{"id":"12c89b73-dd63-485e-abe5-eb5b348c578f","arxiv_id":"2504.20256","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Adding an annealing step that reactivates removed dictionary terms lowers coefficient error for SINDy and hard thresholding pursuit on several nonlinear benchmarks, with mixed gains on experimental projectile data.","lead":"The authors add a cooling-schedule reactivation step to hard thresholding sparse model discovery, and report lower coefficient errors on Lorenz, FitzHugh-Nagumo, logistic, and forced van der Pol benchmarks. The idea is that terms pruned too early can be reintroduced while the algorithm cools.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper does not show that the cooling schedule, rather than the extra random exploration and extra iterations it buys, causes the reported improvements; the schedule and per-problem hyperparameters are also not ablated or sensitivity-tested.","rationale":"Good-faith reading: the paper makes a modest empirical claim, gives multiple synthetic systems and one real dataset, and openly states limitations (annealing insufficient in low noise, sometimes fails). Those honest negatives are real evidence. The load-bearing problem is attribution, not fraud or internal inconsistency. The algorithm is executable and the comparisons are plausible, but the annealed variant differs from its baseline in at least three ways: random perturbation, cooling schedule, and total iteration count. Without an ablation isolating the schedule, the central sentence 'by introducing an annealing scheme ... we are able to improve performance' is not established; the data support 'some stochastic restart variant with extra iterations improves these benchmarks.' The reader's conditional verdict already captures the reproducibility risk, and this stress-test identifies a concrete experimental check rather than a reason to reject. No code or data release and missing derivative-estimation details further justify the condition. Verdict unchanged: CONDITIONAL until the attribution check or equivalent sensitivity analysis is supplied.","tokens_in":7766,"tokens_out":5396,"duration_ms":66909,"concrete_test":"Run SHTreP-A against three controls on the Lorenz and forced van der Pol benchmarks with identical 300 noise draws and identical seeds: (i) constant p in {0.05, 0.2, 0.5} instead of the cooling schedule; (ii) one random reactivation step (p = 1) followed by plain SHTreP; (iii) plain SHTreP run for the full 500 iterations. If any control matches the error distribution of SHTreP-A, the cooling schedule is not the cause of the reported improvement; if none does, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the annealing schedule itself improves hard-thresholding model discovery. The evidence supports the weaker statement that some randomly perturbed hard-thresholding runs, with hand-selected s and lambda and one fixed cooling schedule, beat unperturbed runs on the reported benchmarks. Three gaps make this load-bearing. First, there is no ablation: SHTreP-A is never compared with SHTreP using a constant reactivation probability p, with a single random support perturbation, or with the same total iteration budget. The algorithm section (Eq. 1 and the reactivation step) adds both random exploration and many extra iterations relative to the unannealed baseline; either factor could explain the error reductions. Second, the paper states that the cooling schedule provides another set of hyperparameters and can make a significant effect on the obtained minimizer, yet no sensitivity analysis, selection rule, or seed-averaging over the random reactivation draws is reported. Third, all benchmarks use per-problem choices of s, lambda, and the same hand-written schedule, so the consistent gains could reflect favorable tuning rather than a general property. The authors do report failure cases and honestly bound the claim, which is to their credit, but the central mechanism is not isolated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an annealing scheme for hard thresholding algorithms used in sparse model discovery. The scheme reactivates a random fraction of deactivated library terms, with the fraction decreasing according to a cooling schedule, and is applied to two base algorithms: SINDy and a hard thresholding pursuit variant (SHTreP). The authors compare annealed and unannealed versions on Lorenz, FitzHugh–Nagumo, logistic, forced Van der Pol, and experimental projectile data, measuring relative ℓ1 and ℓ2 coefficient errors under added Gaussian noise. They report that annealing generally improves accuracy, especially for SHTreP in large-library settings, while acknowledging cases where it fails. The central claim is that annealing allows early-pruned terms to be recovered and thus improves sparse model discovery.","tokens_in":8013,"tokens_out":4943,"duration_ms":50461,"significance":"The proposed modification is simple, broadly applicable to hard-thresholding optimizers, and motivated by a real failure mode: early pruning of important terms in noisy or large-library settings. If the improvement is robust, the method could be practically useful, and the authors honestly report cases where annealing does not help (e.g., Figure 5 and the Conclusion). The paper also benefits from benchmarking on several nonlinear systems, including chaotic, excitable, entrained, and real experimental data, and from reporting failure cases rather than overclaiming. However, the evidence is