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REVIEW 4 major objections 6 minor 36 references

An Adaptive Genetic Algorithm for determining optimal structures for atomic clusters

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims an adaptive genetic algorithm finds the lowest-energy Lennard-Jones cluster structures for N=2 to 20 and reports energies whose only limit is machine precision.

desk verdict Eq. (4) inverts the fitness ranking, breaking the method as written; the Poisson-disk/graph-crossover ideas are still worth a referee's time. read the letter →

arxiv 2411.18087 v1 pith:ADU4MN5P submitted 2024-11-27 physics.comp-ph

Brandon Willnecker , Mervlyn Moodley This is my paper · ORCID
classification physics.comp-ph
keywords adaptivegeneticalgorithmLennard-JonesclustersglobaloptimizationatomicclusterstructureBoltzmannselectionbasinhoppingmachineprecisioncrossover
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

The paper proposes an adaptive genetic algorithm (AGA) for locating the global minimum energy structure of atomic clusters and demonstrates it on Lennard-Jones clusters of 2 to 20 particles. The method combines a normalized fitness function, Boltzmann-style selection, elitism, a graph-based crossover that builds child clusters from parent geometries, and Gaussian mutation whose step size adapts during the search. The authors report that the AGA recovers the known lowest-energy clusters and reproduces the Wales–Doye benchmark energies to at least six decimal places, while carrying additional digits so that precision is limited only by floating-point arithmetic. If the claim is right, the algorithm offers a generally applicable optimizer for pair-potential many-body problems that is competitive with basin hopping.

What carries the argument

The load-bearing components are: the normalized fitness F(x_i) = (E_max - E(x_i))/(E_max - E_min), which maps the best energy in a generation to 1 and the worst to 0; Boltzmann selection with probability p(x_i) = $Z^{{-1}}$ $e^{{-β F(x_i)}}$, whose free parameter β is used to sharpen the search; a graph-theoretic crossover in which the union of two parent clusters is represented as a complete graph whose edges carry two-particle energies, and a child is an N-vertex walk with small edge sum; and adaptive Gaussian mutation with variance that can shrink as the population matures. Elitism carries 10% of the fittest individuals unchanged. Together these operators are meant to let the population explore broadly early and fine-tune late, which is the 'adaptive' mechanism the paper credits for the reported convergence.

What would settle it

Instrument the selection operator in a fresh implementation exactly as Eq. (4) is written and count how often each fitness rank is selected; if the least-fit individuals are chosen most often yet the N=2 to 20 minima still converge, then the published formula cannot describe the tested method, and if the code uses the opposite sign, the machine-precision claim must be re-verified for the corrected selector.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a fully adaptive genetic algorithm—one that adjusts its mutation scale and selection pressure during the run—can locate the global minima of Lennard-Jones clusters for all sizes from N = 2 through N = 20. The energies computed by the AGA agree with the Wales–Doye reference values to the six decimal places that the reference table reports, and the authors provide additional digits, asserting that the residual error is at the level of machine precision. Because the same framework is described for any pairwise potential, the result is presented as a general optimization method for many-body configuration problems, not just a Lennard-Jones-specific solver.

Load-bearing premise

The reported convergence assumes the selection step actually prefers lower-energy, higher-fitness clusters, but the written Boltzmann formula gives the worst cluster the largest selection probability, so the algorithm that was run must differ from the one described in Eq. (4).

Editorial extensions

If this is right

  • If the AGA works as claimed, it provides reference-quality energies for Lennard-Jones clusters with more significant digits than the standard benchmark table, which can be used to test other optimizers.
  • Because the method is stated for any pairwise potential, the same code could be adapted to Morse, Sutton–Chen, or embedded-atom clusters without changing the genetic machinery.
  • The reported convergence data (roughly 40, 150, 5000, and 6000 generations for N=5, 10, 15, 20) give a concrete scaling baseline that future adaptive searches could improve upon.
  • The graph-based crossover offers a reusable way to combine two three-dimensional geometries while preserving favorable local packing from both parents.

