{"id":"bb5f1119-f12a-4e2c-80db-6c472fa9c1dd","arxiv_id":"2505.09327","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A stochastic NG-RC controller with an explicit noise feature vector tracks designed trajectories in noisy Van der Pol systems, but its stability guarantee reuses Mao's theorem without verifying it for the implemented closed loop.","lead":"This paper adds a noise term to next-generation reservoir computing to build feedback controllers for stochastic, multi-scale oscillators, and tests the idea on a Van der Pol model and on a simulated model of epileptic EEG activity. A generalist might read it as a candidate data-driven control method for neurostimulation, but the medical claim rests on simulation, not real recordings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The LaSalle-based stability guarantee is never verified for the discrete learned closed loop; Eq. (8) makes error contraction an assumption, not a theorem.","rationale":"The reader's weakest assumption and my stress-test identify the same structural gap: Eq. (8) forces e_{i+1}=K e_i by substituting the learned model, and no evidence is supplied that the learned model is exact or that W_u is invertible. My independent reading confirms this is the most load-bearing point. The generic stochastic LaSalle theorem in Appendix A.1 is internally standard, but it is never connected to the discrete S-NGRC controller: no V, w, γ are exhibited, and conditions (H1), (19)–(21) are not verified for the controlled Van der Pol system or the EEG-derived model. Because the central claim in the Abstract and Section 4 is a rigorous stability guarantee, this disconnect is fatal to the theory as stated. The paper's own Section 5.2.2 admits that control performance deteriorates near t=490 and that long-term errors accumulate, further undermining the 'seizure suppression' claim, although that is secondary relative to the theoretical gap. The recommendation to reject is therefore unchanged; the paper could be repaired by supplying model-error bounds, a residual-aware error recursion, and a genuine verification that the closed loop satisfies the LaSalle hypotheses, but none of these are present.","tokens_in":19835,"tokens_out":2916,"duration_ms":32296,"concrete_test":"Compute, on the 500 test samples of Section 5.1.1, the one-step model residual r_i = X_{i+1}^{true} - (W_X O_{X,i} + W_u O_{u,i} + W_n O_{n,i}) with the control law (8) active, and report sup_i |r_i| together with the condition number of W_u. Then simulate the closed-loop recursion e_{i+1}=K e_i + r_i with ρ(K)<1. If sup|r_i| is not small compared with (1−ρ(K)) times the initial error, or if the condition number of W_u is large enough to make the computed control unreliable, the claimed convergence cannot be inferred from ρ(K)<1 alone. This directly tests whether the guarantee survives model error and invertibility failures.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—asymptotic convergence with a 'rigorous' stochastic LaSalle proof—rests on Proposition 1 in Appendix A.1, a generic continuous-time theorem for controlled SDEs. That proposition is never instantiated for the S-NGRC closed loop: no Lyapunov function V, no function w, and no integrable γ satisfying (19)–(21) are constructed for the actual error dynamics, and no argument shows that the discrete learned recursion (5), with ridge-regression weights and event-triggered updates, is the continuous SDE (18) to which the theorem applies. In contrast, Section 3.2 imposes e_{i+1}=K e_i by solving Eq. (8) for O_{u,i}; this is exact only if the NG-RC model (5) is exact and W_u is invertible. The paper provides neither model-error bounds for its ridge regression nor invertibility conditions on W_u. With a model residual r_i = X_{i+1}^{true} - (W_X O_{X,i}+W_u O_{u,i}+W_n O_{n,i}), the true closed-loop recursion is e_{i+1}=K e_i + r_i, and ρ(K)<1 alone cannot control r_i. Thus the stated guarantee is either a design tautology or an unverified appeal to a theorem whose hypotheses are never checked. The simulations do not close this gap because they never report r_i magnitudes or condition numbers of W_u; the EEG experiment explicitly admits growing long-term error, weakening the empirical support for the general claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a stochastic next-generation reservoir computing (S-NGRC) controller for trajectory tracking in multiscale stochastic dynamical systems. The method trains an NG-RC ridge-regression model that includes noise features, then inverts the learned input weight matrix to compute a control input designed to make the tracking error satisfy e_{i+1}=K e_i with ρ(K)<1. The paper claims that asymptotic stability of this controller is rigorously proved via an extended stochastic LaSalle theorem, and it presents numerical experiments on a stochastic Van der Pol system with additive and multiplicative noise and a seizure-suppression simulation using EEG data.","tokens_in":20111,"tokens_out":5846,"duration_ms":58031,"significance":"If the theoretical guarantee claimed in the abstract were established, the paper would be a valuable contribution connecting data-driven reservoir control with stochastic stability theory. The manuscript does provide a self-contained statement and proof of