{"id":"0d528f19-222d-4634-9667-fe4f1711a2bd","arxiv_id":"2411.16200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A neural network surrogate replaces the explicit energy function in high-index saddle dynamics, and nearby surrogate saddle points are proven to approximate true ones when the network is accurate.","lead":"Scientists train a neural network to imitate an energy function, then run saddle-point dynamics on the learned model to find transition states and build solution landscapes. This matters because many real systems have no explicit energy formula, only data, so the method could bring saddle-search tools to data-driven molecular and biological problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 3.3, the C^3 surrogate-accuracy condition underpinning Theorems 3.7-3.8, is never verified, and value-only training does not control Hessian error; the practical reliability claim rests on an unmeasured quantity.","rationale":"The reader's weakest assumption, Assumption 3.3, is exactly the load-bearing point. Theorems 3.7 and 3.8 are internally consistent: the contraction argument is valid, and the convergence rate follows if the surrogate is C^3-close to the true energy. The problem is that the paper never establishes, empirically or theoretically, that the trained networks satisfy the needed uniform gradient, Hessian, and Hessian-Lipschitz error bounds. Universal approximation in C^3 requires controlling derivatives during training, but the loss function (2.3) only fits values (with an optional gradient term), so it does not constrain Hessian errors. The numerical sections report saddle-point coordinates but not derivative-error diagnostics, leaving the theorem's quantitative predictions disconnected from the experiments. This is not an internal inconsistency, but it is a correctness risk for the claimed guarantee. The ribosomal experiment's 'new transition states' illustrate the practical hazard: the local theorem cannot rule out spurious critical points created by surrogate error, so those states require external validation. My recommendation agrees with the reader's conditional verdict: the core method is plausible and the theory is sound conditional on a strong, unverified assumption; public artifacts and derivative-error diagnostics would be needed before acceptance. I set verdict_should_be to UNCHANGED because this concern does not move the verdict away from CONDITIONAL.","tokens_in":16250,"tokens_out":7969,"duration_ms":196355,"concrete_test":"Use the §4.1 toy model (E known). After training the 3x128 tanh net with 5000 points and 30000 epochs, sample a dense grid (e.g., 200x200) in a ball of radius δ=1 around the true saddle (1.2842, 3.4484) and compute max ||∇E_NN−∇E||, max ||∇²E_NN−∇²E||, and an estimate of the Lipschitz constant of ∇²E_NN−∇²E via finite differences. Compare with the eps needed so that 4 eps/mu is consistent with the observed saddle displacement (~4.5e-3) and with the rate used in Thm 3.8. If the measured Hessian error exceeds this eps (or fails to decay as data increase), Assumption 3.3 is not satisfied and the reported experiments do not fall under the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result is a clean perturbation argument, but its precondition is Assumption 3.3: the trained surrogate must satisfy uniform bounds epsilon on ||grad E_delta||, ||Hess E_delta||, and the Hessian Lipschitz constant in a neighborhood of the saddle. The only justification offered is Remark 3.4, which invokes the universal approximation theorem plus 'thorough training.' That does not follow: the loss (2.3) is value-based (and optionally gradient-based), so it does not control Hessian errors; a network can fit sampled values while oscillating between samples, producing large derivative errors. The experiments never measure these C^3 errors, so the quantitative bounds 4 epsilon/mu and the convergence rate of Thm 3.8 are not connected to any verified quantity. The ribosomal 'new transition states' (Sec 4.5) are particularly exposed: since Thm 3.7 is local, spurious critical points far from true saddles are not excluded, so claims of new conformational states need independent validation. The absence of code/data further prevents a check of whether the reported surrogate accuracy actually meets the hypothesis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes NN-HiSD, a surrogate-based variant of the high-index saddle dynamics method in which a neural network replaces the explicit energy function. The method uses automatic differentiation for gradient and Hessian information, optionally adds heavy-ball or Nesterov momentum, and is intended to locate saddle points and construct solution landscapes for systems with no closed-form energy. The main theoretical result (Theorems 3.7 and 3.8) shows, under an assumption that the surrogate