{"id":"72aac264-20e5-4efc-b4e0-02d61be2d7d3","arxiv_id":"2412.17499","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Adding a diffusion-norm penalty to the latent neural SDE loss restores the effective noise level for constant-diffusion stochastic time series, demonstrated on bistable and multistable conceptual models.","lead":"Latent neural SDEs, an AI approach for learning random dynamics from noisy time series, systematically underestimate the noise level in data. This paper adds a penalty term to the training loss that restores the correct effective noise for constant-diffusion systems, which matters for modeling stochastic climate and other multistable dynamics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim is not yet supported: γ is tuned on the same summary statistics that are later reported, and those statistics do not identify the diffusion function; Fig. 11 shows the trained diffusion still differs from the EBM.","rationale":"The reader's conditional verdict already identifies the constant-diffusion scope and the manual tuning of γ as key weaknesses. My concern goes one step further: even inside the constant-diffusion regime, the reported evaluation metrics do not establish that the diffusion component is captured. The paper's own direct comparison in Fig. 11 shows a mismatch between the trained and true diffusion functions, and the paper explains this away by appealing to KM coefficients. But finite-Δt KM coefficients are not the true diffusion coefficients unless the limit Δt→0 is taken; no such extrapolation is reported. Combined with the fact that γ is selected by matching the same transition-rate statistic that is later reported, the evidence for the abstract's central claim is substantially weaker than the text suggests. This does not require rejection: the method may still be useful as a calibration procedure for constant-diffusion systems, and the authors are transparent about the global scalar form of L_G and the failure on multiplicative noise. However, the paper should either add a direct diffusion-function comparison with proper uncertainty quantification and a held-out γ-selection protocol, or rephrase the central claim to describe matching of selected summary statistics rather than accurate capture of the diffusion component. I therefore keep the reader's conditional verdict unchanged.","tokens_in":15370,"tokens_out":7197,"duration_ms":78665,"concrete_test":"Extract the trained diffusion gθ from the γ = 200 EBM run and compute the normalized L2 error between gθ(x)^2 and the data-generating constant diffusion σ^2 over the observed state range. In addition, simulate long trajectories from both the trained prior SDE and the EBM and estimate the second KM coefficient at Δt ∈ {0.005, 0.01, 0.02, 0.04}, then extrapolate to Δt → 0. If the extrapolated model diffusion does not approach σ^2 within error bars, the central claim should be weakened to 'matches selected summary statistics after calibration on those statistics.'","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that adding the noise penalty L_G enables the latent SDE to 'accurately capture the diffusion component of the data.' The supporting evidence consists of transition rates and finite-Δt Kramers–Moyal coefficients matching between generated trajectories and data. Two problems make this evidence non-identifying for the diffusion. First, γ is selected in Sec. VI by inspecting transition rates on the same test set that is then used for evaluation, so the reported match is a calibration outcome rather than a predictive estimate. Second, these statistics do not uniquely determine the diffusion: for a bistable SDE, the transition rate depends on both the drift barrier and the noise amplitude, and KM coefficients estimated at Δt = 0.01 include finite-step and drift corrections unless extrapolated to Δt→0. The paper's own Fig. 11 shows that at the selected γ = 200, the trained drift and diffusion functions are 'significantly different' from the EBM's, even though the finite-Δt KM plots look similar. That is direct evidence that the trained diffusion function is not the data's diffusion function, contradicting the abstract's wording. The contribution is better described as a one-parameter global dial that can match chosen summary statistics in constant-diffusion systems, not as an accurate estimation of the diffusion component. The explicit admission in Sec. VI should therefore be treated as a limitation of the central claim, not as a harmless explanatory remark. The absence of error bars and the absence of a held-out γ-selection protocol make it impossible to tell whether the matching is robust or an artifact of tuning on the evaluation statistic.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the systematic underestimation of the diffusion (noise) term in latent neural stochastic differential equations (SDEs). It first reproduces this underestimation on a bistable energy balance model (EBM) and analyzes the trade-off between log-likelihood and KL divergence in the loss. The authors then propose adding a noise penalty L_G = ∫ ||gθ(u(t), t)|| dt to the training objective, controlled by a hyperparameter γ, and show that for the EBM and several other constant-diffusion systems, a suitably chosen γ brings the prior SDE's transition rate, marginals, and finite-Δt Kramers-Moyal coefficients closer to the data. The paper claims that this enables the latent neural SDE to accurately capture the diffusion component of the data, and it discloses that the method is only accurate for constant diffusion and fails for multiplicative