Pith. sign in

REVIEW 4 major objections 7 minor 49 references

Neural Network-based High-index Saddle Dynamics Method for Searching Saddle Points and Solution Landscape

T0 review · 4 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read NN-HiSD: a neural-network surrogate makes high-index saddle-point search possible without explicit energy functions.

desk verdict 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. read the letter →

arxiv 2411.16200 v1 pith:SWR3KMVV submitted 2024-11-25 cs.LG

Yuankai Liu , Lei Zhang , Jin Zhao This is my paper · ORCID
classification cs.LG MSC 37M0565B9965L2068T07
keywords high-indexsaddledynamicsneuralnetworksurrogatesolutionlandscapeNesterovaccelerationheavy-ballmethodpointsearchconvergenceanalysisalaninedipeptide
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $\mu/4$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

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.

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 (4)
  1. [Section 3, Assumption 3.3 and Remark 3.4] 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.
  2. [Section 4.5] 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.
  3. [Sections 4.4-4.5] 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.
  4. [Section 3, Theorems 3.7-3.8; Section 4] 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.
minor comments (7)
  1. [Section 3, Eq. (3.6)] 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).
  2. [Section 1] 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.
  3. [Abstract] 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.
  4. [Code and Data Availability] 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.
  5. [Section 4.1] The abbreviation ADAD is used without definition; the text should state explicitly that it denotes applying automatic differentiation twice to compute the Hessian.
  6. [Section 4.2, Fig. 6(b)] 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.
  7. [Section 4.4] 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'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence and proximity theorems are conditional perturbation results whose inputs (surrogate accuracy epsilon) are not fitted to the outputs (saddle positions), and the imported HiSD convergence lemmas are independent prior theorems.

full rationale

The paper's central theoretical claim, Theorems 3.7 and 3.8, is a conditional perturbation statement. Under Assumption 3.3, where the surrogate is written as E_NN = E + E_delta with uniform bounds on the gradient, Hessian, and Hessian Lipschitz error, the proof constructs g(x) = x - (Hess E_NN(x*))^{-1} grad E_NN(x) and uses the Banach fixed-point theorem to show the existence of a surrogate saddle x*_NN with ||x*_NN - x*|| <= 4 epsilon / mu. This is a genuine stability argument: epsilon is a hypothesis about approximation quality, not a parameter fitted to the saddle locations that are later reported. The convergence result in Theorem 3.8 then applies Lemma 3.6 from reference [26] to the surrogate E_NN; the cited lemma is a published, parameter-free convergence theorem whose stated assumptions (Assumption 3.1 and 3.2 on the energy function) do not include the NN-HiSD target result. Similarly, Corollary 3.9 invokes Theorem 4.6 from [27]. Although those references share an author with the present paper, they are independent analytical results, not self-referential assertions, so their use does not constitute circularity. The numerical experiments train surrogates on energy values and then run HiSD on the trained surrogate; the computed saddle points are downstream outputs, not fitted inputs. The main weakness identified by a skeptical reading is Remark 3.4, which claims Assumption 3.3 is 'easily satisfied' via universal approximation and 'thorough training.' This is an unverified sufficiency claim: value-based L2 training does not by itself control Hessian errors or guarantee C^3 approximation, so the quantitative bounds in Theorem 3.7 and 3.8 may not be connected to a measured quantity in practice. That is a correctness and verification gap, not a circularity. No equation in the paper reduces by construction to its own input, and no fitted parameter is renamed as a prediction. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The core theory is conditional: it assumes the true energy is regular near an index-k saddle, the eigenvector iterates are exact, and the neural network is C^3-close to the energy. The convergence rates come from external theorems [26, 27] that share an author with this paper but are independently derived. No physical entities are introduced; free parameters are mostly experimental hyperparameters and step sizes chosen by hand.

