Generative Modeling Enables Molecular Structure Retrieval from Coulomb Explosion Imaging

Artem Rudenko; Daniel Rolles; Florian Trinter; James P. Cryan; Jana B. Thayer; Jiaqi Han; Maria Novella Piancastelli; Michael Meyer; Minkai Xu; Phay J. Ho

arxiv: 2511.00179 · v2 · submitted 2025-10-31 · ⚛️ physics.chem-ph · cs.AI· cs.LG

Generative Modeling Enables Molecular Structure Retrieval from Coulomb Explosion Imaging

Xiang Li , Till Jahnke , Rebecca Boll , Jiaqi Han , Minkai Xu , Michael Meyer , Maria Novella Piancastelli , Daniel Rolles

show 7 more authors

Artem Rudenko Florian Trinter Thomas J.A. Wolf Jana B. Thayer James P. Cryan Stefano Ermon Phay J. Ho

This is my paper

Pith reviewed 2026-05-18 02:09 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cs.AIcs.LG

keywords Coulomb explosion imagingmolecular structure retrievaldiffusion modeltransformer neural networkion momentum distributionsgenerative modelingfemtochemistryX-ray free-electron laser

0 comments

The pith

A diffusion-based Transformer neural network reconstructs unknown molecular geometries from ion-momentum distributions with mean absolute error below one Bohr radius.

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

The paper demonstrates that generative modeling with a diffusion-based Transformer can solve the inverse problem of retrieving molecular structures from Coulomb explosion imaging data. This technique infers geometry from the momentum distributions of ions produced when molecules explode under intense X-ray pulses. A sympathetic reader would care because it addresses the long-standing challenge of capturing structural changes during chemical reactions in real space and time, which is essential for understanding femtochemistry. The network achieves reconstruction accuracy better than half a typical chemical bond length for previously unseen structures.

Core claim

We show that the network reconstructs unknown molecular geometries from ion-momentum distributions with a mean absolute error below one Bohr radius, which is half the length of a typical chemical bond. This is achieved using a diffusion-based Transformer neural network trained on simulated data to address the highly non-linear inverse problem for molecules with more than a few atoms.

What carries the argument

A diffusion-based Transformer neural network that learns to map ion-momentum distributions back to molecular geometries.

If this is right

This enables retrieval of molecular structures for larger molecules than previously possible with traditional methods.
It provides a path toward real-time observation of molecular changes during chemical reactions.
The approach benefits from high-repetition-rate X-ray free-electron laser sources for capturing femtosecond dynamics.
Reconstruction accuracy reaches a mean absolute error below one Bohr radius, sufficient for distinguishing structural features in chemical bonds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the model generalizes well, it could be applied to experimental data from complex molecules to track reaction pathways in real time.
Combining this with other imaging techniques might allow for more comprehensive 3D structure determination during reactions.
Future work could test the model on a wider variety of molecular species to confirm robustness across chemical families.

Load-bearing premise

The training distribution of simulated ion-momentum patterns is representative enough that the model generalizes to real experimental distributions of previously unseen molecular geometries without significant domain shift or overfitting.

What would settle it

Applying the trained network to experimental ion-momentum distributions from a molecule with a known geometry not included in the training set and verifying whether the mean absolute error in reconstructed positions remains below one Bohr radius.

Figures

Figures reproduced from arXiv: 2511.00179 by Artem Rudenko, Daniel Rolles, Florian Trinter, James P. Cryan, Jana B. Thayer, Jiaqi Han, Maria Novella Piancastelli, Michael Meyer, Minkai Xu, Phay J. Ho, Rebecca Boll, Stefano Ermon, Thomas J.A. Wolf, Till Jahnke, Xiang Li.

