{"id":"ef798c78-14a4-4965-9f80-db7f6a7c385f","arxiv_id":"2507.06267","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"HADES-NN estimates parameters of non-autonomous ODEs with discontinuous forcing by iterating between neural-network smoothing of the input and Levenberg-Marquardt parameter fitting, recovering true parameters where standard methods fail.","lead":"A two-stage algorithm, HADES-NN, first replaces a discontinuous external input in a differential equation with a smooth neural-network approximation, then fits the model parameters to data with a standard optimizer, iterating until both converge. The method recovers parameters in simulated Lotka-Volterra, circadian clock, and yeast signaling models where standard optimizers fail.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence theorem assumes global minimization by LM, which the implemented algorithm does not guarantee; the proof's first inequality and parameter-convergence step both lack support.","rationale":"The reader's weakest_assumption already identifies the LM-global-minimizer gap and the compactness issue; my reading agrees and adds that the proof's transition from surrogate-loss convergence to parameter convergence is not justified by Assumption 1 alone. This is internally inconsistent with the algorithm as described, not merely a disagreement with consensus: the theorem's hypothesis (global minimizers) is stronger than what the method's optimization step can guarantee. The empirical demonstrations are suggestive but do not fill the gap, because the circadian and yeast models are exactly the settings where nonconvexity is most plausible. I would not reject the paper: the method is clearly specified, the experiments are reproducible in principle, and the proof may be repairable with an explicit compactness assumption plus a guarantee (or empirical check) that LM reaches a global minimum, or a weaker theorem stated for local minimizers. Since the reader's verdict is already CONDITIONAL, my stress test does not move it.","tokens_in":17574,"tokens_out":4695,"duration_ms":54980,"concrete_test":"At each HADES-NN outer iteration on the smooth-signal Lotka-Volterra benchmark of Fig. 1B, compute the surrogate loss (3.5) at LM's returned p_n and compare it with the global minimum of the same loss obtained by multistart differential evolution (or another global solver). If the two values differ for any iteration, the premise that p_n is a global minimizer—required for the first inequality in Theorem B.4—is violated exactly in the regime the theorem covers; the theorem then does not apply to the implemented algorithm.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence claim (Theorem B.4, Appendix B) is load-bearing but rests on a premise the implemented algorithm does not deliver. Theorem B.4 assumes {p_n} is a sequence of global minimizers of the smooth surrogate loss (3.5), and Section 3.2 states that LM is used to find this global minimizer. However, the cited support for Step 2 (Prop. B.2, from Ref. [49]) bounds only the number of iterations needed to reach a stationary point (gradient norm below tolerance) under inverse-problem conditions; it does not establish global optimality in a nonconvex landscape. LM is a local method, so warm-starting with p_{n-1} does not guarantee p_n = argmin of (3.5). The first inequality in the proof of Theorem B.4—which compares p_n with p* on the surrogate loss—depends exactly on this unverified equality. The proof also moves from loss convergence to parameter convergence without showing {p_n} lies in a compact sublevel set or that the true loss is continuous in p; Assumption 1 (unique minimizer) alone does not provide compactness. For the circadian and yeast examples, no check is reported that LM attained a global minimum at any iteration. The paper's headline guarantee is therefore not established by the supplied argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes HADES-NN, an iterative two-stage method for estimating parameters of non-autonomous ODEs with discontinuous external signals. Stage 1 fits a smooth neural-network approximation of the discontinuous signal; Stage 2 uses this smooth signal in the ODE and estimates parameters with Levenberg-Marquardt. The two stages are alternated and warm-started from the previous iteration. The method is demonstrated on a Lotka-Volterra model with a Markov-switching input, a human circadian pacemaker model driven by wearable light-exposure data, and a yeast mating-response network with experimental data. The paper also states and proves a convergence theorem