is C^3-close to the true energy, that a nearby surrogate saddle exists and that NN-HiSD converges locally to it at a rate inherited from the known HiSD convergence theory. Numerical experiments cover toy potentials, the Müller-Brown potential, a modified Rosenbrock function, the alanine dipeptide free-energy landscape, and a ribosomal assembly pseudo-energy landscape.","tokens_in":16516,"tokens_out":10851,"duration_ms":103946,"significance":"If the main theorem's hypothesis can be verified, the paper would provide a clean and useful conditional guarantee for neural-network-based saddle search: a perturbation argument bounds the surrogate saddle displacement by O(epsilon/mu), and the local convergence proof correctly reduces to the established HiSD theorem. The momentum-accelerated variants and the use of automatic differentiation in place of a dimer method are natural and well-motivated. However, the paper's central practical claim depends on Assumption 3.3, which is never checked, and the data-driven experiments lack uncertainty quantification and independent validation. The theoretical skeleton is sound, but the manuscript currently overstates the reliability of the method for systems without explicit energies.","major_comments":[{"comment":"Assumption 3.3 is the sole bridge between the trained network and the quantitative guarantees, but it is never checked in any experiment. The loss in Eq. (2.3) is a value-matching loss with an optional gradient term, so it does not control the Hessian errors and Hessian-Lipschitz constants required by parts (i)-(iii). Remark 3.4 invokes the universal approximation theorem and 'thorough training', but that theorem is a density statement and does not imply that a particular network trained on finitely many samples has uniform C^3 accuracy on U(x*,delta). Consequently, the bound ||x*_NN - x*|| <= 4epsilon/mu in Theorem 3.7 and the rate in Theorem 3.8 are not connected to any measurable training quantity. The authors should either measure epsilon (gradient, Hessian, and Hessian-Lipschitz errors in the relevant region) for each experiment, or add a training scheme with explicit derivative penalties and report the resulting epsilon; otherwise the claim after Theorem 3.8 that convergence velocity 'predominantly remains unaltered' is unsupported.","section":"Section 3, Assumption 3.3 and Remark 3.4"},{"comment":"The claim that S1 and S3 are 'new potential conformational states' is not supported by the presented results. Theorems 3.7 and 3.8 are local statements: they guarantee closeness to a true saddle only when Assumption 3.3 holds in a neighborhood of that saddle and when the dynamics starts nearby. They do not exclude spurious critical points of the surrogate elsewhere, and the surrogate is trained on only 119 data points with no train/test error reported and no comparison to an independent method. Before presenting S1 and S3 as new biological findings, the authors should validate them by independent means, for example additional MD or cryo-EM analysis, or at least a repeated-training stability check showing that the states persist across network initializations.","section":"Section 4.5"},{"comment":"The data-driven experiments report no error bars, no repeated-training statistics, and no held-out validation of the surrogate. Table 1 reports single runs for each noise level, and the alanine and ribosome landscapes are each based on one trained network; the ribosome case uses 119 points, making overfitting a serious concern. Since the central claim is 'effectiveness and reliability' for systems without explicit energies, the manuscript should provide repeated-training or bootstrap errors, distance-to-reference comparisons for known states, and a discussion of the surrogate error on held-out data.","section":"Sections 4.4-4.5"},{"comment":"The theorem statements and the convergence plots concern convergence to the surrogate saddle x*_NN, not to the original saddle x*. The final error to the true saddle is ||x(n)-x*|| <= ||x(n)-x*_NN|| + ||x*_NN-x*||, where the second term is only bounded by 4epsilon/mu under Assumption 3.3. The experiments plot ||x(n)-x*_NN|| (for example Fig. 8) and compare with known saddle coordinates in Tables 1-2, but they never measure or report epsilon, so the paper does not establish a quantified accuracy of the NN-HiSD landscape with respect to the original energy.","section":"Section 3, Theorems 3.7-3.8; Section 4"}],"minor_comments":[{"comment":"The expression for D has a typographical error: it should read D = 4epsilon/(mu + sqrt(mu^2 - 32Mepsilon)), not D = 4epsilon/mu + sqrt(mu^2 - 32Mepsilon).","section":"Section 3, Eq. (3.6)"},{"comment":"The organization paragraph does not match the actual section numbering: the convergence analysis is in Section 3 and the numerical experiments in Section 4, not