Inverse Design of Amorphous Materials with Targeted Properties

Jilin Hu; Jonas A. Finkler; Morten M. Smedskjaer; Tao Du; Yan Lin

REVIEW 5 major objections 6 minor 85 references

Inverse Design of Amorphous Materials with Targeted Properties

T0 review · 5 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that a diffusion model trained on simulated amorphous structures can generate new glassy atomic configurations conditioned on target properties, and that adding Hamiltonian Monte Carlo refinement to the denoising step is w

desk verdict A serious methods paper with two genuinely new ideas—HMC-refined energy-based denoising and ghost-atom density control—but the claims are only validated within classical force-field MD, so treat the inverse-design results as proof-of-concept, not as a recipe for real glasses. read the letter →

arxiv 2509.13916 v2 pith:XDIHL2LH submitted 2025-09-17 cond-mat.mtrl-sci

Jonas A. Finkler , Yan Lin , Tao Du , Jilin Hu , Morten M. Smedskjaer This is my paper · ORCID

classification cond-mat.mtrl-sci PACS 61.43.-j62.20.de02.70.Ns

keywords inversedesignamorphousmaterialsdiffusionmodelHamiltonianMonteCarloglassgenerativeequivariantgraphneuralnetworkmelt-quenchsimulation

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 introduces AMDEN, a diffusion model that generates atomic structures of amorphous materials conditioned on target properties such as Young's modulus, lithium content, shear modulus, and ring size. Its central claim is that standard denoising cannot reach the low-energy relaxed structures typical of slow-cooled glasses, so the authors add an energy-based score function refined with Hamiltonian Monte Carlo steps during generation. The work also provides three amorphous datasets (multi-element glass, amorphous silicon with three thermal histories, and amorphous silica) to train and test the method. If the claims hold, inverse design of glasses becomes possible at the atomic level, including structures that standard melt-quench simulations cannot produce.

What carries the argument

The central object is AMDEN's score-based diffusion process over atomic positions and element embeddings, with an equivariant graph neural network as the score function. The novel piece is the energy-based variant: instead of predicting noise directly, the network outputs a scalar 'noise energy' per atom; the score is its gradient, and Hamiltonian Monte Carlo steps on that energy are inserted between denoising iterations. This lets the generator equilibrate on the learned distribution and cross barriers that standard denoising cannot, which is what enables relaxed low-energy structures. A 'ghost atom' class controls atomic density without breaking equivariance.

What would settle it

Relax AMDEN-generated structures for a fixed composition with a quantum-mechanical (e.g., density functional theory) relaxation and compare energies and pair distribution functions to the same treatment of melt-quench reference structures; if the generated samples systematically relax to different energies or moduli, the claim that HMC denoising reaches the correct low-energy basin fails. Alternatively, train on a small atomic cluster whose ground-state structure is independently known and check whether HMC denoising finds that known minimum.

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Extended reading notes

Core claim

AMDEN learns the distribution of amorphous atomistic structures from classical molecular dynamics melt-quench simulations using a score-based diffusion process with an equivariant graph neural network. It generates atomic positions and element types, and a 'ghost atom' mechanism lets it adjust density. The paper's key finding is that the standard denoising trajectory reproduces melt-like structures well but fails to produce annealed low-energy structures; a variant that predicts a scalar 'noise energy' and interleaves Hamiltonian Monte Carlo steps during denoising recovers the low-energy structures, with radial distribution functions and potential energies close to training data. On the mult

Load-bearing premise

The training data come from classical force-field melt-quench simulations, so if those potentials misrepresent the real atomic structures or elastic properties of glasses, everything AMDEN learns and validates is offset from reality.