a generic stochastic LaSalle-type theorem for controlled SDEs in Appendix A.1, and the Van der Pol simulations are described with enough detail (training/test splits, RMSE values, parameter sweeps) to be plausible as an empirical study. However, the central stability claim is not supported: the theorem is never instantiated for the proposed control law, and the control design in Eq. (8) enforces error contraction only under exact-model and invertibility assumptions that are neither provable nor verified. The EEG experiment, as the authors acknowledge, exhibits growing long-term error and is carried out on the fitted SDE rather than on recorded dynamics directly.","major_comments":[{"comment":"The derivation of the control law assumes that the learned model (5) is exact and that W_u is invertible. Writing the true closed-loop recursion with residual r_i = X_{i+1}^{true} - (W_X O_{X,i}+W_u O_{u,i}+W_n O_{n,i}), substitution of (8) gives e_{i+1}=K e_i + r_i. The paper provides no bound on r_i and no conditions under which the ridge-regression weight matrix W_u is invertible, so ρ(K)<1 alone does not imply convergence. Thus the tracking guarantee is imposed by design rather than established.","section":"§3.2, Eq. (8)"},{"comment":"Proposition 1 is a continuous-time stability theorem for a generic controlled SDE under hypotheses (H1), (19)-(21). The paper never verifies these hypotheses for the S-NGRC closed loop: no Lyapunov function V, no function w, no integrable γ are constructed, and no argument connects the discrete, event-triggered recursion (5)-(8) to the continuous-time SDE (18) of the proposition. In particular, condition (21) concerns existence of p-th moments of the solution of (18), which depends on the feedback law and is not checked. The abstract's claim of rigorous asymptotic stability is therefore unsupported.","section":"§4 and Appendix A.1, Proposition 1"},{"comment":"The EEG control experiment trains the controller on perturbed trajectories generated from the fitted SDE (16) and then applies the controller to trajectories from the same fitted SDE; this validates control of the model, not of the recorded EEG signals. The paper also states that control performance deteriorates near t=490 and lists accumulation of long-term control errors as a limitation, directly weakening the conclusion of effective seizure suppression.","section":"§5.2.2"}],"minor_comments":[{"comment":"On page 4, 'the the discretization step' contains a duplicated article.","section":"§3.1"},{"comment":"The section heading 'Stablity analysis' should be 'Stability analysis'.","section":"§4"},{"comment":"The text before Eq. (11) spells 'Lassalle' instead of 'LaSalle'.","section":"§5.1.2"},{"comment":"The caption contains the typo 'wihch', which should be 'which'.","section":"Figure 14 caption"},{"comment":"The model is described as one-dimensional even though the data are 23-channel; please clarify that a single channel is selected for modeling.","section":"§5.2.1"},{"comment":"The sentence 'The bottom panel (left two segments: Phase A and B)' appears twice in consecutive paragraphs; remove the duplication.","section":"§5.2.2"}],"recommendation":"reject","confidential_remarks":"For the editor: The numerical study is potentially salvageable as an applied methods paper if the theoretical claims are removed or substantially weakened. As a math.DS submission, the missing connection between the stability theorem and the algorithm is a disqualifying gap. I also note that no code or data are made available, which limits reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the algorithmic core is fine, but the central theoretical claim does not hold up. Adding an explicit noise feature vector to NG-RC is a reasonable, small extension, and the Van der Pol results look credible; the comparison showing conventional NG-RC diverging under sigma1=1, sigma2=2, epsilon=0.5 is the most convincing part of the paper. That said, the abstract promises a rigorous stochastic LaSalle proof, and that proof is not delivered. Appendix A.1 restates a generic theorem from Mao for a controlled SDE, then never verifies any of its hypotheses for the actual discrete-time learned closed loop. No Lyapunov function, no gamma, no w, no moment bound are constructed for the S-NGRC error dynamics. In addition, Eq. (8) chooses the control to cancel the model terms exactly, so e_{i+1}=K e_i holds only if the ridge-regression model is exact and W_u is invertible. The paper provides neither model-error bounds nor invertibility conditions. With any residual the true recursion is e_{i+1}=K e_i + r_i, and rho(K)<1 alone does not control r_i. It is a design tautology or an unverified appeal to a theorem, not a derived guarantee.\n\nWhat is genuinely good: the noise feature vector is a real departure from standard NG-RC, and the robustness experiments over twelve parameter settings give a useful picture of where the method starts to fail. The EEG section is honest in its limitations section—it explicitly admits increasing long-term error and imperfect model fitting. The problem is that the EEG experiment is a simulation on a fitted one-dimensional SDE, not a real-data validation, and the phrase 'seizure suppression' in the conclusions overstates what was done. No code or data are provided, so the numerical claims are not independently checkable.