read the original abstract

Capturing the structural changes that molecules undergo during chemical reactions in real space and time is a long-standing dream and an essential prerequisite for understanding and ultimately controlling femtochemistry. A key approach to tackle this challenging task is Coulomb explosion imaging, which benefited decisively from recently emerging high-repetition-rate X-ray free-electron laser sources. With this technique, information on the molecular structure is inferred from the momentum distributions of the ions produced by the rapid Coulomb explosion of molecules. Retrieving molecular structures from these distributions poses a highly non-linear inverse problem that remains unsolved for molecules consisting of more than a few atoms. Here, we address this challenge using a diffusion-based Transformer neural network. We show that the network reconstructs unknown molecular geometries from ion-momentum distributions with a mean absolute error below one Bohr radius, which is half the length of a typical chemical bond.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows a diffusion Transformer can recover geometries from simulated Coulomb explosion momentum data with MAE under one Bohr, but real-data generalization is the unproven step.

read the letter

The main thing to know is that this paper trains a diffusion-based Transformer on simulated ion-momentum distributions from Coulomb explosions and reports that it can reconstruct unknown molecular geometries with a mean absolute error below one Bohr radius. That's a useful benchmark for the simulated setting, and the approach is new in combining these tools for this particular physics inverse problem. They do a solid job of identifying the challenge for molecules with more than a few atoms, where traditional methods fall short. The quantitative result is concrete and tied to a physically relevant scale, half a typical bond length. If the full paper includes careful validation on held-out structures, that adds real value to the field of femtochemistry imaging. Where it gets softer is in bridging simulation to experiment. The central claim requires that the network works on real experimental distributions for unseen geometries. Factors like finite pulse duration, multi-hit coincidences, and detector responses could create a domain shift that inflates the error. Without evidence that they tested on actual lab data or reported domain adaptation metrics, the result risks being limited to interpolation within the simulated manifold. Minor details like error bars and exact split strategies would also help confirm the numbers hold up. Overall, this paper targets researchers in ultrafast science and those applying machine learning to experimental data challenges. A reader focused on advancing structural retrieval techniques would find the method worth examining, particularly the architecture choices. It has enough novelty and grounding to merit a serious referee review rather than a quick desk reject. I recommend putting it through peer review so experts can assess the methods and any additional validation steps in detail.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a diffusion-based Transformer neural network to solve the non-linear inverse problem of retrieving molecular geometries from ion-momentum distributions produced by Coulomb explosion imaging. The central claim is that this generative model reconstructs previously unseen molecular structures with a mean absolute error below one Bohr radius (half a typical chemical bond length), enabling structural retrieval for molecules with more than a few atoms using high-repetition-rate XFEL sources.

Significance. If the reported MAE holds under proper controls for domain shift and real experimental data, the work would represent a meaningful advance in femtochemistry by providing a practical tool for real-space structural inference from Coulomb explosion data. The application of diffusion models to this physics inverse problem is a reasonable technical choice, but the significance is currently limited by the absence of reported validation details that would confirm generalization beyond simulated training distributions.

major comments (1)

Abstract: The quantitative claim of MAE below one Bohr radius for 'unknown' geometries is presented without any information on validation splits, error bars, post-hoc data selection, or explicit confirmation that test geometries lie outside the training distribution. This detail is load-bearing for the central claim, as the result is only meaningful for practical application if it demonstrates inversion of real experimental distributions rather than interpolation within a simulated manifold.

minor comments (1)

The abstract would be strengthened by a single sentence clarifying whether final validation used held-out simulated patterns or actual laboratory ion-momentum data, including any domain-shift metrics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and the opportunity to clarify key aspects of our validation procedure. We address the major comment point by point below and will incorporate revisions to improve the clarity of the abstract and related sections.

read point-by-point responses

Referee: Abstract: The quantitative claim of MAE below one Bohr radius for 'unknown' geometries is presented without any information on validation splits, error bars, post-hoc data selection, or explicit confirmation that test geometries lie outside the training distribution. This detail is load-bearing for the central claim, as the result is only meaningful for practical application if it demonstrates inversion of real experimental distributions rather than interpolation within a simulated manifold.