Editorial extensions

If this is right

AMDEN can adjust the Young's modulus of generated multi-element glasses while keeping lithium content within a few percent of the targeted value.
Requenched compositions from AMDEN match target moduli better than the raw generated structures, indicating that structural inaccuracy is the dominant source of error.
Standard denoising cannot generate low-energy relaxed amorphous structures; energy-based HMC refinement recovers energy and structure matching the training data for slow-cooled samples.
On fixed-composition silica, conditioning on shear modulus and average ring size produces samples beyond the training range at larger cell sizes, enabling structure-only property control.
The approach opens the route to generating 'forbidden glasses' whose structural features cannot be obtained by traditional melt-quench procedures.

Reading between the lines

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

If the same HMC-refinement scheme were applied to models trained on quantum-mechanical or experimental data, it might correct some of the classical force-field bias; the paper does not test this.
The failure of standard denoising on glassy landscapes suggests a general limitation of diffusion models on rough energy landscapes, not just a flaw of this architecture; the authors hint at this through a spin-glass analogy.
The fixed-cell diffusion process, combined with ghost-atom density control, could be extended to variable cell volumes or open-boundary conditions to target pressure-dependent properties, though the paper does not explore that.
Ring-size conditioning on small training cells may allow targeted generation of medium-range order features, enabling systematic study of structure-property relationships in disordered materials beyond current simulation workflows.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a solid, honest methods paper, and the most useful part is the systematic failure analysis of standard denoising on relaxed amorphous structures. The HMC-refinement variant is a genuine modification over prior diffusion work on glasses, and the ghost-atom mechanism for density control is simple and effective. The paper's own discussion concedes the training-data problem, which is the right thing to do, and it should be read as proof-of-concept for the generative machinery rather than as a direct path to experimental inverse design.

What's new and good: the energy-based score—where the network predicts a scalar noise energy and the score comes from backprop—combined with HMC steps during reverse diffusion is a real contribution. The demonstration that standard denoising fails to reproduce annealed a-Si structures and that HMC fixes both the energy and the RDFs is convincing. The MEG, a-Si, and SiO2 datasets are useful resources, assuming they become public. The inverse design on Young's modulus and Li content is suggestive; the requenching check is a nice independent validation within the same force field.

Soft spots: Fig. 4 only shows charge-balanced samples, which overstates the design capability; the main text needs the MAE or R2 for all samples, and a justification for the exclusion. There are no error bars anywhere, and no quantitative comparison against existing generative models for amorphous materials (e.g., ref 35). The extrapolation claims—larger cells, properties beyond the training range—rest on a few samples and should be softened. The bigger issue is the force-field ground truth: all training and validation uses BMP, Stillinger-Weber, and Tersoff-Munetoh potentials, and the requench check uses the same potential, so it validates the model's ability to match the force-field's composition-property mapping, not reality. The authors acknowledge this and call for better data, which is fine, but the framing sometimes overreaches.

The paper deserves a serious referee. The core claims about generation quality and HMC refinement are internally supported, and the limitations are mostly disclosed. A referee should push for full data and code release, error bars and goodness-of-fit statistics, inclusion of unbalanced samples, and at least one baseline comparison. A DFT cross-check on a handful of generated structures would also go a long way toward addressing the force-field concern.

Referee Report

5 major / 6 minor

Summary. The paper introduces AMDEN, a diffusion-model framework for inverse design of amorphous materials. The model operates on atomic positions and element embeddings in a periodic cell, uses an equivariant graph neural network as the score backbone, and supports conditioning on macroscopic properties (Young's modulus, shear modulus) and structural features (average ring size). A 'ghost atom' mechanism is intended to control density during generation. The authors also propose an energy-based variant in which the score is obtained as the gradient of a learned noise energy E_theta, and they couple this with Hamiltonian Monte Carlo (HMC) refinement during denoising. Training and validation use three new classical-MD datasets: a multi-element glass (MEG) dataset with 11 elements, three amorphous silicon datasets with different thermal histories, and an amorphous SiO2 dataset with varied cell sizes. The paper reports that standard denoising fails to generate low-energy relaxed structures, that HMC denoising recovers energies and structures close to the training distributions, and that property-conditioned generation can adjust Young's modulus and Li content in the MEG system and shear modulus/ring size in SiO2 beyond the training range. The authors acknowledge several limitations, including the lack of high-quality training data and the reliance on classical force fields.