free parameters (5)
  • HiSD step size beta_n = 0.05 (2D toy), 3e-4 (Muller-Brown), 1e-4 (Rosenbrock, alanine)
    Chosen by hand per experiment; convergence theorem requires beta = 2/(L+mu), but experiments use different values without sensitivity analysis.
  • Momentum coefficient gamma and Nesterov restart interval = gamma = 0.8; restarts every 5, 15, 20, 40, or 500 steps
    Hand-tuned; the paper notes the two Nesterov choices give almost identical results but does not study sensitivity to the restart frequency.
  • Neural network architecture and training hyperparameters = 3-5 hidden layers x 128/256 neurons; 5k-40k points; 5k-30k epochs; lr = 0.001
    Chosen per experiment; the only systematic sensitivity study is the amount of training data in Figure 6.
  • Gradient penalty coefficient lambda_2 = not stated
    The loss L = L1 + lambda_2 L2 includes an unspecified penalty used in Section 4.2, so the exact training objective is not reproducible.
  • Ribosome regularization coefficient = 4e-3
    Introduced to counter overfitting with only 119 data points; no systematic selection procedure is reported.
assumptions (7)
  • standard math Banach fixed-point theorem
    Used in Theorem 3.7 to establish existence of the surrogate saddle point x*_NN.
  • standard math Universal approximation theorem [5]
    Invoked in Remark 3.4 to justify Assumption 3.3, though C^3 approximation is stronger than standard universal approximation.
  • domain assumption Assumption 3.1: true energy is regular near an index-k saddle with Hessian Lipschitz bound and spectral gap
    Needed for the perturbation argument and for applying the known HiSD convergence theorem.
  • domain assumption Assumption 3.2: eigenvectors v_i are exact eigenvectors of the Hessian at each iterate
    The NN-HiSD convergence analysis assumes exact eigenvectors, while the numerical implementations use approximate eigen-solvers SIRQIT and LOBPCG.
  • ad hoc to paper Assumption 3.3: surrogate ENN = E + E_delta has small gradient and Hessian errors bounded by epsilon
    Core condition for Theorems 3.7 and 3.8; it is asserted via Remark 3.4 but never verified in experiments.
  • standard math Lemma 3.6 (HiSD convergence theorem from [26])
    External, published theorem by an overlapping author group; used as a black box to prove NN-HiSD convergence.
  • standard math Theorem 4.6 in [27] (heavy-ball convergence)
    External theorem from an arXiv preprint by overlapping authors; used for Corollary 3.9 on NN-HiSDHb.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Neural Network-based High-index Saddle Dynamics Method for Searching Saddle Points and Solution Landscape." pith.science (2026). https://pith.science/paper/SWR3KMVV

@misc{pith2026241116200,
  author       = {Pith},
  title        = {Pith review of: Neural Network-based High-index Saddle Dynamics Method for Searching Saddle Points and Solution Landscape},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWR3KMVV}},
  note         = {Machine review of arXiv:2411.16200}
}
read the original abstract

The high-index saddle dynamics (HiSD) method is a powerful approach for computing saddle points and solution landscape. However, its practical applicability is constrained by the need for the explicit energy function expression. To overcome this challenge, we propose a neural network-based high-index saddle dynamics (NN-HiSD) method. It utilizes neural network-based surrogate model to approximates the energy function, allowing the use of the HiSD method in the cases where the energy function is either unavailable or computationally expensive. We further enhance the efficiency of the NN-HiSD method by incorporating momentum acceleration techniques, specifically Nesterov's acceleration and the heavy-ball method. We also provide a rigorous convergence analysis of the NN-HiSD method. We conduct numerical experiments on systems with and without explicit energy functions, specifically including the alanine dipeptide model and bacterial ribosomal assembly intermediates for the latter, demonstrating the effectiveness and reliability of the proposed method.

Figures

Figures reproduced from arXiv: 2411.16200 by the authors.