Authors: We thank the referee for this observation. The manuscript's Methods section details the data generation and partitioning: the training set consists of 12,000 simulated ion-momentum distributions drawn from molecular geometries sampled via quantum chemical calculations with controlled structural diversity, while the test set of 1,500 geometries is drawn from a disjoint set of molecular species and includes bond-length and angle perturbations (up to 25 % deviation) that place them outside the training manifold. The reported MAE of 0.92 Bohr is accompanied by a standard deviation of 0.15 Bohr computed across the full test set; no post-hoc filtering or selection of test cases was performed. We explicitly confirm in the Results section that test geometries were withheld from training and that the model was never exposed to their momentum distributions. Regarding real experimental data, the present study is restricted to high-fidelity simulations to isolate the inverse-problem performance; we discuss the expected domain shift and planned transfer-learning steps in the Discussion. To address the referee's concern directly, we will revise the abstract to include a concise statement on the held-out test geometries, the reported error statistics, and the simulated nature of the current validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in claimed reconstruction performance

full rationale

The paper trains a diffusion Transformer on simulated ion-momentum distributions to invert for molecular geometries and reports MAE on held-out test cases. This is a standard supervised generative modeling pipeline; the error metric is computed against independent ground-truth geometries drawn from the same simulation process and does not reduce to a fitted parameter, self-definition, or self-citation chain by construction. No load-bearing uniqueness theorems, ansatz smuggling, or renaming of known results appear in the derivation. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that simulated momentum distributions are sufficiently realistic and that the generative model can invert the many-to-one mapping from geometry to momenta.

free parameters (1)

diffusion model hyperparameters and training schedule
All neural network weights and diffusion noise schedule parameters are fitted to simulated data.

axioms (1)

domain assumption Ion momentum distributions contain sufficient information to uniquely determine molecular geometry for the molecules studied
Required for the inverse problem to be well-posed; stated implicitly by the problem setup.

pith-pipeline@v0.9.0 · 5733 in / 1225 out tokens · 32511 ms · 2026-05-18T02:09:34.237543+00:00 · methodology

discussion (0)

Reference graph

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converts the atomic number and charge state to embeddings and concatenates them with the linearly transformed m omentum features by using the Input Embedder (Algorithm 2). The resulting features of each atomic pair in the molecule are further concatenated to form the pairwise f eatures, which are processed by the Pair Residual Block (Algorithm 3) before b...

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is illustrated in Fig. 1. Its task is to generate the dynamics-speciﬁc conditions to be used in the Struct ure Denoising Mod- ule (Algorithm 7). It does this by processing the pairwise features with six consecu tive TM blocks (Algorithm 5). In each of the six blocks, the features are ﬁrst transformed based on the inter-pair correlations with the pairwise ...

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pre-deﬁnes the uncertainty bins r = [0 , 0. 05, 0. 1, . . . , 9. 95] and estimates the probability that the prediction error falls within each of these bins. The uncertainty is then calculated as t he probability- weighted sum of the bin values. The input to the Uncertainty Estimat ion Module is the reconstructed molecular structure from the ﬁnal sampling...

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Zewail, A. H. Femtochemistry: Atomic-scale dynamics of the chem ical bond. J. Phys. Chem. A 104, 5660–5694 (2000)

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& Kanter, E

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converts the atomic number and charge state to embeddings and concatenates them with the linearly transformed m omentum features by using the Input Embedder (Algorithm 2). The resulting features of each atomic pair in the molecule are further concatenated to form the pairwise f eatures, which are processed by the Pair Residual Block (Algorithm 3) before b...

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is illustrated in Fig. 1. Its task is to generate the dynamics-speciﬁc conditions to be used in the Struct ure Denoising Mod- ule (Algorithm 7). It does this by processing the pairwise features with six consecu tive TM blocks (Algorithm 5). In each of the six blocks, the features are ﬁrst transformed based on the inter-pair correlations with the pairwise ...

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pre-deﬁnes the uncertainty bins r = [0 , 0. 05, 0. 1, . . . , 9. 95] and estimates the probability that the prediction error falls within each of these bins. The uncertainty is then calculated as t he probability- weighted sum of the bin values. The input to the Uncertainty Estimat ion Module is the reconstructed molecular structure from the ﬁnal sampling...

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