Significance. If the claims are substantiated, this would be a useful contribution to generative modeling of amorphous materials, an area where diffusion models are less mature than for crystals and molecules. The paper's strengths are the introduction of multiple new MD datasets, the explicit requenching validation for the composition-to-modulus mapping, the forward-noising ground-truth check for the noise-energy evaluation, and the interesting proposal to use HMC on the learned noise energy to counteract the rugged potential-energy-landscape problem. The 'structure-only' control of shear modulus and ring size, and the idea of generating 'forbidden glasses,' are conceptually valuable. However, the quantitative evidence is currently incomplete: key figures lack error bars and correlation metrics, main plots exclude a disclosed subset of samples, there is no comparison against published baselines, and the entire framework inherits the accuracy of the classical force fields with no independent DFT or experimental cross-check. The paper also does not yet provide code or trained models, so reproducibility cannot be assessed.

major comments (5)

[§II E, Fig. 4] The main plots in Fig. 4 include only perfectly charge-balanced samples, while unbalanced samples are relegated to Supp. Fig. S3. The text reports mean absolute charge errors (0.0025 e and 0.0096 e per atom) but does not report the fraction of generated samples excluded, nor the R²/MAE for the correlations, nor error bars. Because the central claim is that AMDEN adjusts shear modulus and ring size purely through structure, this selection could bias the apparent correlation. Please report the yield of charge-balanced samples, the metrics for all generated samples, and the metrics with and without the charge-balance filter.
[§II B, Fig. 2] The inverse-design result for the MEG dataset is described qualitatively as 'decent correlation' and 'within few percent' Li content, but no quantitative metrics (R², MAE, standard error) or error bars are given. The requenching comparison uses the same BMP potential as the training data, so it validates composition-to-modulus mapping only within that potential. Moreover, no comparison is made to existing generative models for amorphous materials, such as the diffusion model of Yang & Schwalbe-Koda (2025) or earlier GAN/VAE approaches. Please add numerical error metrics, confidence intervals, and a baseline comparison so the reader can assess whether AMDEN improves over the state of the art.
[§II D and §IV B 4] The HMC refinement is validated against forward noising trajectories of the same model and against potential energies computed with the same Stillinger-Weber potential used for training. This is a self-consistency check, not an independent validation. More importantly, the comparison between standard denoising and HMC denoising is confounded by compute budget: standard denoising uses n=200 steps, while HMC denoising uses n=2000 steps and sets wcond=0. The reported improvement may come from the larger number of diffusion steps, the absence of CFG, or the HMC sampling itself. Please include an ablation with standard denoising at 2000 steps, and report HMC acceptance rates and wall-clock cost. A comparison to a short MD relaxation (as used in prior work) would also clarify the contribution of HMC.
[§IV A and Discussion] The entire pipeline—training data, property labels, and validation—is based on classical force fields (BMP-shrm for MEG, Stillinger-Weber for a-Si, Tersoff-Munetoh for SiO2). The paper itself notes that universal ML force fields can be unreliable for high-energy melt-quench structures and that 'the biggest challenge' is a lack of high-quality training data, yet the title and abstract claim inverse design of amorphous materials without qualification. No DFT or experimental cross-check is provided for any generated structure, RDF, ring statistic, or elastic modulus. This is an external-correctness risk. Please either add DFT validation of a representative subset of generated samples (e.g., RDFs and elastic moduli for a few MEG and SiO2 structures) or explicitly reframe the claims as 'in silico inverse design within the accuracy of the chosen force fields.'
[§II A and Fig. 1] The ghost atom mechanism is described as enabling density control, but the paper never quantitatively evaluates whether the target density is achieved. No results show achieved versus target density, and it is unclear how the ghost fraction interacts with property conditioning. Since density is a physically important variable, particularly for shear modulus, please report the achieved densities for the generated MEG and SiO2 samples, or state explicitly if density was not controlled in those experiments.