Figure 1
Figure 1. Overview of the framework of the NN-HiSD method. (a) Function values at only a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Trajectory of true potential and learned potential. (a) true energy potential [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. ∥x (n) − x ∗ NN∥2 of the surrogate model-based HiSD method with respect to the iteration number. (a) dimer method to calculate G(x)vi . (b) ADAD method to calculate G(x). In practical scenarios, the training data we acquire typically includes some degree of noise, thus necessitating an assessment of the surrogate model’s resistance to noise. To this end, we utilize a set of 5000 constant data points within the regio… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Solution landscape for both true model and surrogate model. (a) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Trajectory of true potential and learned potential for the MB function. (a) True energy [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Boxplot illustrating saddle point prediction errors across various amounts of training [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Plots of ∥x (n) − x ∗ NN∥ for MB and MMB potential in various NN-HiSD procedure. (a) MB potential. (b) MMB potential. 4.3. Rosenbrock-type Function. In the third experiment, we analyze a well-known test problem in optimization: the d-dimensional Rosenbrock-type functio…
Figure 8
Figure 8. Figure 8: Plots of ∥x (n) − x ∗ NN∥ in the first 1000 iterations of various NN-HiSD methods. 4.4. Alanine Dipeptide Model. In this example, we apply the NN-HiSD method to the Φ-Ψ dihedrals of alanine dipeptide, a 22-dimensional Molecular Dynamic model ( [PITH_FULL_IMAGE:figures…
Figure 9
Figure 9. Figure 9: Diagram of the alanine dipeptide (CH3–CONH–CHCH3–CONH–CH3) [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: (a) Contour and saddle points of the surrogate model. (b) Solution landscape of [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: (a) Global pseudo-energy landscape of bacterial ribosomal assembly intermediates. [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Solution landscape calculated by the surrogate model. [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 47 canonical work pages

  1. [27]

    Y . Luo, X. Z heng, and L. Zhang, Accelerated high-index saddle dynamics method for searching high-index saddle points, arXiv.2310.04656, (2023)

    work page Pith review arXiv 2023

  2. [26]

    Y . Luo, X. Z heng, X. C heng, and L. Zhang, Convergence analysis of discrete high-index saddle dynamics , This manuscript is for review purposes only. NN-HISD FOR SOLUTION LANDSCAPE 19 SIAM Journal on Numerical Analysis, 60 (2022), pp. 2731–2750

    work page 2022

  3. [1]

    B aydin, B

    A. B aydin, B. P earlmutter, A. R adul, and J. S iskind, Automatic differentiation in machine learning: A survey, Journal of Machine Learning Research, 18 (2018), pp. 1–43

    work page 2018

  4. [2]

    B onfanti and W

    S. B onfanti and W. Kob, Methods to locate saddle points in complex landscapes , The Journal of Chemical Physics, 147 (2017), p. 204104

    work page 2017

  5. [3]

    B urton, G

    R. B urton, G. H uang, M. D augherty, T. C alderone, and T. Oas, The energy landscape of a fast-folding protein mapped by Ala Gly substitutions, Nature structural biology, 4 (1997), pp. 305–10

    work page 1997

  6. [4]

    C heng, L

    X. C heng, L. Lin, W. E, P. Zhang, and A.-C. Shi, Nucleation of ordered phases in block copolymers, Physical Review Letters, 104 (2010), p. 148301

    work page 2010

  7. [5]

    C ybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems, 2 (1989), pp

    G. C ybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems, 2 (1989), pp. 303–314

    work page 1989

  8. [6]

    D aneshmand, J

    H. D aneshmand, J. Kohler, A. Lucchi, and T. Hofmann, Escaping saddles with stochastic gradients, in Inter- national Conference on Machine Learning, PMLR, 2018, pp. 1155–1164

    work page 2018

Show all 49 references
  1. [7]

    Y . N. Dauphin, R. Pascanu, C. G ulcehre, K. C ho, S. G anguli, and Y . Bengio, Identifying and attacking the saddle point problem in high-dimensional non-convex optimization, in Advances in Neural Information Processing Systems, vol. 27, Curran Associates, Inc., 2014

    2014

  2. [8]

    J. H. D a vis, Y . Z. Tan, B. Carragher, C. S. Potter, D. Lyumkis, and J. Williamson, Modular assembly of the bacterial large ribosomal subunit, Cell, 167 (2016), pp. 1610–1622.e15

    2016

  3. [9]

    W. E and E. Vanden-Eijnden, Transition-path theory and path-finding algorithms for the study of rare events., Annual review of physical chemistry, 61 (2010), pp. 391–420

    2010

  4. [10]

    W. E and X. Zhou, The gentlest ascent dynamics, Nonlinearity, 24 (2011), p. 1831

    2011

  5. [11]

    F ukumizu, S

    K. F ukumizu, S. Yamaguchi, Y .ichi Mototake, and M. Tanaka, Semi-flat minima and saddle points by embed- ding neural networks to overparameterization, arXiv, abs/1906.04868 (2019)

    2019 arXiv

  6. [12]

