{"id":"324d6f43-8d22-4aa7-928c-26fedda89048","arxiv_id":"2411.14468","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A differential-equation-based training rule with distributed PID control is proposed for a symmetric Wuxing neural network and tested on MNIST.","lead":"This paper proposes a training method for a neural network built from symmetric differential equations, replacing backpropagation's chain rule with differential equation signal propagation and adding a distributed PID control scheme. The author tests it on MNIST and reports faster training and improved accuracy, but the evidence is preliminary and lacks comparison to standard baselines.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The update rule in Eq 4.10 rests on an unvalidated correlation proxy; no ablation isolates whether G1 actually drives the reported improvement.","rationale":"The reader's weakest_assumption identifies the G1 correlation as load-bearing, and my analysis agrees. Even granting reversibility, the update law is where the actual training behavior is determined, and it is the least justified component: there is no derivation, no convergence argument, and no ablation isolating the correlation. The proposed random-sign ablation would directly settle whether Eq 4.10 works for the stated reason or merely through an unprincipled parameter drift. Since the central empirical claim is unsupported without such a test, the reader's REJECT verdict stands unchanged.","tokens_in":8963,"tokens_out":6331,"duration_ms":72245,"concrete_test":"Re-run the same network and training loop but replace G2 in Eq 4.10 with a random sign ±G2 while preserving magnitude, and separately with a shuffled assignment of G2 values across K3 parameters, keeping all other settings identical. If either variant matches the original accuracy, then the specific G1–parameter correlation is not the source of improvement; additionally, compare the sign of -G2 with a finite-difference gradient of the classification loss with respect to K3 over one epoch, and if sign agreement is at chance, Eq 4.10 is not a signal-directed descent rule.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the update law K3_new = K3_old * exp(-G2_k3) in Eq 4.10 improves classification, which in turn requires that G1 in Eq 4.8/3.6 is a valid proxy for the desired parameter change. This is never derived. Section 3.1 offers only an analogy (dX=aY) and asserts a 'strong correlation' between forward and reverse signals, but no theorem or experiment connects the sign or magnitude of G1 to a decrease in any loss. Worse, G1 is the product of time-integrals of signed deviations D(t) and D_hat(t); its sign depends on the DC offsets of these integrals, and no normalization or error-dependent sign convention (linking to Eq 3.3/3.4) is established. The experiments in Figures 3 and 5 compare variants of the same heuristic (adjusting K3 only vs K1 and K3, or different named PID strategies), but all use the same G1 correlation, so they cannot detect whether the correlation itself is doing the work. A control that breaks the G1 signal is missing. If the correlation is wrong, the reported accuracy gains could arise from the exponential schedule, parameter drift, or initialization, and the method would have no reason to generalize.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9263,"tokens_out":3544,"duration_ms":35151,"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":[{"comment":"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.","section":"Section 3.2 and Section 4.3 (Eqs. 3.6-3.8 and 4.8-4.10)"},{"comment":"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.","section":"Section 4.1 and Figures 3 and 5"},{"comment":"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.","section":"Section 2.2 and Section 3.1"},{"comment":"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.","section":"Section 4.3, Figure 5"}],"minor_comments":[{"comment":"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.","section":"Equation (3.2)"},{"comment":"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.","section":"Section 4.1"},{"comment":"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.","section":"Figures 3 and 5"},{"comment":"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.","section":"Equation (2.2)"},{"comment":"The statement that 'we have identified four other effective training strategies' is unverifiable because no details, results, or references are provided.","section":"Section 5"}],"recommendation":"reject","confidential_remarks":"The paper relies on a prior preprint [1] for central properties such as symmetry, reversibility, and fixed-point behavior, without providing independent verification in this manuscript. The experimental section is minimal and lacks basic baselines or error bars, and the theoretical connection between the correlation variable and error reduction is absent. Given the strong claims in the abstract, I am concerned about the novelty and completeness of the submission for a machine learning venue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper describes a training method for the author's 'Wuxing' symmetric differential-equation network, using the correlation between forward and reverse signal integrals to update parameters, with a PID-inspired split across three parameter sets. The idea is interesting and the presentation is honest about its own limits, but the central claim — faster and more accurate MNIST training — isn't backed by the experiments shown.