minor comments (6)

[Fig. 3] The figure has panels a–f, but the text refers to panel 'e' for noise energy and panel 'd' for potential energies; the caption and in-text references should be checked for consistency. Also, the RDF panels in a and c have the same x-axis and y-axis labels but different data; consider adding legends and clarifying the line styles.
[Eq. (21)] The learnable scale factor γ is said to balance position-based and atomic energies, but no initialization, regularization, or final value is given. Please provide these details, since the magnitude of γ directly affects the score and the HMC dynamics.
[Eq. (24)] The Metropolis–Hastings acceptance criterion is written in terms of total energy E_tot, which includes momenta. It would be helpful to state explicitly that momenta are refreshed after each acceptance/rejection and that only positions are updated, and to clarify whether element embeddings are also updated during HMC.
[§IV B 4] After Eq. (19), the text states 'kB × T = 1' but does not define the separate values of kB and T. Since these are arbitrary normalization constants, a single constant would be clearer and would avoid confusion with the physical Boltzmann constant.
[Data Availability] The paper states that source code and training data 'will be made available upon final publication.' For a computational methods paper, it would be preferable to provide at least trained model checkpoints and dataset statistics in a repository or supplement during the review process, to enable independent verification of the central claims.
[References] Several references are preprints or lack page numbers (e.g., [35], [17], [20], [33]). If final versions are available, they should be updated; otherwise, the preprint status should be explicitly noted.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the paper's property predictions and HMC-refinement claims are checked against independent molecular-dynamics computations (requenching, Stillinger-Weber energies, stress-tensor moduli) rather than by the model's own outputs; the two self-citations are non-load-bearing background.

full rationale

The central claims—inverse design of amorphous structures with targeted Young's modulus, shear modulus, and ring size, and the ability of HMC denoising to generate relaxed low-energy structures—are not circular because the evaluation metrics are computed independently of the trained model. Training labels are generated via classical MD melt-quench workflows (BMP for MEG, Stillinger-Weber for Si, Tersoff-Munetoh for SiO2) and properties are computed with physics-based finite-difference stress-strain calculations and ring analyses, not by the model itself. At inference, the Young's modulus of generated samples is obtained by the same stress-tensor computation, and the MEG validation requenches the entire generated composition with the original melt-quench workflow, providing an external ground truth (Fig. 2b); the correlation with targets is imperfect, which is evidence that the metric is not forced. For the Si relaxation claims, HMC-generated samples are evaluated by their Stillinger-Weber potential energies (Fig. 3d) and radial distribution functions relative to the training data—quantities not used as training targets—so the reduction in potential energy under HMC is an empirical finding, not an identity. The noise-energy comparison along forward-noising and denoising trajectories (Fig. 3e) is a self-consistency diagnostic, since E_hat is trained on forward-noised samples, but this diagnostic is not the load-bearing evidence. Self-citations (Refs [31], [41]) appear only in background and application-motivation contexts and do not ground the central claim; no uniqueness theorem from the authors is invoked. The genuine limitation is external validity: all training, conditioning, and validation data come from unvalidated classical force fields, which the paper itself flags as needing further validation and as a lack of high-quality training data. That is a correctness risk, not a circularity of the derivation chain.