\n\nWho should read this: anyone working on data-driven control of stochastic oscillators might steal the noise-feature idea and the experimental setup, but they should not cite the stability result as a theorem. The paper deserves a serious referee—the gap is identifiable and potentially repairable if the authors add model-error bounds or instantiate the LaSalle conditions for the actual closed loop. I would send it to peer review with the expectation of major revision, not desk reject it.","headline":"The noise-feature trick is a genuine but small improvement to NG-RC, and the Van der Pol experiments are plausible; the advertised stability proof, however, is never actually connected to the closed loop, and the convergence guarantee is mostly a design assumption.","tokens_in":20710,"tokens_out":2029,"would_cite":false,"duration_ms":21722,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93E15","93C40","60H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims a data-driven feedback controller, built from next-generation reservoir computing with explicit noise features, that asymptotically tracks desired trajectories in multiscale stochastic systems and proves convergence via…","keywords":["stochastic dynamical systems","next-generation reservoir computing","adaptive control","multiscale systems","stochastic LaSalle theorem","event-triggered control","Van der Pol oscillator","EEG seizure suppression"],"falsifier":"Run the closed loop at parameter settings where the reported Y-component RMSE exceeds 1.0 (for example, $\\varepsilon=0.1$, $\\sigma=2$) and check whether the tracking error keeps decreasing toward zero over a long horizon; if the error saturates or diverges despite the claimed LaSalle conditions, the asymptotic convergence guarantee would be falsified.","tokens_in":19546,"feed_emoji":"🎛️","tokens_out":12069,"duration_ms":102332,"temperature":0.7,"pith_summary":"This paper is trying to establish that a next-generation reservoir computing controller, modified to include an explicit noise feature vector, can make a multiscale stochastic dynamical system asymptotically track a prescribed target trajectory even when the underlying drift and diffusion terms are unknown. The claimed guarantee is that the tracking error converges almost surely to an invariant set, established through an extended stochastic LaSalle theorem rather than through model-based stochastic control. If the claim holds, this gives a computationally cheap, data-driven route to stabilizing noisy oscillatory systems, with the numerical experiments showing rapid tracking on a stochastic Van der Pol oscillator under additive and multiplicative noise and partial suppression of seizure-like EEG oscillations.","feed_headline":"Stochastic NG-RC tracks noisy multiscale systems with a proof","feed_subtitle":"A data-driven controller adds noise features to reservoir computing and can modulate EEG seizures.","key_machinery":"The load-bearing object is the stochastic feature stack $O_{\\mathrm{total}}=\\{O_{\\mathrm{lin},i},O_{\\mathrm{nonlin},i},O_{u,i},O_{n,i}\\}$, containing one-step delayed states, monomials up to third order, the control input, and the noise increment; ridge regression produces the output weights $W_{\\mathrm{out}}$, and the controller inverts the $W_u$ block to enforce the error recursion. The stability argument is carried by the operator $\\mathcal{L}$ of the controlled error SDE and a Lyapunov function $V(e,t)$ satisfying $\\mathcal{L}V\\le\\gamma(t)-w(e)$, with $\\gamma$ integrable and $w$ a nonnegative continuous function whose zero set is the attractor.","core_discovery":"The central discovery is a closed-loop control law for stochastic systems learned entirely from data. After training the ridge-regression model $X_{i+1}=W_X O_{X,i}+W_u O_{u,i}+W_n O_{n,i}$ on a perturbed version of the system, the controller sets $O_{u,i}=W_u^{-1}[X_{\\mathrm{des},i+1}-W_X O_{X,i}-W_n O_{n,i}+K e_i]$ with $\\rho(K)<1$, so that the tracking error would evolve as $e_{i+1}=K e_i$ if the model were exact. The paper proves, for the associated continuous-time controlled SDE, a stochastic LaSalle proposition: under local Lipschitz and linear-growth conditions, a Lyapunov inequality $\\mathcal{L}V\\le\\gamma(t)-w(e)$ with $\\gamma\\in L^1$, and bounded $p$-th moments, the error converges almost surely to the invariant set $\\{e:w(e)=0\\}$. The experiments then show this controller tracks amplitude-modulated targets in a multiscale stochastic Van der Pol system under additive and multiplicative noise and reduces seizure-like oscillations in an EEG-derived one-dimensional SDE.","pith_inferences":["Because the stability proof is carried out in continuous time while the controller operates in discrete time, a natural next step is a discrete-time stochastic Lyapunov argument or an explicit discretization-error bound; the paper does not close that gap.","If the learned model is not exact, the error recursion becomes $e_{i+1}=K e_i+\\eta_i$ with an unmodeled residual $\\eta_i$, so adding a robustifying term or an online update of the output weights would be a testable extension toward a guarantee that survives model error.","On the EEG task, control performance degrades near