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An effective nucleation reaction coordinate only needs to separate basins; it need not match the committor pointwise.

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2026-07-10 11:18 UTC pith:JGCR55XZ

load-bearing objection Clean Ising demonstration that basin separation, not pointwise committor fidelity, is what MSMs need for nucleation rates; both learned p_B and LGCS recover brute-force rates.

arxiv 2607.08207 v1 pith:JGCR55XZ submitted 2026-07-09 physics.comp-ph

Learned Committors as Reaction Coordinates for Nucleation Rates

Hubert J. Naguszewski , David Quigley This is my paper
classification physics.comp-ph
keywords committornucleation ratesMarkov state modelIsing modelreaction coordinatecollective variablegeometric cluster sizemachine learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Finding good low-dimensional reaction coordinates for nucleation is hard. The ideal choice is the committor—the probability a configuration reaches the stable phase before returning to the metastable one—but computing it has long been too expensive for routine use. This paper trains a convolutional neural network on brute-force committor labels for the two-dimensional Ising model, then uses that learned proxy as the coordinate of a Markov state model. The network recovers brute-force magnetisation-reversal rates across a range of temperatures and fields. Surprisingly, the simple largest geometric cluster size does the same, even though it is a poor pointwise predictor of the true committor. The practical claim is therefore clear: for rate calculations it is enough that a coordinate cleanly separates the two basins; it does not have to preserve the committor for every microstate. That distinction matters for how collective variables are chosen in rare-event nucleation simulations more generally.

Core claim

Markov state models built on a neural-network proxy for the committor recover brute-force nucleation rates for magnetisation reversal in the two-dimensional Ising model across thermodynamic conditions. The largest geometric cluster size recovers the same rates even though it fails as a pointwise committor predictor, showing that reliable basin separation, not pointwise fidelity, is the requirement for accurate rate estimation.

What carries the argument

p_B-NN, a convolutional neural network trained on brute-force committor labels and used directly as the reaction coordinate of a Markov state model whose rates are obtained from mean first-passage times.

Load-bearing premise

The chosen basin boundaries and the lag times at which the longest implied timescale plateaus are assumed to give Markovian dynamics whose mean first-passage times equal the true nucleation rates.

What would settle it

Find a thermodynamic condition of the same Ising model at which an MSM on the largest geometric cluster size (or on p_B-NN) yields a nucleation rate that disagrees with an independent brute-force mean first-passage time by more than statistical error.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper trains a convolutional neural network (p_B-NN) as a high-accuracy proxy for the committor of magnetisation reversal in the 2D Ising model, using brute-force shooting labels (n=4096) on microstates sampled from nucleating trajectories. Separate MSMs are then built with p_B-NN and with largest geometric cluster size (LGCS) as the reaction coordinate; both recover independent brute-force nucleation rates across fast, intermediate and slow regimes (Fig. 1) and a range of (β,h). Pointwise accuracy is quantified (Figs. 2–3): p_B-NN stays within a ±2.5% band for >95% of states while LGCS (and FK) scatter substantially, yet LGCS still yields correct rates. Saliency maps confirm that p_B-NN attends to clusters and their boundaries. The central claim is that an effective CV for nucleation rates must separate the metastable and stable basins but need not preserve the committor pointwise for every microstate.

Significance. The result cleanly separates two requirements that are often conflated in the nucleation literature: pointwise fidelity to the committor versus the ability of a coordinate to support accurate mean-first-passage-time rates. Because both a near-ideal learned committor and a deliberately imperfect geometric size recover the same brute-force rates (and match earlier Brendel et al. values where available), the work supplies concrete evidence that simple cluster-size CVs remain serviceable for rate calculations even when they fail histogram or pointwise tests. Strengths include GPU-enabled ground-truth labels at ±0.01 accuracy, explicit lag-time convergence via the implied-timescale plateau, saliency-based interpretability, and a public data release. The distinction has immediate practical value for the choice of collective variables in rare-event methods for nucleation.

minor comments (6)
  1. Methods: Basin boundaries are defined via the LGCS peak (parent) and LGCS = 0.5 L^{2} (stable) for both coordinates. While the stable cut is justified by p_B = 1, a short note confirming that pure p_B-based cuts (e.g. 0.01/0.99) leave the p_B-NN rates unchanged would remove any residual hybrid character.
  2. Methods / Results: Lag times are chosen as the shortest value at which the longest implied timescale plateaus. Including the implied-timescale curves (even in SI) for a few representative (β,h), especially the fast regime where LGCS is slightly worse, would make the Markovianity claim fully transparent.
  3. Fig. 1 caption and text: The division into regimes A/B/C is clear visually but never stated quantitatively (e.g. by rate decade or free-energy barrier). A one-sentence definition would help readers.
  4. Fig. 3: The sigmoid mappings used for LGCS and FK are not characterised (parameters or goodness-of-fit). Reporting them, or noting that the scatter is insensitive to the precise sigmoid, would strengthen the pointwise comparison.
  5. Notation: The symbol appears as p_B-NN, p B-NN and pB-NN in different places; a single consistent form would improve readability.
  6. Section IV: The broader implications for off-lattice systems with shape fluctuations or polymorphs are asserted rather than argued. A brief caveat that the basin-separation sufficiency has so far been demonstrated only for 2D Ising spin-flip dynamics would keep the claim proportionate.