in Appendix B claiming that the sequence of parameter estimates converges to the minimizer of the original non-smooth loss.","tokens_in":17791,"tokens_out":3851,"duration_ms":43456,"significance":"If the theoretical claim and numerical comparisons hold, HADES-NN would be a practically useful and conceptually simple solution to a real obstacle in fitting non-autonomous models with abrupt inputs: the non-smoothness of the loss landscape that defeats standard gradient-based and direct-search optimizers. The paper's strengths are its breadth of applications, the use of experimentally measured input signals and data in the circadian and yeast examples, and the systematic comparison with six alternative optimizers across 100 random initializations. The numerical demonstrations are internally consistent and show a clear practical effect. However, the central convergence proof in Appendix B has a genuine gap concerning the global-optimality assumption on the LM stage, and the final step from loss convergence to parameter convergence is not justified by the stated assumptions. These issues do not invalidate the empirical contribution, but they do mean the headline theoretical guarantee is not established as written.","major_comments":[{"comment":"Theorem B.4 assumes that p_n is a global minimizer of the smooth surrogate loss (3.5), but the implemented Stage 2 uses the Levenberg-Marquardt algorithm, which is a local method. Proposition B.2, cited from Ref. [49], only guarantees that a stationary point with gradient norm below tolerance is reached within O(eps^{-2}) iterations; it does not establish that p_n equals argmin of (3.5) in a nonconvex landscape. The first inequality in the proof of Theorem B.4 relies exactly on p_n being a global minimizer, so the proof does not cover the algorithm as implemented. The authors should either add conditions under which LM finds the global minimum, verify global optimality numerically for the reported examples, or weaken the theorem and adjust the claims in the abstract and introduction accordingly.","section":"Appendix B, Theorem B.4; Section 3.2, Eq. (3.5)"},{"comment":"The proof moves from convergence of the surrogate loss values to convergence of the parameter vectors, but this step is not justified. From lim_n ||y_obs - y(.; p_n, S_tilde_n)|| <= ||y_obs - y(.; p*, S)|| one cannot conclude p_n -> p* unless {p_n} is known to lie in a compact set and the true loss is continuous in p (or some equivalent identifiability-plus-compactness condition holds). Assumption 1 (uniqueness of p*) alone does not provide compactness. Moreover, the displayed inequality bounds the surrogate loss with S_tilde_n, not the true loss with S; an additional triangle-inequality step is needed before loss convergence can be related to parameter convergence. Please state and verify the missing compactness and continuity assumptions.","section":"Appendix B, Theorem B.4, final paragraph"},{"comment":"The proof of Theorem B.4 assumes ||S_tilde_n - S||_{L2({t^o_j})} < eps for the observation times, but Stage 1 trains the network to match S at the signal-observation times {t^s_i} using Eq. (3.3). The manuscript does not state that {t^o_j} is contained in {t^s_i}, nor does it provide an argument transferring L2 convergence from one finite grid to another. This is a missing hypothesis in the convergence theorem and should be stated explicitly or proved.","section":"Section 3.2 and Appendix B, Theorem B.4"}],"minor_comments":[{"comment":"The paragraph beginning 'We compared the performance of HADES-NN with six popular optimization algorithms' appears twice verbatim; please remove the duplicate.","section":"Section 2.2"},{"comment":"There are typos in this proposition: 'respoect' should be 'respect' and 'dependnt' should be 'dependent'.","section":"Proposition B.1"},{"comment":"In inequality (14), the constant C2 from the Lipschitz-in-S assumption appears in a term that bounds the difference F(y1,p,S2) - F(y2,p,S2); this term should involve the Lipschitz constant C1 for y (or a generic constant C(T)). This does not affect the argument because the constants are generic, but the displayed formula is misleading.","section":"Lemma B.3, Eq. (14)"},{"comment":"The proof mixes the notations S_tilde_n and S_n for the same smooth approximation; please use one notation consistently.","section":"Appendix B, Theorem B.4"},{"comment":"The displayed formula for the loss function contains apparent typesetting artifacts with extra parentheses and a misplaced backslash; please fix the LaTeX so that the L2 norm notation is