the end of the interval and depends on the random perturbation used in training, suggesting that structured perturbation design, rather than random forcing, is likely the key lever for clinical translation; this is an inference, not a result tested in the paper."],"forward_implications":["In the low-noise single-scale case, the controlled system reaches the desired trajectory within 1-2 time steps, with a control RMSE of $0.1654$.","At $\\sigma_1=1$, $\\sigma_2=2$, $\\varepsilon=0.5$, the conventional NG-RC controller diverges at step 302 while the stochastic NG-RC keeps the control signal bounded and achieves total RMSE $0.3632$.","Under multiplicative noise ($\\sigma_1=0.8$, $\\sigma_2=1$, $\\varepsilon=0.1$), the slow variable tracks closely and the total RMSE is $0.2359$.","On EEG-derived resting and seizure dynamics, the controller reduces large-amplitude seizure oscillations toward the resting-state distribution over a 100-step interval, with control RMSE $0.0752$."],"supporting_citations":[{"why":"Introduces next-generation reservoir computing, the base method the paper extends with an explicit noise feature vector.","marker":"[19]"},{"why":"Supplies the one-step delay feature $O_{\\mathrm{lin},i}$ and the NG-RC control design for chaotic maps that the paper adapts to SDEs.","marker":"[21]"},{"why":"Provides the conventional NG-RC controller used as the comparison baseline, which diverges at step 302 in the experiment.","marker":"[20]"},{"why":"Gives the discrete-time nonlinear control framework from which the error-recursion control law (8) is adopted.","marker":"[34]"},{"why":"LaSalle's theorem, the classical invariant-set stability result the paper extends to the stochastic controlled setting.","marker":"[35]"},{"why":"Supplies the stochastic LaSalle theorem and the Lyapunov and moment conditions that Proposition 1 adapts for the controlled SDE.","marker":"[38]"},{"why":"Kolmogorov-Centsov theorem used in Lemma 1 to show that the stochastic integral is almost surely uniformly continuous.","marker":"[39]"},{"why":"Moment inequality for stochastic integrals used to bound the noise term in proving the error process's uniform continuity.","marker":"[40]"}],"fun_headline_variants":["Proven data-driven control for stochastic multiscale systems","Stochastic NG-RC controller: stability proof for noisy dynamics","Adaptive control from data, with a stochastic LaSalle guarantee","Tame noisy multiscale systems with a proven data-driven controller","EEG seizure tamed by stochastic NG-RC with mathematical proof"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tracking guarantee assumes the ridge-regression model (5) is an exact representation of the controlled system and that the learned input-weight matrix $W_u$ is invertible, because the control law (8) is derived by substituting the model so that the error follows $e_{i+1}=K e_i$ exactly.","fun_headline_variants_meta":{"raw":{"variants":["Proven data-driven control for stochastic multiscale systems","Stochastic NG-RC controller: stability proof for noisy dynamics","Adaptive control from data, with a stochastic LaSalle guarantee","Tame noisy multiscale systems with a proven data-driven controller","EEG seizure tamed by stochastic NG-RC with mathematical proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000343,"raw_usage":{"total_tokens":1894,"prompt_tokens":960,"completion_tokens":934,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":846}},"tokens_in":576,"tokens_out":934,"duration_ms":8113,"temperature":1.0,"reasoning_tokens":846,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:35:02.155895+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the closed loop at parameter settings where the reported Y-component RMSE exceeds 1.0 (for example, $\\varepsilon=0.1$, $\\sigma=2$) and check whether the tracking error keeps decreasing toward zero over a long horizon; if the error saturates or diverges despite the claimed LaSalle conditions, the asymptotic convergence guarantee would be falsified.","supporting_citations":[{"cited_title":"Controlling chaotic maps us- ing next-generation reservoir computing.Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(2), 2024","cited_arxiv_id":null,"evidence_quote":"Supplies the one-step delay feature $O_{\\mathrm{lin},i}$ and the NG-RC control design for chaotic maps that the paper adapts to SDEs."},{"cited_title":"Controlling chaos using edge computing hardware.Nature Communications, 15(1):3886, 2024","cited_arxiv_id":null,"evidence_quote":"Provides the conventional NG-RC controller used as the comparison baseline, which diverges at step 302 in the experiment."},{"cited_title":"CRC press, 2018","cited_arxiv_id":null,"evidence_quote":"Gives the discrete-time nonlinear control framework from which the error-recursion control law (8) is adopted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"LaSalle's theorem, the classical invariant-set stability result the paper extends to the stochastic controlled setting."},{"cited_title":"Stochastic versions of the lasalle theorem.Journal of Differential Equations, 153:175–195, 1999","cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic LaSalle theorem and the Lyapunov and moment conditions that Proposition 1 adapts for the controlled SDE."}],"review_version":1}