entirely empirical, and the current experiments do not isolate the proposed mechanism from confounds such as additional iterations and random exploration, nor do they provide code, data, or sensitivity analysis. These gaps prevent the central claim from being conclusively established.","major_comments":[{"comment":"The annealed algorithm differs from the baseline in two confounded ways: it adds random support perturbations and it runs for many more iterations (the full cooling schedule plus up to 500 iterations). The reported comparisons in Figures 2–8 therefore do not establish that the cooling schedule, rather than the extra random exploration or the larger iteration budget, is responsible for the error reductions. The authors should ablate SHTreP-A against SHTreP with a constant reactivation probability, against SHTreP with a single initial random support perturbation, and against SHTreP run for the same total number of iterations as the annealed variant.","section":"Description of Algorithm, Eq. (1) and the annealing step"},{"comment":"The paper states that the cooling schedule 'provides another set of hyperparameters for the optimization scheme and can make a significant effect on the obtained minimizer,' but no sensitivity analysis or principled selection rule is given for the schedule, and each benchmark uses a different hand-chosen schedule with per-problem sparsity s and threshold λ. Without systematic variation of the schedule and without seed-averaging over the random reactivation draws, the consistent gains could reflect favorable tuning rather than a general property of annealing.","section":"Description of Algorithm (annealing schedule)"},{"comment":"The numerical experiments omit essential reproducibility details: the derivative approximation used to form Ẋ is not specified (only 'a numerical approximation' is mentioned), the termination condition is not stated, and no code or data are provided. Because the paper's evidence is entirely empirical, these omissions prevent verification of the central claim and of Table I.","section":"Comparisons between original and annealed algorithms"},{"comment":"Figures 2–8 report standard deviations over 300 noise instances, but SHTreP-A and SINDy-Anne are stochastic algorithms; the text does not state how many algorithm runs per noise instance are used or how the random reactivation draws are seeded. The shaded widths are therefore ambiguous: if each noise instance yields a single stochastic trajectory, the reported standard deviations do not capture the algorithm's own variance.","section":"Lorenz System and Figures 2–8"}],"minor_comments":[{"comment":"There is a typo in this section: 'becuase' should be 'because'.","section":"Logistic"},{"comment":"The annealing schedule is written as '{1,.99,...,.8,.7...,.1,.09,.08,...,.01,0}', which is ambiguous; please provide an explicit sequence or pseudocode.","section":"Description of Algorithm"},{"comment":"References [36] and [37] appear to be the same paper by van Breugel, Kutz, and Brunton with near-identical titles; please merge or disambiguate.","section":"References"},{"comment":"The sentence 'all the models fixate on incorrect solutions in the absence of sampling noise but actually improve their fit significantly when sampling noise is added' is surprising and unexplained; either provide a mechanism or soften the claim.","section":"Lorenz System"},{"comment":"The projectile-motion models are reported without any uncertainty or stability assessment across the stochastic runs; a statement about how representative the shown equations are would strengthen the comparison.","section":"Projectile Motion, Table I"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about failure cases and the idea is incrementally novel, but the central empirical claim is not yet isolated from confounds. I recommend asking for an ablation study, sensitivity analysis of the cooling schedule, and code/data release before further consideration. The manuscript fits the journal's scope as an optimization heuristic paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a modest, honestly-reported algorithmic variation — annealed random reactivation of pruned terms in SINDy and hard thresholding pursuit — and the experiments suggest it helps in some noisy, large-library settings. The problem is that the paper never actually tests whether the cooling schedule, rather than the extra random exploration and extra iterations, is doing the work.\n\nWhat's new: I'm not aware of another paper putting a simulated-annealing-style reactivation schedule into SINDy or hard thresholding pursuit. The comparisons are systematic: 300 noise realizations per setting, multiple systems including Lorenz, FitzHugh-Nagumo, logistic, forced Van der Pol with a time-dependent library, and a real projectile dataset. They report failure cases (low-noise Lorenz, logistic x0=10) and say so explicitly. That is good practice.\n\nSoft spots, in order. First, no ablation. SHTreP-A differs from SHTreP in two ways: random support perturbations and a much longer iteration count (the schedule runs for many steps, then 500 more iterations). Neither is controlled. A constant-p random reactivation, a single perturbation, or a matched iteration budget would isolate the mechanism. Second, the paper itself says the cooling schedule is another set of hyperparameters with significant effect, but there is no sensitivity analysis, no seed averaging over the random draws, and no code or data shipped. Third, hyperparameters (sparsity s, thresholds lambda, polynomial degree, smoothing passes) are chosen per problem with no stated rule. So the consistent gains could reflect favorable tuning rather than a general property.