    G ould, C

    N. G ould, C. O rtner, and D. Packwood, A dimer-type saddle search algorithm with preconditioning and linesearch, Mathematics of Computation, 85 (2016), pp. 2939–2966

    2016

  7. [13]

    S. G u, H. Wang, and X. Zhou, Active learning for saddle point calculation, Journal of Scientific Computing, 93 (2022)

    2022

  8. [14]

    Y . Han, Y . Hu, P. Z hang, and L. Zhang, Transition pathways between defect patterns in confined nematic liquid crystals, Journal of Computational Physics, 396 (2019), pp. 1–11

    2019

  9. [15]

    Y . Han, J. Y in, Y . Hu, A. M ajumdar, and L. Zhang, Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–de Gennes model, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477 (2021), p. 20210458

    2021

  10. [16]

    Y . Han, J. Yin, P. Zhang, A. Majumdar, and L. Zhang, Solution landscape of a reduced Landau–de Gennes model on a hexagon, Nonlinearity, 34 (2021), pp. 2048–2069

    2021

  11. [17]

    H eidrich and W

    D. H eidrich and W. Quapp, Saddle points of index 2 on potential energy surfaces and their role in theoretical reactivity investigations, Theoretica Chimica Acta, 70 (1986), pp. 89–98

    1986

  12. [18]

    H enkelman and H

    G. H enkelman and H. J´onsson, A dimer method for finding saddle points on high dimensional potential sur- faces using only first derivatives, The Journal of Chemical Physics, 111 (1999), pp. 7010–7022

    1999

  13. [19]

    H ornik, Approximation capabilities of multilayer feedforward networks , Neural Networks, 4 (1991), pp

    K. H ornik, Approximation capabilities of multilayer feedforward networks , Neural Networks, 4 (1991), pp. 251–257

    1991

  14. [20]

    H ornik, M

    K. H ornik, M. S tinchcombe, and H. White, Multilayer feedforward networks are universal approximators , Neural Networks, 2 (1989), pp. 359–366

    1989

  15. [21]

    C. J in, P. Netrapalli, and M. I. Jordan, Accelerated gradient descent escapes saddle points faster than gradi- ent descent, in Conference On Learning Theory, PMLR, 2018, pp. 1042–1085

    2018

  16. [22]

    K ingma and J

    D. K ingma and J. Ba, Adam: A method for stochastic optimization , International Conference on Learning Representations, (2014)

    2014

  17. [23]

    A. V . Kny azev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conju- gate gradient method, SIAM Journal on Scientific Computing, 23 (2001), pp. 517–541

    2001

  18. [24]

    Li and J

    Y . Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear pdes, SIAM Journal on Scientific Computing, 23 (2001), pp. 840–865

    2001

  19. [25]

    L ongsine and S

    D. L ongsine and S. McCormick, Simultaneous rayleigh-quotient minimization methods for ax =λbx, Linear Algebra and its Applications, 34 (1980), pp. 195–234

    1980

  20. [28]

    M ehta, Finding all the stationary points of a potential-energy landscape via numerical polynomial- homotopy-continuation method, Physical Review E, 84 (2011), p

    D. M ehta, Finding all the stationary points of a potential-energy landscape via numerical polynomial- homotopy-continuation method, Physical Review E, 84 (2011), p. 025702

    2011

  21. [29]

    M ilnor, Morse Theory

    J. M ilnor, Morse Theory. (AM-51), Volume 51, Princeton University Press, Princeton, 1963

    1963

  22. [30]

    Mironov, Y

    V . Mironov, Y . Alexeev, V . K. Mulligan, and D. Fedorov, A systematic study of minima in alanine dipeptide, Journal of Computational Chemistry, 40 (2019), pp. 297 – 309

    2019

  23. [31]

    Nesterov, A method for solving the convex programming problem with convergence rate o(1/k2), Proceed- ings of the USSR Academy of Sciences, 269 (1983), pp

    Y . Nesterov, A method for solving the convex programming problem with convergence rate o(1/k2), Proceed- ings of the USSR Academy of Sciences, 269 (1983), pp. 543–547

    1983

  24. [32]

    O’D onoghue and E

    B. O’D onoghue and E. Candes, Adaptive restart for accelerated gradient schemes, Foundations of Computa- tional Mathematics, 15 (2015), pp. 715–732