Reading between the lines

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

  • A natural next test is to apply the same AGA to N=21–30 Lennard-Jones clusters, where structural transitions and deeper local-minima traps appear, to see whether the machine-precision claim survives beyond the demonstrated range.
  • One could define explicit schedules for β and the mutation variance in terms of population diversity, turning the qualitative 'adaptive' description into a reproducible protocol and allowing fair compute-budget comparisons with basin hopping and simulated annealing.
  • The graph-theoretic crossover could be generalized to non-pairwise potentials by weighting edges with incremental many-body energy contributions, although the paper does not attempt this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes an adaptive genetic algorithm (AGA) for locating the lowest-energy structures of Lennard-Jones clusters. The algorithm combines a Poisson-disk initialization, a normalized energy-based fitness function, Boltzmann selection, elitism, a graph-based crossover that forms child clusters via minimum-edge-sum walks on the union graph of two parents, and Gaussian mutation with possibly generation-dependent standard deviations. Results are reported for clusters with N = 2 to 20 particles and are compared with the benchmark values of Wales and Doye. The authors claim that the AGA achieves energies that are 'more accurate' than those in the literature and that the precision of the results is limited only by machine precision.

Significance. If the method worked as described, it would be a useful demonstration of a genetic algorithm for cluster structure optimization. However, the scope is very modest: the global minima for N ≤ 20 Lennard-Jones clusters are long-established benchmarks, and the paper provides no timing, scalability, or comparison against other modern methods. The paper's main strengths are that it reproduces the known benchmark energies to six decimal places and that the algorithm is conceptually simple. Yet the significance is undermined by a load-bearing internal inconsistency in the selection operator, a conflation of precision and accuracy in the central claim, and the absence of any specification of the supposedly adaptive parameters. As presented, the paper does not support its claims of novelty or superior accuracy.

major comments (4)
  1. [II.C, Eq. (4)] The selection probability is defined as p(x_i) = Z^{-1} e^{-β F(x_i)}, with F normalized so that the fittest individual has F = 1 and the least fit has F = 0. For any β > 0, this assigns the smallest selection probability to the fittest individual and the largest to the least fit, directly contradicting the sentence immediately above Eq. (4) and the subsequent discussion that increasing β increases the likelihood of lower-energy systems being chosen. Selection is the core mechanism that biases reproduction toward fitter individuals, so the algorithm as written would be expected to drift away from the global minimum, not converge to it. The reported results can be attributed to this method only if the implemented code used a different selection rule, but no code or correction is provided.
  2. [II.C, Eq. (5)] The normalization constant Z is written as a sum over j of e^{-β F(x_i)}, using x_i inside the sum. It should sum over x_j to be consistent with the definition of Z as the denominator of p(x_i). This is an indexing error, but it is part of the same selection formula whose sign is already inverted, and both issues must be corrected before the method can be reproduced.
  3. [III, Table I and Conclusion] The claim that the AGA results are 'more accurate' than those of Wales and Doye and that precision is 'only governed by machine precision' is not supported. The values in Table I agree with the Wales-Doye benchmark to six decimal places and then show additional digits, but this is simply higher numerical precision in evaluating the energy of a structure, not evidence of greater accuracy. No energy lower than the literature global minimum is reported, no multiple-run statistics or uncertainties are given, and no independent verification of the printed digits beyond the benchmark comparison is presented. The conclusion overstates what the data show.
  4. [II.C, II.E, and III] The paper describes the algorithm as 'adaptive' because β and σ_i can change, but it never specifies the adaptation schedule, initialization values, update rules, or even the ranges used in the reported runs. Since the adaptation mechanism is the central methodological novelty claimed in the title and abstract, omitting these details makes the results irreproducible and prevents any assessment of whether the adaptive behavior contributes to the reported convergence. The population size, maximum generation count, mutation probability, and Poisson-disk minimum distance r0 are also never stated, so the experimental setup is incomplete.
minor comments (6)
  1. [II.C, Eq. (5)] The index error in the normalization sum should be fixed by writing Z = sum_j e^{-β F(x_j)}; this is a typographical/indexing issue separate from the sign error in Eq. (4).
  2. [II.A] The phrase 'Poisson disk dampling algorithm' appears to be a typo for 'Poisson disk sampling algorithm'.
  3. [II.D] The crossover description says to 'Generate an N-vertex walk with a minimum edge sum' on a complete graph of 2N vertices, but no algorithm or complexity bound is given for this step, and it is unclear how the random starting vertex interacts with the minimum-edge-sum requirement. This is an underspecified operation that needs a concrete definition for reproducibility.
  4. [III, Table II] The 'Convergence Point' column is not defined; the text mentions convergence rates but does not state the tolerance or stopping criterion used to identify when a run has converged.
  5. [III, Fig. 3] The convergence plots are described qualitatively, but no axis labels, population sizes, parameter values, or multiple-run ensembles are shown, so the reader cannot judge the robustness of the convergence behavior.
  6. [Introduction] The reference formatting is inconsistent, e.g., Ref. [16] is given as 'D. Wales and J. Doye (1997), J. Phys. Chem. A, 101, 5111 (1997)' with a doubled year; the reference list should be standardized.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the GA optimizes an external Lennard-Jones potential and is validated against the independent Wales–Doye benchmark; the Eq. (4) sign issue is a correctness problem, not circularity.