Assumptions & free parameters 8 free parameters · 4 assumptions · 2 invented entities

The central result rests on classical-MD-generated datasets, on the standard score-matching assumption, and on the validity of using the learned noise energy as an effective Hamiltonian for HMC. The ghost-atom mechanism is a computational device with no physical counterpart. The main free parameters are hand-chosen noise schedules, normalization, CFG weights, and HMC timesteps; they affect the generated structures but are disclosed. No new physical particles or forces are introduced.

free parameters (8)

sigma_X_max (position noise schedule amplitude) = 1.7 A (MEG), 1.5 A (others)
Chosen by hand to set the scale of position noise in the forward SDE; controls starting noise and affects generated structures.
sigma_E_max (element embedding noise scale) = 1.5
Chosen by hand for the VP element noise schedule.
cutoff radius r_cut = 6.5 A
Chosen to include bonded and strongly interacting atoms while keeping the graph sparse; affects medium-range order representation.
nnorm normalization factor = 40
Hand-set for numerical stability in EGNN message aggregation.
CFG condition weight wcond = 0.25 (MEG), 1.0 (SiO2)
Manually chosen per dataset for conditioned score strength.
HMC timestep dt0 = 0.2 (MEG), 0.4 (Si and SiO2)
Adjusted per dataset to maintain acceptance rate near 0.5.
HMC diffusion time interval [tHMC_min, tHMC_max] = [0.0, 0.5]
Chosen to reduce cost and apply HMC where the learned energy landscape is informative.
loss weight lambda = 0.5
Hand-set balance between position and element noise losses.

assumptions (4)

domain assumption Classical interatomic potentials (BMP, Stillinger-Weber, Tersoff-Munetoh) accurately represent amorphous structures and the target properties for the studied systems.
All training data and property labels come from MD simulations with these potentials (Methods IV A); no DFT or experimental cross-check is provided.
standard math The SDE score-matching framework with VE/VP noise schedules is a valid generative model for the atomic position/element distribution.
The method assumes the learned score function approximates the gradient of log density of amorphous samples (Eq. 18-20).
ad hoc to paper The learned noise energy E_theta is smooth enough and close enough to the true glass distribution that Hamiltonian Monte Carlo on it samples relaxed structures.
The HMC denoising variant treats E_theta as an effective Hamiltonian with k_B T = 1 (Section IV B 4), an assumption validated only by internal comparisons.
domain assumption Disorder fluctuations in small amorphous silica cells (80 to 250 atoms) can be used to learn structure-property relationships that extrapolate to larger cells.
The SiO2 experiment relies on natural finite-size structural variation to condition shear modulus and ring size; the paper acknowledges extrapolation and charge-balance limitations.

invented entities (2)

Ghost atoms
purpose: A special element class that lets the diffusion model vary the number of real atoms in a fixed-size cell and thereby control density during generation.
Ghost atoms are removed after denoising; they are a computational device, not a physical species, and have no falsifiable experimental handle.
Noise energy field E_theta
purpose: A learned scalar atomic energy used to compute scores via gradient and to drive HMC refinement during denoising.
The paper explicitly states E_theta is usually not the potential energy; only for Boltzmann-distributed unconditional data at t=0 does it recover the potential. It is a model construct validated only internally.

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Cite this review

Pith. "Pith review of Inverse Design of Amorphous Materials with Targeted Properties." pith.science (2026). https://pith.science/paper/XDIHL2LH

@misc{pith2026250913916,
  author       = {Pith},
  title        = {Pith review of: Inverse Design of Amorphous Materials with Targeted Properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDIHL2LH}},
  note         = {Machine review of arXiv:2509.13916}
}

read the original abstract

Disordered (amorphous) materials, such as glasses, are emerging as promising candidates for applications within energy storage, nonlinear optics, and catalysis. Their lack of long-range order and complex short- and medium-range orderings, which depend on composition as well as thermal and pressure history, offer a vast materials design space. To this end, relying on machine learning methods instead of trial and error is promising, and among these, inverse design has emerged as a tool for generating materials with desired properties. Although inverse design methods based on diffusion models have shown success for crystalline materials and molecules, similar methods targeting amorphous materials remain less developed, mainly because of the limited availability of large-scale datasets and the requirement for larger simulation cells. In this work, we propose and validate an inverse design method for amorphous materials, introducing AMDEN (Amorphous Material DEnoising Network), a diffusion model-based framework that generates structures of amorphous materials. These low-energy configurations are typically obtained through a thermal motion-driven random search-like process that cannot be replicated by standard denoising procedures. We therefore introduce an energy-based AMDEN variant that implements Hamiltonian Monte Carlo refinement for generating these relaxed structures. We further introduce several amorphous material datasets with diverse properties and compositions to evaluate our framework and support future development.