noise.","tokens_in":15640,"tokens_out":3345,"duration_ms":30801,"significance":"If the central claim held, the proposed penalty would be a simple, practical remedy for a known limitation of latent neural SDEs, with direct application to stochastic climate and energy-balance models. The paper is clearly written, makes its code and data publicly available, and includes several illustrative examples (EBM, FitzHugh-Nagumo, Ornstein-Uhlenbeck, triple-well). However, the evidence as presented does not support the strong claim of 'accurately capturing the diffusion component': the headline metric (transition rate) is used both to select γ and to evaluate the fit, and the paper's own direct comparison of the learned diffusion function with the true diffusion (Fig. 11) shows substantial disagreement. The contribution is better characterized as a one-parameter global noise dial that can match selected summary statistics in constant-diffusion systems, which is useful but weaker than the abstract's claim.","major_comments":[{"comment":"The hyperparameter γ is selected by inspecting transition rates on the same data set that is later used for evaluation ('Having found a seemingly reasonable value γ = 200 by looking at the transition rates'). The reported transition-rate match is therefore a calibration outcome, not an independent prediction. The abstract's claim that the model 'accurately captures the diffusion component' is not supported by this metric, since the selection criterion and the reported statistic coincide. Please either select γ on a validation split and evaluate on a held-out set, or explicitly frame the reported match as calibration and temper the claim accordingly.","section":"Sec. VI, Fig. 7"},{"comment":"The paper's own direct comparison shows that at γ = 200 the learned drift and diffusion functions are 'significantly different' from the EBM's, even though the finite-Δt KM coefficients (Fig. 10) look similar. The KM coefficients are computed at Δt = 0.01 without extrapolation to Δt → 0, so the second KM coefficient includes finite-step and drift corrections and is not the true diffusion function. Consequently, matching finite-Δt KM coefficients does not demonstrate that the diffusion component is accurately captured. Either provide Δt → 0 extrapolated KM coefficients or an alternative direct estimate of the diffusion, or revise the central claim to say that the method matches finite-step effective statistics rather than the diffusion function.","section":"Sec. VI, Fig. 11"},{"comment":"The paper states that 'the hyperparameter γ only fits diffusion terms globally in its current formulation' and that the result 'is thus only accurate for data sets with constant diffusion.' Appendix F confirms that the method fails for linear (multiplicative) noise. This is an honest and important limitation, but it directly contradicts the abstract's unqualified claim that the model 'accurately captures the diffusion component of the data.' The contribution should be framed as a global noise penalty for constant-diffusion systems, not as a general diffusion estimator.","section":"Sec. VI, Appendix F"},{"comment":"All reported results, including the β-sweep in Fig. 3 and the γ-sweep in Fig. 7, appear to be single training runs without multiple seeds or error bars. Since the paper makes quantitative claims about matching transition rates and Wasserstein distances, it is important to show that the results are stable across random initializations; otherwise, the observed matches could be fortuitous. Please add at least a small number of independent runs and report the spread, or clearly state that the results are single-seed demonstrations.","section":"Sec. IV and Sec. VI"}],"minor_comments":[{"comment":"The phrase 'W e' appears in the Introduction with a stray space; please correct the typo.","section":"Abstract / Introduction"},{"comment":"The caption reads 'Comparison of Wasserstein distane...'; 'distane' should be 'distance'.","section":"Fig. 4 caption"},{"comment":"The sentence 'The first two Kramers-Moyal coefficients are quantification of an SDE's drift and diffusion terms given data output by the system' is grammatically awkward; consider rephrasing.","section":"Sec. IV D"},{"comment":"The reference to 'GitHub issues of packages that implement them' is vague; a specific issue link or repository would be helpful.","section":"Sec. V"},{"comment":"The caption 'Replaced actual trained latent neural SDE's diffusion with constant value σlatent...' is hard to parse; consider rewriting for clarity.","section":"Fig. 6 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper has a useful empirical observation and a simple remedy, and the authors are transparent about the constant-diffusion limitation. However, the abstract and introduction substantially overstate what is demonstrated: the evidence does not support the claim that the diffusion component is 'accurately captured.' A revision that recalibrates the claims, adds independent validation of the diffusion estimate (or explicitly frames the procedure as calibration of summary statistics), and includes multiple seeds would make the contribution sound. This is within the manuscript's scope and can be fixed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper gives the clearest account I've seen of why latent neural SDEs shrink the diffusion: a KL/likelihood balance that pushes the noise down, demonstrated with a clever replaced-diffusion experiment. Second, the proposed fix is a global penalty on the diffusion norm, and it can make generated trajectories match transition rates and Kramers-Moyal coefficients for constant-diffusion systems. That's a real, useful result for people who want to fit bistable dynamics with a simple tweak.