\n\nNone of this kills the paper. The central empirical observation may well be true. But as presented, the evidence supports only the weaker statement that some randomly-perturbed hard-thresholding runs, with hand-picked settings, beat the unperturbed baselines on these benchmarks.\n\nWho this is for: readers working on SINDy variants or sparse dictionary learning who want a cheap idea to try. It deserves a serious referee — the idea is simple, the comparisons are mostly careful, and the honesty about failures is a plus — but the referee should ask for code, an ablation, and sensitivity analysis. I would not cite it as a demonstrated general improvement, but I would cite it as a plausibly useful trick if it holds up under that scrutiny.","headline":"A modest, honest paper showing annealed random reactivation can improve SINDy and hard thresholding pursuit on several benchmarks, but the cooling schedule itself is never isolated from extra random exploration and extra iterations.","tokens_in":8531,"tokens_out":2101,"would_cite":false,"duration_ms":23074,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65K10","90C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding an annealing step that reactivates a shrinking fraction of thresholded library terms improves hard-thresholding sparse model discovery in noisy, large-library settings.","keywords":["sparse model discovery","SINDy","hard thresholding pursuit","simulated annealing","cooling schedule","dictionary learning","dynamical systems","coefficient error"],"falsifier":"Run SHTreP-A and SHTreP with a common total iteration budget on the same Lorenz, FitzHugh–Nagumo, logistic, and forced van der Pol ensembles, selecting the cooling schedule, $s$, and $\\lambda$ for both methods by cross-validation at each noise level; if SHTreP-A's relative $\\ell^1$ coefficient error is not below SHTreP's, the claimed improvement is falsified.","tokens_in":7562,"feed_emoji":"🌡️","tokens_out":10572,"duration_ms":93809,"temperature":0.7,"pith_summary":"Hard-thresholding sparse discovery algorithms, such as SINDy (sparse identification of nonlinear dynamics), permanently remove library terms whose coefficients fall below a threshold, so a term dropped early under noisy derivative estimates can never come back. This paper introduces an annealing modification: after each thresholding step, a fraction of the removed terms is randomly reactivated into the candidate set, and that fraction follows a cooling schedule that decreases from 1 to 0. The resulting algorithms, SHTreP-A and SINDy-Anne, are compared with their non-annealed counterparts on the Lorenz, FitzHugh–Nagumo, logistic, and forced van der Pol systems, and on experimental projectile-motion data. The annealed versions generally achieve lower relative coefficient errors, particularly in noisy settings with large function libraries. The paper's goal is to show that a simple stochastic reactivation mechanism makes hard-thresholding optimization more reliable without changing the underlying sparse model-selection problem.","feed_headline":"Annealing resurrects dropped terms in sparse model discovery","feed_subtitle":"Reintroducing thresholded terms with decreasing probability lowers coefficient error on noisy, large-library benchmarks.","key_machinery":"The load-bearing mechanism is the annealed hard-thresholding update. In SHTreP-A, each iteration computes $\\tilde{\\xi}^k = H_s(\\xi^k + \\Theta(X)^T(\\dot{X}-\\Theta(X)\\xi^k))$, where $H_s$ keeps the $s$ largest-magnitude entries; annealing then selects each zero entry with probability $p_k$ and adds it to the candidate support set $S_k$, followed by the debiasing step $\\xi^{k+1} = \\arg\\min_{\\mathrm{supp}(x)\\subseteq S_k}\\|\\Theta(X)x-\\dot{X}\\|_2$. The cooling schedule $\\{1,0.99,\\dots,0.8,0.7,\\dots,0.1,0.09,\\dots,0.01,0\\}$ controls how many removed terms are reconsidered over time. This random reactivation is what distinguishes SHTreP-A and SINDy-Anne from their base algorithms: it acts as a 'heat bath' that lets the optimization escape the trap of a prematurely pruned support set.","core_discovery":"The central claim is that annealing the hard-thresholding step improves sparse model discovery. Instead of discarding thresholded coefficients forever, the optimizer re-admits them: at iteration $k$ it reactivates each currently zero coefficient with probability $p_k$, where $p_k$ follows a cooling schedule (starting at 1 and decaying to 0), then solves the restricted least-squares problem on the resulting support set. This allows terms whose early elimination was an artifact of noise or of an incomplete support set to return once the coefficient estimate has improved. The authors report that SHTreP-A outperforms plain hard thresholding pursuit (SHTreP) on noisy ensembles of the Lorenz, FitzHugh–Nagumo, logistic, and forced van der Pol systems, and that SINDy-Anne matches or improves SINDy depending on the problem; on the experimental projectile-motion data, only SHTreP-A produces a physically plausible $x$-equation. The discovery is that allowing lost support to be recovered can matter as much as enforcing sparsity, and a simple stochastic reactivation step provides that recovery.","pith_inferences":["This reader's inference: the reactivation