the other way around.","section":"Section 1"},{"comment":"There are minor grammatical issues in the abstract, such as 'to approximates' and 'a neural network-based surrogate model to approximates the energy function'; these should be corrected.","section":"Abstract"},{"comment":"Stating that data and code are 'available from the corresponding author upon reasonable request' is not sufficient for reproducibility; the authors should provide a repository with scripts, trained models, and data-processing steps.","section":"Code and Data Availability"},{"comment":"The abbreviation ADAD is used without definition; the text should state explicitly that it denotes applying automatic differentiation twice to compute the Hessian.","section":"Section 4.1"},{"comment":"The gradient-correction experiment is not fully reproducible because the penalty coefficient lambda_2 and the exact training protocol for the additional 5000 epochs are not specified.","section":"Section 4.2, Fig. 6(b)"},{"comment":"The sentence 'We calculate all possible critical points of the alanine dipeptide model' is too strong without a completeness argument; the downward/upward search procedure is heuristic, so it should be rephrased as 'all critical points found by our search procedure'.","section":"Section 4.4"}],"recommendation":"major_revision","confidential_remarks":"I see no circularity concern: Theorems 3.7 and 3.8 are new conditional results that reduce to the known HiSD convergence theorems in [26] and [27]. The main gap is that the key assumption (C^3-accurate surrogate) is not verified and the real-data experiments are presented without uncertainty quantification. These issues are fixable within the paper's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does a sensible thing: replace an explicit energy with a trained NN surrogate and run HiSD on it. The new theoretical piece is Theorem 3.7, a clean contraction-mapping argument showing that if the surrogate is uniformly close in gradient and Hessian (Assumption 3.3), its saddle lies within 4 epsilon/mu of the true one. That proof is fine. What is not fine is that the assumption is never checked, and the experiments don't supply the missing evidence.\n\nWhat is genuinely new: the parametric surrogate formulation (energy as a function of x and alpha) and the epsilon-proximity bound. The momentum accelerations are imported from [27] and the convergence backbone from [26]; that's fine, they cite correctly. The numerical studies are broad: toy models, Muller-Brown, Rosenbrock, alanine dipeptide, and a ribosomal assembly dataset. The alanine dipeptide experiment is a reasonable demonstration that a surrogate can be used to build a solution landscape from noisy free-energy data.\n\nThe load-bearing weakness is Assumption 3.3. The loss (2.3) is value-based (with an optional gradient term), so it does not control Hessian error. Remark 3.4's appeal to universal approximation plus 'thorough training' does not bridge that gap. A network can fit values while oscillating between samples, producing large second derivatives. The paper never reports or estimates empirical Hessian error, so the 4 epsilon/mu bound is not connected to any measured quantity. This matters most for the ribosomal example: with 119 points and regularization, the surrogate is a rough fit, and Theorem 3.7 is local. Calling S1 and S3 'new potential conformational states' is not supported; they could be artifacts. The Nesterov variant has no convergence proof (only heavy-ball is covered by Corollary 3.9), and the experiments treat it empirically, which is okay but should be labeled. Code and data are not released, only 'available upon request,' which makes it hard to check the surrogate accuracy claims.\n\nThe stress-test note is on target; I don't think it overstates. Who this is for: people working on saddle search and solution landscapes who want a data-driven entry point. With public artifacts and a real check of the Hessian error (or a theory of when value training controls it), the method would be more credible. As is, I'd send it to review but expect major revision.","headline":"Useful extension of HiSD to data-only energies via NN surrogates, with a clean but conditional proximity theorem; the unverified C^3 surrogate assumption and thin experimental validation mean the practical claims outrun the evidence.","tokens_in":17029,"tokens_out":2153,"would_cite":false,"duration_ms":45077,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M05","65B99","65L20","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"NN-HiSD: a neural-network surrogate makes high-index saddle-point search possible without explicit energy functions.","keywords":["high-index saddle dynamics","neural network surrogate","solution landscape","Nesterov acceleration","heavy-ball method","saddle point