\n\nThe paper deserves credit for the diagnosis and for shipping code across several examples (EBM, OU, FHN, triple-well) plus an honest appendix showing the method fails for multiplicative noise. The claim in the abstract, though, is overstrong. 'Accurately captures the diffusion component' is not what the evidence shows. Figure 11 compares the learned diffusion function directly to the ground truth at the selected gamma and they are clearly different. What matches are the finite-time KM coefficients and transition rates, which are summary statistics of the path distribution, not the diffusion function itself. The stress-test is right about the tuning: gamma is chosen by inspecting transition rates on the same test set used for evaluation, so the headline match is a calibration outcome. There are no error bars anywhere, so it's hard to know how robust any of the fitted curves are.\n\nThat said, the paper is unusually candid. It states that gamma only fits globally and that the method is only for constant diffusion. It shows the linear-noise failure itself. The KM match and marginals provide some independent support, and the transition-rate tuning is disclosed rather than hidden.\n\nMy bottom line: the mechanism analysis is a genuine contribution, and the penalty is a simple dial that practitioners may find useful. But 'accurate estimation of the diffusion component' should be softened to 'matching user-selected path statistics under constant diffusion.' The missing seeds and the tuned-on-eval-statistic protocol are fixable in revision but need to be addressed.\n\nRecommendation: worth a serious referee. Not a desk reject. The authors should be pushed for repeated seeds, a held-out gamma selection procedure, and a revised abstract that says what the method actually delivers.","headline":"A clear diagnosis of why latent neural SDEs underestimate noise, plus a simple global penalty that can match path statistics for constant-diffusion systems; the abstract overclaims, but the core analysis is worth refereeing.","tokens_in":16199,"tokens_out":1811,"would_cite":true,"duration_ms":17656,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Latent neural SDEs underestimate the noise in stochastic time series, and adding a penalty on the diffusion term to the loss restores the correct noise level for constant-diffusion systems.","keywords":["latent neural SDE","noise estimation","diffusion term","stochastic time series","Kramers-Moyal coefficients","bistable dynamics","variational inference","noise regularization"],"falsifier":"Use the method on data generated by an SDE with linear state-dependent diffusion, such as $dT = (a_1 + a_2 \\tanh(T-T_0) - a_3 T^4)dt + 0.135 T \\, dB_t$, and compare the recovered diffusion function with the true one. The paper's own appendix reports that the recovered diffusion is nearly constant while the truth is linear, which marks exactly where the central claim stops holding.","tokens_in":15167,"feed_emoji":"🎲","tokens_out":7101,"duration_ms":59268,"temperature":0.7,"pith_summary":"Latent neural stochastic differential equations (SDEs) are generative models for stochastic time series, but they systematically underestimate the noise level in the data, which makes them miss the true probability distribution of multistable systems. The paper identifies the cause: the diffusion size emerges from a trade-off between the reconstruction likelihood, which prefers zero diffusion, and the KL divergence to the prior, so the trained diffusion is a measure of model uncertainty rather than data noise. It proposes adding a noise penalty $L_G = \\int \\|g_\\theta(u(t),t)\\|\\,dt$, weighted by a hyperparameter $\\gamma$, to the training objective, and shows that with a suitable $\\gamma$ the latent neural SDE recovers the correct diffusion level. This is demonstrated on a bistable energy balance model, where transition rates, marginals, and the first two Kramers-Moyal coefficients of the generated data match the ground truth, and on FitzHugh-Nagumo, Ornstein-Uhlenbeck, and triple-well systems. The method is explicitly limited to data with constant diffusion.","feed_headline":"Adding a noise penalty lets latent neural SDEs match true noise levels","feed_subtitle":"A tunable penalty on the diffusion term restores correct transition rates and Kramers-Moyal coefficients in generated trajectories.","key_machinery":"The central object is the noise penalty term $L_G = \\int_0^T \\|g_\\theta(u(t),t)\\|\\, dt$ added to the latent neural SDE objective, giving $\\max_{\\theta,\\phi}\\mathbb{E}_{x_{\\text{data}}}(L_E - \\beta L_{KL} + \\gamma L_G)$. It is a scalar, computed by integrating the Euclidean norm of the diffusion network's output along the posterior trajectory, so it taxes the total amount of noise the model injects. In the loss, it counteracts the likelihood's pressure toward zero diffusion, and its strength is controlled by the hyperparameter $\\gamma$, which the paper tunes by matching summary statistics such as transition rates.","core_discovery":"The paper claims that the previously observed noise underestimation in latent neural SDEs is not an incidental training failure but a structural consequence of the loss function, and that it can be corrected by a simple additive term. In the objective $\\max_{\\theta,\\phi}\\mathbb{E}_{x_{\\text{data}}}(L_E - \\beta L_{KL})$, the diffusion $g_\\theta$ is pulled in two directions: the