Figures

Figures reproduced from arXiv: 2509.13916 by the authors.

**Figure 2.** a, the generated Li concentration is in good agreement with the targeted value (within few percent), despite only a very small fraction of the training data containing a similar ratio of Li atoms. The Young’s modulus of the generated structures is shown in Fig. 2b. While a decent correlation between the target and generated modulus is observed, the generated modulus falls short of the target, especially for larger… view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 11 linked inside Pith

[1]

Ini- tial structures were generated from varying compositions of the glass formers SiO 2 and P 2O5, and the modifiers Al2O3 Li2O, BeO, K 2O, CaO, TiO 2, BaO and ZnO

Multi Element Glass dataset The Multi Element Glass (MEG) Dataset consists of 9,027 glass samples, containing 11 different elements. Ini- tial structures were generated from varying compositions of the glass formers SiO 2 and P 2O5, and the modifiers Al2O3 Li2O, BeO, K 2O, CaO, TiO 2, BaO and ZnO. The samples were then obtained using a simulation workflow...
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Each of the three datasets consisted of 10,000 samples containing 256 Si atoms

Amorphous Si dataset Three amorphous silicon datasets were generated using the Stillinger-Weber potential [ 43]. Each of the three datasets consisted of 10,000 samples containing 256 Si atoms. All samples were created using the same simula- tion workflow but using a different random seed ensuring a unique atomic structure. Samples in themeltdataset were o...
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Amorphous SiO 2 dataset Samples of the amorphous silica (SiO 2) dataset vary in size, with the number of atoms being uniformly dis- tributed between 80 and 250 atoms. Simulations were performed using the Tersoff potential parameterized by Munetohet al.[ 66]. Samples were initially melted at 3500 K and the immediately quenched using a local ge- ometry opti...
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The forward diffusion process provides ground truth for noisy samples at intermediate stepsx t and noise compo- nents to supervise the score functions θ

Materials diffusion process Following the SDE framework [ 36], our materials diffu- sion process is composed of two subprocesses: a forward diffusion process and a reverse denoising diffusion process. The forward diffusion process provides ground truth for noisy samples at intermediate stepsx t and noise compo- nents to supervise the score functions θ. Th...
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The graph G = (V,E ) provided to EGNN is a graph of the atoms in each sample x, calculated with a cutoff radius rcut = 6.5 ˚A

Equivariant graph neural network The EGNN serves as the equivariant backbone of the score function. The graph G = (V,E ) provided to EGNN is a graph of the atoms in each sample x, calculated with a cutoff radius rcut = 6.5 ˚A. The cutoff radius was chosen to ensure that all bonded and strongly interacting atoms share a direct edge connection while still k...
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Score function The score function extended the EGNN framework to predict noise components in both positions and el- ement embeddings during the reverse denoising diffu- sion process, optionally conditioned on desired properties. The score function sθ mapped a noisy material sample xt = (C,X t,E t), diffusion step t, and desired properties yto position and...
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The key difference between the standard score function and the energy-based variant lies in how noise components are predicted

Hamiltonian Monte Carlo refinement The energy-based score functionwas introduced to support the prediction of noise energy utilized in Sec- tion II D. The key difference between the standard score function and the energy-based variant lies in how noise components are predicted. While the standard score function directly predicted position and element nois...
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The total system noise energy was given by, ˆEθ(x) = 1 σ(t) nX i=1 eX i +γ·e atom i ,(21) where γ is a learnable scale factor balancing the contribu- tion of atomic energies

position-based energy [ 73]: eX i = 1 2 ∥Xi −X (L) i ∥2 , derived from squared distances between original posi- tions and EGNN-output positions; and 2) atomic energy: eatom i =H (L) i , EGNN-output node features. The total system noise energy was given by, ˆEθ(x) = 1 σ(t) nX i=1 eX i +γ·e atom i ,(21) where γ is a learnable scale factor balancing the cont...