search","convergence analysis","alanine dipeptide"],"falsifier":"Take one of the trained surrogates used in the numerical sections, evaluate $\\nabla E_\\delta$ and $\\nabla^2 E_\\delta$ at the reported saddle point and its neighborhood against a high-accuracy reference, and check whether $\\|\\nabla E_\\delta\\|\\le\\varepsilon$ and $\\|\\nabla^2 E_\\delta\\|\\le\\varepsilon$ hold with the $\\varepsilon$ required by condition (3.3). If the gradient or Hessian error exceeds that bound, or the surrogate Hessian has a different number of negative eigenvalues than the true Hessian, the predicted $4\\varepsilon/\\mu$ closeness and the convergence guarantee do not apply.","tokens_in":16015,"feed_emoji":"⛰️","tokens_out":9405,"duration_ms":75173,"temperature":0.7,"pith_summary":"This paper proposes NN-HiSD, a method for finding saddle points and building solution landscapes when the energy function is not available in closed form or is expensive to evaluate. The idea is to train a neural network surrogate for the energy, then run the existing high-index saddle dynamics (HiSD) on that surrogate, using automatic differentiation for gradients and Hessians. The paper proves that a sufficiently accurate surrogate has a saddle point within $4\\varepsilon/\\mu$ of the true one, and that NN-HiSD converges to it at essentially the same linear rate as HiSD on the exact energy. Momentum versions (heavy-ball and Nesterov) are included and shown to preserve convergence. If correct, the method turns data-only energy samples into saddle-point and transition-path predictions on systems such as alanine dipeptide and ribosomal assembly intermediates.","feed_headline":"Surrogate networks find saddles without the energy formula","feed_subtitle":"NN-HiSD: a trained neural net can replace an explicit energy and still converge to nearby saddles and landscapes.","key_machinery":"The load-bearing device is the decomposition $E_{\\mathrm{NN}}=E+E_\\delta$ with the three error bounds (Hessian Lipschitz constant, Hessian norm, and gradient norm all at most $\\varepsilon$), combined with the fixed-point map $g(x)=x-(\\nabla^2 E_{\\mathrm{NN}}(x^*))^{-1}\\nabla E_{\\mathrm{NN}}(x)$ used to prove existence and closeness of the surrogate saddle. On the algorithmic side, the machinery is the HiSD update $\\dot{x}=\\beta(I-2\\sum_{i=1}^k v_i v_i^\\top)F(x)$ and $\\dot{v}_i=-\\gamma(I-v_i v_i^\\top-2\\sum_{j<i}v_j v_j^\\top)G(x)v_i$, which orients motion along the $k$ unstable directions and is run entirely on the surrogate using automatic differentiation, with optional heavy-ball or Nesterov momentum. The same machinery builds solution landscapes by downward and upward searches between saddles of adjacent index.","core_discovery":"The central claim is that replacing an explicit energy $E$ by a trained surrogate $E_{\\mathrm{NN}}=E+E_\\delta$ does not break saddle-point search, provided $E_\\delta$ is uniformly small in gradient and Hessian on a neighborhood of the saddle. Under that assumption (Assumption 3.3), Theorem 3.7 shows via a contraction-mapping argument that the surrogate has a critical point $x^*_{\\mathrm{NN}}$ within distance $4\\varepsilon/\\mu$ of the true saddle $x^*$, with the same Morse index; Theorem 3.8 then shows the discrete NN-HiSD iteration converges to $x^*_{\\mathrm{NN}}$ at rate $(1-2/(\\tilde{\\kappa}+3))^n$, where $\\tilde{\\kappa}=(L+\\varepsilon)/(\\mu-\\varepsilon)$, so a well-trained surrogate leaves the convergence speed essentially unchanged. The paper also proves the analogous result for the heavy-ball accelerated variant. The numerical sections support the claim on toy potentials, the Müller-Brown potential, a 7D Rosenbrock function, alanine dipeptide free-energy data, and a 119-map ribosomal assembly dataset.","pith_inferences":["Inference: the proved bound is controlled by gradient and Hessian error rather than value error, so adding derivative information to the training loss should be the standard practice; the theory predicts this is what makes $4\\varepsilon/\\mu$ small.","Inference: the ribosomal experiment's transition states that were 'not observed in the experiments' are a testable prediction: cryo-EM or molecular-dynamics follow-up could check whether those states appear as real conformational intermediates.","Inference: the same surrogate-plus-dynamics recipe could extend to non-gradient systems by fitting the force field directly, which the paper lists as future work; the convergence proof would then need a different fixed-point argument.","Inference: the bound suggests a practical stopping criterion: train until the estimated gradient and Hessian error in the region of interest is below the desired saddle-location tolerance times $\\mu/4$."],"forward_implications":["For any system where energy values (and optionally gradients) can be sampled, NN-HiSD yields