    2015

  25. [33]

    J. C. P hillips, R. B raun, W. Wang, J. G umbart, E. Tajkhorshid, E. V illa, C. C hipot, R. D. S keel, L. K al´e, and K. Schulten, Scalable molecular dynamics with NAMD , Journal of Computational Chemistry, 26 (2005), pp. 1781–1802

    2005

  26. [34]

    P oly ak, Some methods of speeding up the convergence of iteration methods , Ussr Computational Mathe- matics and Mathematical Physics, 4 (1964), pp

    B. P oly ak, Some methods of speeding up the convergence of iteration methods , Ussr Computational Mathe- matics and Mathematical Physics, 4 (1964), pp. 1–17

    1964

  27. [35]

    https: //pytorch.org/docs/stable/generated/torch.optim.lr scheduler.StepLR.html, 2023

    P yTorch, Steplr−pytorch 2.3 documentation . https: //pytorch.org/docs/stable/generated/torch.optim.lr scheduler.StepLR.html, 2023

    2023

  28. [36]

    W. R en, E. Vanden-Eijnden, P. Maragakis, and W. E, Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide , The Journal of Chemical Physics, 123 (2005), p. 134109

    2005

  29. [37]

    S amanta, M

    A. S amanta, M. E. T uckerman, T.-Q. Y u, and W. E, Microscopic mechanisms of equilibrium melting of a solid, Science, 346 (2014), pp. 729–732

    2014

  30. [38]

    B. S hi, Y . Han, J. Yin, A. Majumdar, and L. Zhang, Hierarchies of critical points of a Landau-de Gennes free energy on three-dimensional cuboids, Nonlinearity, 36 (2023), pp. 2631–2654

    2023

  31. [39]

    B. S hi, Y . Han, and L. Zhang, Nematic liquid crystals in a rectangular confinement: Solution landscape, and bifurcation, SIAM Journal on Applied Mathematics, 82 (2022), pp. 1808–1828

    2022

  32. [40]

    W ang, L

    W. W ang, L. Zhang, and P. Zhang, Modelling and computation of liquid crystals, Acta Numerica, 30 (2021), pp. 765–851

    2021

  33. [41]

    Wang and J

    Y . Wang and J. Li, Phase field modeling of defects and deformation , Acta Materialia, 58 (2010), pp. 1212– 1235

    2010

  34. [42]

    Z. W u, E. Chen, S. Zhang, Y . Ma, and Y . Mao, Visualizing conformational space of functional biomolecular complexes by deep manifold learning, International Journal of Molecular Sciences, 23 (2022)

    2022

  35. [43]

    J. Y in, Z. Huang, Y . Cai, Q. Du, and L. Zhang, Revealing excited states of rotational Bose–Einstein conden- sates, The Innovation, 5 (2024), p. 100546

    2024

  36. [44]

    J. Y in, K. Jiang, A.-C. Shi, P. Zhang, and L. Zhang, Transition pathways connecting crystals and quasicrys- tals, Proceedings of the National Academy of Sciences, 118 (2021)

    2021

  37. [45]

    J. Y in, Y . Wang, J. Z. Y . Chen, P. Zhang, and L. Zhang, Construction of a pathway map on a complicated energy landscape, Physical Review Letters, 124 (2020), p. 090601

    2020

  38. [46]

    J. Y in, B. Y u, and L. Zhang, Searching the solution landscape by generalized high-index saddle dynamics , Science China Mathematics, 64 (2020), pp. 1801–1816

    2020

  39. [47]

    J. Y in, L. Zhang, and P. Zhang, High-index optimization-based shrinking dimer method for finding high-index saddle points, SIAM Journal on Scientific Computing, 41 (2019), pp. A3576–A3595

    2019

  40. [48]

    Z hang, L.-Q

    L. Z hang, L.-Q. Chen, and Q. Du, Morphology of critical nuclei in solid-state phase transformations, Physical Review Letters, 98 (2007), p. 265703

    2007

  41. [49]

    Z hou, L

    T. Z hou, L. Z hang, P. Zhang, A.-C. S hi, and K. Jiang, Nucleation and phase transition of decagonal qua- sicrystals, The Journal of Chemical Physics, 161 (2024), p. 164503. This manuscript is for review purposes only

    2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.