\n\nWhat's new: the PID framing (K1 integral, K2 differential, K3 proportional) and the explicit update rules in Eqs. 4.2–4.10 go beyond the earlier self-cited preprint, which only sketched training. The author also makes a sensible design choice: recognizing that adjusting K1 and K3 simultaneously causes instability, and recommending integral+proportional over differential. That kind of empirical honesty is welcome.\n\nWhere it falls short: the update rule is asserted, not derived. Eq. 4.10 (K3_new = K3_old * exp(-G2_k3)) assumes G1 — the product of time-integrals of forward and reverse deviations — points in the direction that reduces classification error. No theorem, no ablation, no control that breaks the G1 signal. The experiments are a single MNIST run with no baselines, no error bars, and accuracy far below mainstream results. The load-bearing reversibility/fixed-point properties are inherited from the author's own prior preprint, so the foundation is not independently verified here. The PID labels read more as analogy than controller design; there's no stability or convergence argument.\n\nIs this fatal? For the claim 'this is a validated training method,' yes — the evidence doesn't support it. But as an early-stage proposal, it's coherent and testable. A reader working on biologically plausible learning or neuromorphic systems might find the signal-propagation idea worth exploring, provided they treat the reported numbers as preliminary.\n\nMy recommendation: send it to peer review. It deserves a serious referee who can demand proper baselines and a derivation or ablation of the update rule. It should not be accepted without substantial revision and stronger evidence.","headline":"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.","tokens_in":9710,"tokens_out":1958,"would_cite":false,"duration_ms":21917,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Wuxing neural network","symmetric differential equations","distributed PID control","backpropagation-free training","reversible systems","signal propagation","MNIST"],"falsifier":"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.","tokens_in":8736,"feed_emoji":"🧠","tokens_out":5987,"duration_ms":53514,"temperature":0.7,"pith_summary":"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.","feed_headline":"PID control trains a neural net without backpropagation","feed_subtitle":"Forward and reverse signal integrals replace the chain rule and speed up MNIST training.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Wuxing symmetric differential-equation network and establishes its reversibility and fixed-point properties, the foundation the training method relies on.","marker":"[1]"},{"why":"Provides the motivating comparison of a biologically motivated backpropagation alternative, positioning this method as a further candidate.","marker":"[2]"},{"why":"Supplies the classical PID control theory that the distributed PID strategy adapts.","marker":"[3]"},{"why":"Supplies the prior distributed-PID-for-networks idea that the paper adapts to a closed dynamical system.","marker":"[7]"},{"why":"Supports the claim that differential control is unstable in practice, which the paper uses to justify preferring integral and proportional modes.","marker":"[8]"}],"fun_headline_variants":["PID control replaces backprop in neural nets","Distributed PID tunes symmetric neural networks","Neural training via reversible PID signals","Backprop-free training with distributed PID","Symmetric net learns via PID feedback"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["PID control replaces backprop in neural nets","Distributed PID tunes symmetric neural networks","Neural training via reversible PID signals","Backprop-free training with distributed PID","Symmetric net learns via PID feedback"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00011,"raw_usage":{"total_tokens":1042,"prompt_tokens":920,"completion_tokens":122,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":60}},"tokens_in":536,"tokens_out":122,"duration_ms":2092,"temperature":1.0,"reasoning_tokens":60,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:01:42.442776+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Wuxing symmetric differential-equation network and establishes its reversibility and fixed-point properties, the foundation the training method relies on."},{"cited_title":"For more details, please refer to our previous article[1]","cited_arxiv_id":null,"evidence_quote":"Provides the motivating comparison of a biologically motivated backpropagation alternative, positioning this method as a further candidate."},{"cited_title":"3.1 Training theory In neural network training, the traditional backpropagation algorithm has been highly successful","cited_arxiv_id":null,"evidence_quote":"Supplies the classical PID control theory that the distributed PID strategy adapts."},{"cited_title":"Chong, and Y","cited_arxiv_id":null,"evidence_quote":"Supports the claim that differential control is unstable in practice, which the paper uses to justify preferring integral and proportional modes."}],"review_version":1}