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Ini- tial structures were generated from varying compositions of the glass formers SiO 2 and P 2O5, and the modifiers Al2O3 Li2O, BeO, K 2O, CaO, TiO 2, BaO and ZnO

Multi Element Glass dataset The Multi Element Glass (MEG) Dataset consists of 9,027 glass samples, containing 11 different elements. Ini- tial structures were generated from varying compositions of the glass formers SiO 2 and P 2O5, and the modifiers Al2O3 Li2O, BeO, K 2O, CaO, TiO 2, BaO and ZnO. The samples were then obtained using a simulation workflow...

[2] [2]

Each of the three datasets consisted of 10,000 samples containing 256 Si atoms

Amorphous Si dataset Three amorphous silicon datasets were generated using the Stillinger-Weber potential [ 43]. Each of the three datasets consisted of 10,000 samples containing 256 Si atoms. All samples were created using the same simula- tion workflow but using a different random seed ensuring a unique atomic structure. Samples in themeltdataset were o...

[3] [3]

Simulations were performed using the Tersoff potential parameterized by Munetohet al.[ 66]

Amorphous SiO 2 dataset Samples of the amorphous silica (SiO 2) dataset vary in size, with the number of atoms being uniformly dis- tributed between 80 and 250 atoms. Simulations were performed using the Tersoff potential parameterized by Munetohet al.[ 66]. Samples were initially melted at 3500 K and the immediately quenched using a local ge- ometry opti...

[4] [4]

The forward diffusion process provides ground truth for noisy samples at intermediate stepsx t and noise compo- nents to supervise the score functions θ

Materials diffusion process Following the SDE framework [ 36], our materials diffu- sion process is composed of two subprocesses: a forward diffusion process and a reverse denoising diffusion process. The forward diffusion process provides ground truth for noisy samples at intermediate stepsx t and noise compo- nents to supervise the score functions θ. Th...

[5] [5]

The graph G = (V,E ) provided to EGNN is a graph of the atoms in each sample x, calculated with a cutoff radius rcut = 6.5 ˚A

Equivariant graph neural network The EGNN serves as the equivariant backbone of the score function. The graph G = (V,E ) provided to EGNN is a graph of the atoms in each sample x, calculated with a cutoff radius rcut = 6.5 ˚A. The cutoff radius was chosen to ensure that all bonded and strongly interacting atoms share a direct edge connection while still k...

[6] [6]

Score function The score function extended the EGNN framework to predict noise components in both positions and el- ement embeddings during the reverse denoising diffu- sion process, optionally conditioned on desired properties. The score function sθ mapped a noisy material sample xt = (C,X t,E t), diffusion step t, and desired properties yto position and...

[7] [7]

The key difference between the standard score function and the energy-based variant lies in how noise components are predicted

Hamiltonian Monte Carlo refinement The energy-based score functionwas introduced to support the prediction of noise energy utilized in Sec- tion II D. The key difference between the standard score function and the energy-based variant lies in how noise components are predicted. While the standard score function directly predicted position and element nois...

[8] [8]

The total system noise energy was given by, ˆEθ(x) = 1 σ(t) nX i=1 eX i +γ·e atom i ,(21) where γ is a learnable scale factor balancing the contribu- tion of atomic energies

position-based energy [ 73]: eX i = 1 2 ∥Xi −X (L) i ∥2 , derived from squared distances between original posi- tions and EGNN-output positions; and 2) atomic energy: eatom i =H (L) i , EGNN-output node features. The total system noise energy was given by, ˆEθ(x) = 1 σ(t) nX i=1 eX i +γ·e atom i ,(21) where γ is a learnable scale factor balancing the cont...