likelihood term favors a small diffusion because it lets the posterior track the data closely, while the KL term favors a larger diffusion proportional to the discrepancy between posterior and prior drifts. The trained diffusion therefore reflects this balance point, not the data's noise. Adding $\\gamma L_G$ with $L_G = \\int_0^T \\|g_\\theta(u(t),t)\\|\\, dt$ shifts this balance so that, for constant-diffusion data, the prior SDE's diffusion size matches the data noise. The paper supports this by replacing the trained diffusion with constant values and showing the optimum of the loss components aligns with the final trained diffusion, and by demonstrating matching transition rates and Kramers-Moyal coefficients on several test systems. It also reports that the same penalty fails for multiplicative noise because it only adjusts the global diffusion magnitude.","pith_inferences":["If the loss-balance explanation is right, the same underestimation should appear in any variational latent SDE with shared prior and posterior diffusion, and a state-dependent penalty $\\int \\gamma(u)\\|g_\\theta\\|\\, dt$ would be a natural next step to handle multiplicative noise.","The transition-rate mismatch between data and prior SDE could serve as a cheap diagnostic for noise underestimation in applied settings, before any hyperparameter tuning.","Because the data are normalized before training, the constant-diffusion assumption applies to the normalized coordinates; a system with homogeneous noise in physical units may acquire state-dependent effective diffusion once normalized, which would narrow the method's practical scope.","An automatic selection of $\\gamma$, for example by targeting a summary statistic, would remove the manual tuning and make the method more usable; the paper leaves this as future work."],"forward_implications":["With a correctly tuned $\\gamma$, the prior SDE reproduces the data's transition rates and marginal distributions on the bistable energy balance model, not just the mean path.","The first two Kramers-Moyal coefficients (the estimated drift and diffusion) of generated trajectories match those of the data for constant-diffusion systems, so the generative model can be used as a simulator of the stochastic dynamics.","The penalty can be tuned from summary statistics of the data alone, without knowing the underlying drift or diffusion.","The method transfers to a two-dimensional FitzHugh-Nagumo system observed only in one dimension, to an Ornstein-Uhlenbeck process, and to a triple-well model.","For multiplicative noise the global penalty cannot recover the diffusion function, so constant diffusion is a boundary of the method's validity."],"supporting_citations":[{"why":"introduces latent neural SDEs and the ELBO-style objective ($L_E - \\beta L_{KL}$) that this paper extends with the noise penalty.","marker":"[7]"},{"why":"documents the noise-underestimation behaviour of latent neural SDEs that motivates the investigation.","marker":"[8]"},{"why":"supplies the path-space KL divergence between prior and posterior SDEs used in the training objective.","marker":"[23]"},{"why":"provides the variational autoencoder / ELBO interpretation that frames the latent neural SDE objective.","marker":"[24]"},{"why":"provides the Kramers-Moyal coefficient estimation used to compare drift and diffusion of generated trajectories with the data.","marker":"[32]"},{"why":"supplies the FitzHugh-Nagumo parameterization used as a second test system in the experiments.","marker":"[33]"}],"fun_headline_variants":["Noise penalty fixes latent SDEs' diffusion estimates","Simple loss tweak corrects SDE noise bias","Latent SDEs learn true noise with added penalty","One penalty term restores SDE noise accuracy","Tune noise penalty to match true diffusion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The data's diffusion is constant (or effectively constant) across state space, so one global number $\\gamma$ can correct the estimated noise; the paper itself notes the result is only accurate for constant-diffusion data and shows the penalty fails for linear noise.","fun_headline_variants_meta":{"raw":{"variants":["Noise penalty fixes latent SDEs' diffusion estimates","Simple loss tweak corrects SDE noise bias","Latent SDEs learn true noise with added penalty","One penalty term restores SDE noise accuracy","Tune noise penalty to match true diffusion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000511,"raw_usage":{"total_tokens":2458,"prompt_tokens":890,"completion_tokens":1568,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":1493}},"tokens_in":506,"tokens_out":1568,"duration_ms":9056,"temperature":1.0,"reasoning_tokens":1493,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:26:52.834877+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Use the method on data generated by an SDE with linear state-dependent diffusion, such as $dT = (a_1 + a_2 \\tanh(T-T_0) - a_3 T^4)dt + 0.135 T \\, dB_t$, and compare the recovered diffusion function with the true one. The paper's own appendix reports that the recovered diffusion is nearly constant while the truth is linear, which marks exactly where the central claim stops holding.","supporting_citations":[{"cited_title":"Li , author T.-K","cited_arxiv_id":null,"evidence_quote":"documents the noise-underestimation behaviour of latent neural SDEs that motivates the investigation."},{"cited_title":"Lohmann \\ and\\ author P","cited_arxiv_id":null,"evidence_quote":"supplies the FitzHugh-Nagumo parameterization used as a second test system in the experiments."}],"review_version":1}