Circularity Check

0 steps flagged

No significant circularity: MSM rates from learned and geometric coordinates are validated against independent brute-force nucleation rates, not recovered by construction from the training labels.

full rationale

The paper generates ground-truth committor labels by independent shooting (n=4096 trajectories per microstate) from configurations sampled along nucleating paths, trains p_B-NN as a supervised proxy, then builds separate MSMs on the discretized p_B-NN and LGCS coordinates. Rates are extracted as mean first-passage times after lag-time selection via the implied-timescale plateau and are compared to a distinct set of brute-force magnetisation-reversal rates (Fig. 1; agreement also with Brendel et al.). Pointwise accuracy is assessed separately (Figs. 2–3) and is not used to force the rate results. Basin boundaries are defined from the LGCS distribution and a fixed LGCS=0.5 L^{2} threshold (where p_B=1 by construction for the studied range), but this is a conventional absorbing-boundary choice, not a circular redefinition of the rates. No parameter fitted to rate data is re-presented as a prediction; no uniqueness theorem or ansatz is imported via self-citation to force the central claim; and the LGCS rate success despite poor pointwise fidelity is an empirical observation, not a tautology. The derivation chain is therefore externally falsifiable and self-contained.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 1 invented entities

The central claim rests on standard rare-event and MSM machinery plus a small set of modelling choices (basin cut-offs, lag-time plateau criterion, network capacity, label budget). No new physical entities are postulated; the p_B-NN is an empirical proxy trained on external labels. Free parameters are the usual discretisation and architecture knobs that control numerical resolution rather than the physics itself.

free parameters (5)
  • MSM bin counts = 20 (LGCS), 40 (p_B-NN)
    20 states for LGCS and 40 states for p_B-NN are chosen by hand to achieve convergence of implied timescales and ~2.5% committor resolution; rates depend on this discretisation.
  • Lag time τ = condition-dependent (shortest plateau)
    Selected as the shortest lag at which the longest implied timescale plateaus; the reported rate is taken at that lag.
  • Committor label sample size n = 4096
    n=4096 independent trajectories per microstate sets the ground-truth label accuracy target of ±0.01.
  • Training set size per (β,h) = ~2000
    Average of ~2000 labelled microstates per thermodynamic condition; sufficient for RMSE matching label noise.
  • CNN capacity = order 10^5 parameters
    Three residual CNN blocks + two residual linear blocks (~10^5 parameters) chosen to reach label-limited accuracy.
axioms (4)
  • domain assumption The committor p_B is the ideal reaction coordinate whose projection yields dynamics optimally close to Markovian.
    Invoked in the Introduction as the theoretical justification for learning p_B; standard modern transition-state theory (E & Vanden-Eijnden, Peters).
  • domain assumption Mean first-passage times extracted from a lag-converged MSM transition matrix equal the nucleation rates of interest.
    Methods section: rates are obtained from MFPT from parent to stable basin after implied-timescale plateau.
  • ad hoc to paper Parent basin boundary = peak of LGCS distribution in the metastable phase; stable boundary = LGCS = 0.5 L^{2} (beyond which p_B = 1).
    Explicit basin definitions in Methods; required for both label generation and MSM rate extraction.
  • domain assumption Single-spin-flip Metropolis dynamics on the 2D nearest-neighbour Ising model with L=64 adequately samples the nucleation process under study.
    Standard model choice stated in Methods; literature comparisons (Brendel et al.) assume the same dynamics.
invented entities (1)
  • p_B-NN (convolutional neural-network proxy for the committor) independent evidence
    purpose: Provides a fast, differentiable surrogate for the expensive brute-force committor that can be used as an MSM coordinate.
    The network is an empirical function approximator trained on external labels; it is not a new physical degree of freedom. Independent evidence is the held-out pointwise RMSE matching label noise and the rate agreement with brute force.

pith-pipeline@v1.1.0-grok45 · 12642 in / 3014 out tokens · 35163 ms · 2026-07-10T11:18:33.364338+00:00 · methodology

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read the original abstract

A central challenge in the analysis of first-order phase transitions is the identification of optimal reaction coordinates. In principle, the committor is the ideal choice; however, its computational cost has historically made it intractable. Here, we train a convolutional neural network ($p_B$-NN) as a proxy for the committor on brute-force committor labels and use it directly as the coordinate of a Markov state model. Applied to magnetisation reversal in the two-dimensional Ising model, $p_B$-NN reproduces brute-force nucleation rates across a range of thermodynamic conditions. The largest geometric cluster size also recovers accurate rates despite providing a poor pointwise predictor of the committor. These results demonstrate that an effective reaction coordinate for nucleation rate calculation must reliably separate the metastable and stable basins, but need not preserve the committor pointwise for every microstate. We stress that this distinction has direct implications for the choice of collective variable in rare-event simulations of nucleation more broadly.

Figures

Figures reproduced from arXiv: 2607.08207 by David Quigley, Hubert J. Naguszewski.

Figure 1
Figure 1. Figure 1: FIG. 1. Nucleation rates across a range of ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fraction of committor predictions outside a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Predicted committor values compared with brute [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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