saddle points whose error is proportional to the surrogate's local gradient and Hessian error, so training quality directly controls landscape accuracy.","The convergence-rate bound is essentially unchanged from exact-energy HiSD, and Corollary 3.9 extends the accelerated rate to the heavy-ball variant; the Nesterov variant is validated numerically.","Since the surrogate is a smooth tanh network, automatic differentiation supplies exact Hessian-vector products, removing the dimer finite-difference step in the classical HiSD implementation.","The alanine dipeptide and ribosomal-assembly experiments show construction of full solution landscapes, including index-2 saddles and the connections between saddles and minima, from noisy data alone."],"supporting_citations":[{"why":"Defines the high-index saddle dynamics equations that NN-HiSD runs on the surrogate model.","marker":"[47]"},{"why":"Supplies the discrete HiSD convergence theorem (Lemma 3.6) that Theorem 3.8 transfers to the surrogate saddle.","marker":"[26]"},{"why":"Provides the heavy-ball accelerated HiSD convergence result used to prove Corollary 3.9 for NN-HiSDHb.","marker":"[27]"},{"why":"Invoked through the universal approximation theorem to justify that a sufficiently trained network can satisfy the required C^3 approximation assumption.","marker":"[5]"},{"why":"Supplies the automatic differentiation machinery used to obtain gradients and Hessians of the surrogate without the dimer approximation.","marker":"[1]"},{"why":"Provides the bacterial ribosomal assembly pseudo-energy landscape and experimental cluster assignments used in Section 4.5.","marker":"[42]"}],"fun_headline_variants":["Neural nets replace energy functions in saddle search","No explicit energy? Neural surrogate still finds saddles","Train a net to hunt saddle points, drop the formula","Surrogate NN finds saddles and landscapes, no formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proximity and convergence argument rests on Assumption 3.3: the trained network must approximate the energy with gradient and Hessian errors uniformly bounded by $\\varepsilon$ in a neighborhood of the saddle, plus a Hessian Lipschitz bound. The paper takes value-only training on sampled points, together with the universal approximation theorem and adequate training, to be sufficient for this third-derivative-level accuracy.","fun_headline_variants_meta":{"raw":{"variants":["Neural nets replace energy functions in saddle search","No explicit energy? Neural surrogate still finds saddles","Train a net to hunt saddle points, drop the formula","Surrogate NN finds saddles and landscapes, no formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1730,"prompt_tokens":942,"completion_tokens":788,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":724}},"tokens_in":558,"tokens_out":788,"duration_ms":8207,"temperature":1.0,"reasoning_tokens":724,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:23:07.181923+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take one of the trained surrogates used in the numerical sections, evaluate $\\nabla E_\\delta$ and $\\nabla^2 E_\\delta$ at the reported saddle point and its neighborhood against a high-accuracy reference, and check whether $\\|\\nabla E_\\delta\\|\\le\\varepsilon$ and $\\|\\nabla^2 E_\\delta\\|\\le\\varepsilon$ hold with the $\\varepsilon$ required by condition (3.3). If the gradient or Hessian error exceeds that bound, or the surrogate Hessian has a different number of negative eigenvalues than the true Hessian, the predicted $4\\varepsilon/\\mu$ closeness and the convergence guarantee do not apply.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the high-index saddle dynamics equations that NN-HiSD runs on the surrogate model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the discrete HiSD convergence theorem (Lemma 3.6) that Theorem 3.8 transfers to the surrogate saddle."},{"cited_title":"Accelerated high-index saddle dynamics method for searching high-index saddle points","cited_arxiv_id":"2310.04656","evidence_quote":"Provides the heavy-ball accelerated HiSD convergence result used to prove Corollary 3.9 for NN-HiSDHb."},{"cited_title":"C ybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems, 2 (1989), pp","cited_arxiv_id":null,"evidence_quote":"Invoked through the universal approximation theorem to justify that a sufficiently trained network can satisfy the required C^3 approximation assumption."},{"cited_title":"B aydin, B","cited_arxiv_id":null,"evidence_quote":"Supplies the automatic differentiation machinery used to obtain gradients and Hessians of the surrogate without the dimer approximation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bacterial ribosomal assembly pseudo-energy landscape and experimental cluster assignments used in Section 4.5."}],"review_version":1}