unambiguous.","section":"Eq. (3.2)"},{"comment":"No code or data availability statement is included. Given that the method is computational and the yeast and circadian examples use real data, a statement on code and data availability would substantially aid reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The empirical contribution is interesting and likely useful, but the theoretical convergence claim in the abstract and introduction is stronger than what the proof in Appendix B establishes. The revision should either close the global-optimality and compactness gaps or explicitly reframe the theoretical claim as a heuristic convergence guarantee under verified local optimization. I would also encourage the editor to ask for the code and data, as the numerical benchmarks are central to the paper's claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, HADES-NN is a practical, well-demonstrated algorithm for estimating parameters of ODEs with discontinuous inputs; the yeast example with 23 parameters and real experimental data is a strong selling point. Second, the convergence theorem in Appendix B does not actually prove what the paper claims. The proof assumes the LM stage finds the global minimizer of the surrogate loss at each iteration, which LM doesn't guarantee, and the last step moves from loss convergence to parameter convergence without any compactness assumption. That gap is real and load-bearing.\n\nWhat's new: the two-stage warm-started loop—neural network smoothing of the signal with cosine layers, then LM fitting, with the previous parameter estimate feeding the next iteration—is a sensible combination of known pieces (universal approximation, LM, continuation) but I don't know of a prior description of this exact loop for this problem class. The numerical comparisons are thorough: seven methods, three models, repeated random restarts. The circadian example with wearable-device light signals and the yeast mating-response example with only GFP measurements are convincing practical demonstrations.\n\nSoft spots: The proof gap is the main one. The paper should either repair the theorem (possibly via an additional triangle-inequality argument and an explicit compactness/regularity assumption) or soften the claim to a heuristic justification. Also, no code or data is provided, so the reader can't check the implementation; the synthetic experiments are noise-free, which avoids noise sensitivity; and the duplicated paragraph in Section 2.2 should be fixed. NN training details (learning rate, epochs, stopping criteria) are sparse, which seems fixable.\n\nWho this is for: anyone doing parameter inference in non-autonomous ODEs with switch-like inputs—circadian modeling, cell signaling, engineered control—will find the method useful even before the theory is cleaned up.\n\nRecommendation: It deserves a serious referee. I'd send it to review and ask for a major revision: repair or moderate the theorem, release code and data, and supply implementation details. The empirical core is good enough that I wouldn't desk-reject it, but I wouldn't accept it as-is with the current proof.","headline":"HADES-NN is a practical, well-demonstrated estimation method for ODEs with discontinuous inputs, but its convergence proof has real gaps and shouldn't be taken as established.","tokens_in":18383,"tokens_out":2970,"would_cite":false,"duration_ms":34467,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C60","92B05","68T07","93C15","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"By replacing a discontinuous external signal with a sequence of smooth neural-network surrogates, HADES-NN turns a nonsmooth parameter-estimation problem into smooth ones whose minimizers converge to the true minimum, recovering accurate…","keywords":["non-autonomous differential equations","discontinuous external signals","neural network smoothing","parameter estimation","Levenberg-Marquardt","nonsmooth optimization","circadian clock","Lotka-Volterra model"],"falsifier":"On the circadian benchmark, fix the smoothed signal from a late HADES-NN iteration and exhaustively search the four-parameter box for the global minimum of that smoothed loss; if the warm-started Levenberg-Marquardt estimate is not that global minimum, the proof's central premise fails even if the final estimates look accurate.","tokens_in":17316,"feed_emoji":"📈","tokens_out":12002,"duration_ms":124294,"temperature":0.7,"pith_summary":"Fitting a non-autonomous differential equation becomes hard when the external input changes abruptly, because the loss landscape is nonsmooth and standard