full rationale

Walking the derivation chain: the fitness F(x_i) in Eq. (1) is a normalized external energy, not a quantity defined by the target answer. Selection, crossover, and mutation operate on this fitness; the reported table is compared to the independent Wales–Doye [16] values rather than being read off from them. The additional decimal digits are produced by the energy evaluation, and matching to six decimals against an external benchmark is a genuine test, so the central claim does not reduce to its inputs. The only self-citation, Ref. [28] for reduced units, is definitional and non-load-bearing. The adaptive parameters beta and sigma_i are left unspecified, but nothing in the text says they were fitted to the Wales–Doye energies. The apparent inversion of selection in Eq. (4) (F=1 for best, F=0 for worst, p proportional to e^{-beta F}) would make the written algorithm anti-selective; this is a serious reproducibility/correctness defect but not circularity, because the output would not be equivalent to the input by construction—it would simply be unexplained. Accordingly, no circular steps are identified.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim rests on hand-chosen control parameters and unproven heuristic design choices. The most consequential is the selection operator, where the written formula contradicts the stated goal. Because the adaptive schedules are unspecified, the reported energies cannot be reproduced or attributed to the described algorithm.

free parameters (5)
  • β (selection inverse temperature) = not specified
    Called a free parameter in Eq. (4); controls selection pressure, but no schedule or value is given despite the 'adaptive' claim.
  • σ_i (mutation standard deviations) = not specified
    Gaussian mutation widths δ_i ~ N(0, σ_i); an adaptation rule over generations is mentioned but never defined.
  • Elitism factor = 0.1
    Explicitly set to 10% of the population with no sensitivity analysis or justification.
  • Population size and maximum generations = not specified
    Needed to reproduce the algorithm; only convergence generation counts for a sample appear in Table II.
  • Poisson disk minimum separation r0 = not specified
    Enforced minimum interparticle distance in initialization; the value affects initial population diversity and is not given.
assumptions (3)
  • domain assumption The Lennard-Jones pairwise potential in Eq. (3) is the objective for atomic cluster ground-state search.
    Standard model choice for the demonstration, but the claimed generalization to any pair potential is not demonstrated.
  • ad hoc to paper An N-vertex walk with minimum edge sum on the union graph of two parents creates a child likely fitter than at least one parent.
    Stated in Sec. II.D with no proof, statistical test, or comparison to standard crossover operators.
  • domain assumption Boltzmann selection with normalized fitness maps to a canonical distribution that favors low-energy clusters.
    Invoked in Sec. II.C; however, Eq. (4) as written favors low fitness, so the assumption is internally contradicted.

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Cite this review

Pith. "Pith review of An Adaptive Genetic Algorithm for determining optimal structures for atomic clusters." pith.science (2026). https://pith.science/paper/ADU4MN5P

@misc{pith2026241118087,
  author       = {Pith},
  title        = {Pith review of: An Adaptive Genetic Algorithm for determining optimal structures for atomic clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADU4MN5P}},
  note         = {Machine review of arXiv:2411.18087}
}
read the original abstract

The implementation of adaptive genetic algorithms (AGA) for optimization problems has proven to be superior than many other methods due to its nature of producing more robust and high quality solutions. Considering the complexity involved in many-body simulations, a novel AGA is proposed for applications to such systems and is specifically used to determine the lowest energy structures of various sized atomic clusters. For demonstrative purposes, we apply our method to various sized Lennard-Jones clusters and show that our results are more accurate than those found in the literature employing different methods.

Figures

Figures reproduced from arXiv: 2411.18087 by the authors.

Figure 1
Figure 1. FIG. 1. A complete graph with 6 vertices and 15 edges formed [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The Lennard-Jones particle clusters for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Plots of the AGA convergence for 5, 10, 15 and 20 particle clusters. The horizontal line shows the minimum energy [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

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