported accuracy is on training or test data. The abstract's claim of faster training and improved accuracy is therefore not evaluable from the reported experiments.
  3. [Section 2.2 and Section 3.1] The fixed point B0 = (K1 - K2)/K3 in Eq. 2.3 is derived under the assumption that the parameters within each K set are equal. Training updates make these parameters unequal, and the statement in Section 2.2 that the adjustment method remains effective in that regime is asserted without proof. Additionally, the reversibility of the system, which is the foundation of the proposed training method, is inherited from reference [1] and is not independently derived or verified here. Both properties are essential to the method's claimed validity.
  4. [Section 4.3, Figure 5] The text states that in actual operation the differential control strategy was not enabled simultaneously with the other strategies 'mainly because the differential method has great instability.' Since the abstract and introduction claim a distributed PID control approach, the experiments do not actually validate the full PID method; they validate only the integral and proportional components (or combinations) under an unspecified enabling schedule. This gap directly affects the central claim.
minor comments (5)
  1. [Equation (3.2)] The variable 'Leb' is used without definition; it should be stated explicitly whether this is a Lebesgue integral notation or an abbreviation for a cumulative signal measure.
  2. [Section 4.1] The text says 'The method of adjusting K1 can refer to formula 3.5,' but Eq. 3.5 defines D_hat(t), not an update rule; the cross-reference should be corrected or expanded.
  3. [Figures 3 and 5] The y-axis label 'Correct percentage' is inconsistent with the text's use of 'accuracy rate'; the figures should state clearly whether these are training accuracies, test accuracies, or validation accuracies.
  4. [Equation (2.2)] The notation in Eq. 2.2 is garbled (e.g., '0 0 1 00 002' and the superscripted offset 'E^{-1}'), making it hard to parse the intended differential equations; a rewrite with explicit indices would improve readability.
  5. [Section 5] The statement that 'we have identified four other effective training strategies' is unverifiable because no details, results, or references are provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the training rule is an unproven heuristic, but MNIST accuracy is an external metric and no claim reduces to its own inputs.

full rationale

The paper's central update rule (Eq. 3.6–3.8, restated as Eq. 4.8–4.10) is presented as a proposed training method, not as a derivation from a loss function. G1 is defined as the dot product of forward and reverse signal integrals, and the parameter update K3_new = K3_old * exp(-G2) is posited and then evaluated empirically on MNIST. The accuracy metric (Eq. 3.2 and the target comparison in Eqs. 3.3–3.4) is external to the update rule, so the reported improvement is not the update rule by construction. The reversibility premise is asserted via the symmetric equations given in Section 2 rather than imported solely from the author's prior preprint, and the paper explicitly states that the PID design 'is not the only one,' so no uniqueness theorem is invoked. The PID labels are descriptive of arithmetic operations (summation for integral, input-node masking for differential, direct product for proportional). The admitted limitations—no proof that G1 is a valid descent signal, no backpropagation baseline, and the statement 'We have no plans to discuss these issues in this article'—are evidence gaps, not circular reductions. No fitted parameter is relabeled as a prediction, and no load-bearing claim collapses into a self-citation chain. Therefore, no significant circularity is found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central training result rests on hand-chosen scalars (kt and targets), on the inherited symmetry and reversibility of the Wuxing system from the author's own prior paper, and on an unproven equivalence between a signal correlation and the correct parameter update direction. No new physical entities are introduced.

free parameters (4)
  • kt (adjustment parameter) = not reported
    Scales the arctangent in Eqs. 3.7, 4.3, 4.6, and 4.9; no value or tuning procedure is given, yet it controls update magnitude.
  • target1 and target2 = not reported
    Predefined bounds for the Leb output components in Eqs. 3.3 and 3.4; they define the error signal and are hand-chosen without justification.
  • initial K1, K2, K3 values = K1=1, K2=0.5, K3=0.5
    Hand-chosen starting parameter values in Section 4.1; training dynamics depend on them.
  • network architecture = 784, 839, 283, 96, 32, 10
    The six-layer layout is chosen without search or rationale, and all results depend on it.
assumptions (4)
  • domain assumption The Wuxing differential equation system is completely symmetric and reversible, allowing backward signal propagation to trace causal effects.
    Stated in Sections 1 and 3.1, based on the author's prior paper; no proof is given in this manuscript.
  • ad hoc to paper The fixed point B0 = (K1 - K2)/K3, derived under the assumption that parameters within each K set are equal, remains an effective adjustment target when parameters are unequal.
    Section 2.2 asserts this without derivation; the update rules rely on moving the system relative to B0.
  • ad hoc to paper The product or sum of forward and reverse signal integrals, G1, is a meaningful measure of how a parameter should change.
    Core of the training update in Eq. 3.6; no derivation from an objective function.
  • domain assumption Standard PID control concepts transfer to parameter updates inside this closed system.
    Section 4.3 assigns integral, differential, and proportional roles to K1, K2, and K3 by analogy, citing PID literature but not proving equivalence.

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Cite this review

Pith. "Pith review of A Neural Network Training Method Based on Distributed PID Control." pith.science (2026). https://pith.science/paper/HB63FYP6

@misc{pith2026241114468,
  author       = {Pith},
  title        = {Pith review of: A Neural Network Training Method Based on Distributed PID Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HB63FYP6}},
  note         = {Machine review of arXiv:2411.14468}
}
read the original abstract

In the previous article, we introduced a neural network framework based on symmetric differential equations. This novel framework exhibits complete symmetry, endowing it with perfect mathematical properties. While we have examined some of the system's mathematical characteristics, a detailed discussion of the network training methodology has not yet been presented. Drawing on the principles of the traditional backpropagation algorithm, this study proposes an alternative training approach that utilizes differential equation signal propagation instead of chain rule derivation. This approach not only preserves the effectiveness of training but also offers enhanced biological interpretability. The foundation of this methodology lies in the system's reversibility, which stems from its inherent symmetry,a key aspect of our research. However, this method alone is insufficient for effective neural network training. To address this, we further introduce a distributed Proportional-Integral-Derivative (PID) control approach, emphasizing its implementation within a closed system. By incorporating this method, we achieved both faster training speeds and improved accuracy. This approach not only offers novel insights into neural network training but also extends the scope of research into control methodologies. To validate its effectiveness, we apply this method to the MNIST dataset, demonstrating its practical utility.

Figures

Figures reproduced from arXiv: 2411.14468 by the authors.

Figure 1
Figure 1. a illustrates the logical structure underpinning the differential equations, while Figure 1b [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 5
Figure 5. a Integral strategy, differential strategy and proportional strategy are used to adjust K1, K2 and K3 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Neural Network Training Method Based on Neuron Connection Coefficient Adjustments

    cs.NE 2025-01 reject novelty 4.0 of 10

    A backward signal propagation rule updates connection coefficients in the Wuxing neural network, but the MNIST validation lacks quantitative results and reproducibility.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages · cited by 1 Pith paper

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