step is independent of the specific optimizer, so the same annealing scheme could be bolted onto any hard-thresholding or greedy sparse-regression method, including weak-form variants, without further changes.","This reader's inference: the benefit appears to grow with library size and noise, which suggests annealing is most valuable when the dictionary is deliberately over-complete and many irrelevant terms compete for the top-$s$ slots.","This reader's inference: the cooling schedule is a discrete probability path, so a natural testable extension is to tune it automatically on a validation trajectory, or to make the temperature adaptive, raising it whenever the support set changes rapidly.","This reader's inference: because the paper follows classic simulated-annealing guidance, a direct quantitative comparison would be to run simulated annealing over the coefficient vector itself and ask whether support-set annealing matches its recovery rate."],"forward_implications":["On the Lorenz, FitzHugh–Nagumo, and forced van der Pol benchmarks, SHTreP-A lowers both the average and the standard deviation of relative $\\ell^1$ coefficient error compared with SHTreP as sampling noise grows.","In problems with small libraries and low sparsity, such as FitzHugh–Nagumo, annealing can give no measurable advantage: SINDy and SINDy-Anne perform the same.","On the logistic equation with a sigmoid trajectory, SHTreP-A outperforms all other algorithms as noise increases, but with exponential-decay data annealing plays no significant role.","For the forced van der Pol system with a 44-term time-dependent library, SHTreP and SHTreP-A both achieve accurate models while the SINDy variants struggle, and the paper reports that chaotic motion aids learning.","On the projectile-motion data, only SHTreP-A recovers a physically reasonable $x$-equation: no spurious constant force and no positive $x$-acceleration."],"supporting_citations":[{"why":"Defines the baseline SINDy algorithm whose hard-thresholding step permanently removes terms; the paper modifies this step.","marker":"[20]"},{"why":"Supplies the hard thresholding pursuit iteration that SHTreP and SHTreP-A are built on.","marker":"[40]"},{"why":"Provides the iterative hard thresholding framework motivating the sparsity-projection update.","marker":"[41]"},{"why":"Gives the simulated-annealing cooling principles that the reactivation schedule follows.","marker":"[39]"},{"why":"Gives convergence guarantees for SINDy in low-noise settings, the regime the annealed variant seeks to extend.","marker":"[27]"},{"why":"Provides the projectile-motion dataset context and the physics-based interpretation used in the experimental comparison.","marker":"[33]"}],"fun_headline_variants":["Cooling schedule revives dropped terms for better sparse fits","Annealing reanimates thresholded terms, lifting model accuracy","Don't kill terms: anneal them back for sparser, truer models","Sparse discovery meets resurrection: anneal to revive dropped terms","Let them back in: annealing for sparse discovery"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the hand-selected cooling schedule, sparsity level, and threshold values generalize across problems; the paper provides no sensitivity analysis or automated tuning rule, so if the gains vanish when all methods are tuned fairly, the central claim collapses.","fun_headline_variants_meta":{"raw":{"variants":["Cooling schedule revives dropped terms for better sparse fits","Annealing reanimates thresholded terms, lifting model accuracy","Don't kill terms: anneal them back for sparser, truer models","Sparse discovery meets resurrection: anneal to revive dropped terms","Let them back in: annealing for sparse discovery"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1570,"prompt_tokens":882,"completion_tokens":688,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":602}},"tokens_in":498,"tokens_out":688,"duration_ms":7273,"temperature":1.0,"reasoning_tokens":602,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:33:53.551564+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run SHTreP-A and SHTreP with a common total iteration budget on the same Lorenz, FitzHugh–Nagumo, logistic, and forced van der Pol ensembles, selecting the cooling schedule, $s$, and $\\lambda$ for both methods by cross-validation at each noise level; if SHTreP-A's relative $\\ell^1$ coefficient error is not below SHTreP's, the claimed improvement is falsified.","supporting_citations":[{"cited_title":"Finally we will compare these four algorithms in the following section","cited_arxiv_id":null,"evidence_quote":"Defines the baseline SINDy algorithm whose hard-thresholding step permanently removes terms; the paper modifies this step."},{"cited_title":"Kay, The risk of bias in denoising methods: Examples from neuroimaging, PLoS One 17, e0270895 (2022)","cited_arxiv_id":null,"evidence_quote":"Supplies the hard thresholding pursuit iteration that SHTreP and SHTreP-A are built on."},{"cited_title":"Blumensath and M","cited_arxiv_id":null,"evidence_quote":"Provides the iterative hard thresholding framework motivating the sparsity-projection update."},{"cited_title":"Geman and D","cited_arxiv_id":null,"evidence_quote":"Gives the simulated-annealing cooling principles that the reactivation schedule follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the projectile-motion dataset context and the physics-based interpretation used in the experimental comparison."}],"review_version":1}