optimizers stall. The paper's claim is that this difficulty can be bypassed by smoothing: HADES-NN (Harmonic Approximation of Discontinuous External Signals using Neural Networks) uses a neural network with cosine and smooth hidden activations to produce a sequence of smooth approximations $\\tilde S_n(t)$ to the discontinuous signal $S(t)$, and a least-squares optimizer fits the parameters $\\vec p_n$ to each smoothed model. The paper proves that the minimizers of these smoothed loss functions converge to the minimizer of the original nonsmooth loss as the approximations improve. On a Lotka-Volterra system, a human circadian pacemaker driven by wearable light data, and a yeast mating-response network, HADES-NN recovers accurate parameter estimates where L-BFGS, SLSQP, LM, Nelder-Mead, differential evolution, and NeuralODE do not. The payoff would be a general and computationally cheap route to fitting realistic models whose inputs change abruptly.","feed_headline":"Smoothing jumps recovers true parameters in driven ODE models","feed_subtitle":"HADES-NN alternates a smoothing network with least-squares fitting and recovers parameters standard optimizers miss.","key_machinery":"The load-bearing object is the iterated surrogate loss. Instead of minimizing the nonsmooth loss $\\|\\vec y_{\\mathrm{obs}} - \\vec y(\\cdot,\\vec p,S)\\|$ directly, HADES-NN minimizes $\\|\\vec y_{\\mathrm{obs}} - \\vec y(\\cdot,\\vec p,\\tilde S_n)\\|$ for a neural-network-smoothed input $\\tilde S_n(t)$, then uses the resulting $\\vec p_n$ to warm-start the next round of smoothing and fitting. Four ingredients carry the proof: the network architecture makes $\\tilde S_n(t)$ smooth; sensitivity equations make the surrogate loss differentiable in $\\vec p$; a Levenberg-Marquardt convergence bound makes each smooth subproblem solvable; and a Grönwall-type inequality shows trajectory differences are bounded by input differences, so surrogate minima approach the true minimum.","core_discovery":"The central discovery is that the nonsmooth optimization problem caused by a discontinuous input $S(t)$ can be replaced by a sequence of smooth problems that carry the same global minimum. HADES-NN does this in two iterated stages: a fully connected network with a cosine signal layer and five hidden layers of Exponential Linear Units approximates $S(t)$ by a smooth $\\tilde S_n(t)$, and a Levenberg-Marquardt least-squares fit estimates $\\vec p_n$ for the model driven by $\\tilde S_n(t)$. The sequence $\\vec p_n$ converges to the minimizer of the original loss; the proof combines differentiability of each smoothed loss, solvability of each smoothed subproblem, and a Grönwall-type stability bound showing that close signal approximations give close trajectories. In the paper's benchmarks, the method recovered the true parameters of the Lotka-Volterra, circadian, and yeast models, and was the only tested method to do so reliably in the circadian case.","pith_inferences":["Inference: the same alternating smoothing-and-fitting scheme should transfer to any abruptly changing input that can be approximated in $L^2$, such as on-off drug dosing, electrical switching, or sudden policy changes, without changing the core machinery.","Inference: because each iteration yields a point estimate, the sequence of iterates could be interpreted as an empirical distribution for uncertainty quantification, an extension the paper mentions only in passing.","Inference: the proof's reliance on universal approximation suggests the guarantee should extend beyond step-like Markov signals to a broad class of discontinuous inputs with finite $L^2$ norm; this is a testable prediction."],"forward_implications":["With 20 or more observations from the switching Lotka-Volterra system, HADES-NN and NeuralODE both approach the true parameters, and HADES-NN has the smaller mean absolute percentage error.","In the circadian pacemaker benchmark with a wearable light signal, only HADES-NN among the seven tested methods reliably recovers the four individual-specific parameters.","In the yeast mating-response network with 23 parameters, HADES-NN narrows the estimated parameter ranges relative to the original evolutionary-algorithm study while matching the experimental GFP trajectories.","The convergence theorem implies that improving the signal approximation directly improves parameter accuracy, so collecting more signal observations should help as well as more state observations."],"supporting_citations":[{"why":"Supplies the universal-approximation theorem used to assert the neural-network signals $\\tilde S_n$ approach $S$ in $L^2$.","marker":"[51]"},{"why":"Supplies the Levenberg-Marquardt convergence and iteration-complexity bound used to justify solving each smoothed subproblem.","marker":"[49]"},{"why":"Provides the sensitivity-equation result showing the smoothed loss is differentiable in the parameters.","marker":"[47]"},{"why":"Provides the generalized Grönwall inequality used to bound trajectory differences by input differences in Lemma B.3.","marker":"[50]"},{"why":"Supports the claim that deep networks can approximate non-smooth functions effectively, motivating the neural smoothing stage.","marker":"[18]"},{"why":"Defines the human circadian pacemaker model used as the central biomedical benchmark.","marker":"[4]"},{"why":"Defines the yeast mating-response network and supplies the experimental data for that benchmark.","marker":"[6]"},{"why":"Supplies the NeuralODE baseline that HADES-NN is compared against in the parameter-recovery experiments.","marker":"[12]"},{"why":"Defines the Levenberg-Marquardt least-squares algorithm used in the parameter-estimation stage.","marker":"[11]"}],"fun_headline_variants":["HADES-NN smooths abrupt signals to fit ODE parameters","Neural net smoothing recovers parameters from jumpy inputs","Two-step HADES-NN fits models despite signal discontinuities","Smooth the jump, fit the equations: HADES-NN does both","Neural smoothing unlocks ODE fitting from discontinuous signals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that at every iteration the least-squares optimizer actually finds the global minimum of the smoothed model, and that successive smoothed minimizers stay in a bounded region where close loss values force close parameters; if either assumption fails, the proof no longer supports the claimed convergence.","fun_headline_variants_meta":{"raw":{"variants":["HADES-NN smooths abrupt signals to fit ODE parameters","Neural net smoothing recovers parameters from jumpy inputs","Two-step HADES-NN fits models despite signal discontinuities","Smooth the jump, fit the equations: HADES-NN does both","Neural smoothing unlocks ODE fitting from discontinuous signals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1577,"prompt_tokens":899,"completion_tokens":678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":588}},"tokens_in":515,"tokens_out":678,"duration_ms":8386,"temperature":1.0,"reasoning_tokens":588,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:25:15.389893+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the circadian benchmark, fix the smoothed signal from a late HADES-NN iteration and exhaustively search the four-parameter box for the global minimum of that smoothed loss; if the warm-started Levenberg-Marquardt estimate is not that global minimum, the proof's central premise fails even if the final estimates look accurate.","supporting_citations":[{"cited_title":"Convergence and complexity analysis of a levenberg–marquardt algorithm for inverse problems","cited_arxiv_id":null,"evidence_quote":"Supplies the Levenberg-Marquardt convergence and iteration-complexity bound used to justify solving each smoothed subproblem."},{"cited_title":"Sensitivity analysis of ordinary differential equation systems—a direct method","cited_arxiv_id":null,"evidence_quote":"Provides the sensitivity-equation result showing the smoothed loss is differentiable in the parameters."},{"cited_title":"A generalized gronwall inequality and its application to a fractional differential equation","cited_arxiv_id":null,"evidence_quote":"Provides the generalized Grönwall inequality used to bound trajectory differences by input differences in Lemma B.3."},{"cited_title":"Deep neural networks learn non-smooth functions effectively","cited_arxiv_id":null,"evidence_quote":"Supports the claim that deep networks can approximate non-smooth functions effectively, motivating the neural smoothing stage."},{"cited_title":"A simpler model of the human circadian pacemaker","cited_arxiv_id":null,"evidence_quote":"Defines the human circadian pacemaker model used as the central biomedical benchmark."},{"cited_title":"A predictive model of gene expression reveals the role of network motifs in the mating response of yeast","cited_arxiv_id":null,"evidence_quote":"Defines the yeast mating-response network and supplies the experimental data for that benchmark."},{"cited_title":"The levenberg-marquardt algorithm: implementation and theory","cited_arxiv_id":null,"evidence_quote":"Defines the Levenberg-Marquardt least-squares algorithm used in the parameter-estimation stage."}],"review_version":1}