Generative Inversion of Spectroscopic Data for Amorphous Structure Elucidation

Jiawei Guo; Daniel Schwalbe-Koda

Figure 1: Architecture of GLASS: Generative Learning of Amorphous Structures from Spectra. A. Conceptual overview, where forward and backward diffusion map between training data and a prior. Conditional generation samples structures consistent with a given spectra by starting from the prior distribution. B. Score model used to learn the structural prior. Atomic environments

i

are described by positions

\mathbf{r}_{ij}

and distances

d_{ij}

toward neighbors

j

and atomic identity

z_{i}

. The time feature

t

is encoded with Gaussian random Fourier features (RFF). Feed-forward neural networks (FFNN) and a graph neural network (GNN) are used to learn the final score. C. Conditional reverse-time denoising process. At each timestep, atomic coordinates are updated using a combination of the predicted score, gradients from differentiable spectroscopic observables, and stochastic noise, according to the score-based denoising approach. D , Differentiable spectroscopic modules enabling gradient-based conditional sampling. Real-space descriptors (PDF and ADF) are computed using smooth density estimators over two- and three-body terms. Diffraction observables (XRD and ND) are evaluated using differentiable Debye-style pair summations, and X-ray absorption spectra (XANES and EXAFS) are predicted using GNN-based surrogate models.

Figure 2: Comparing reconstruction of amorphous structures with multi-modal spectroscopy. A. Schematic illustration of the evaluation metrics. Spectral diversity quantifies the variability of spectra across independent replicas, while error measures the deviation between spectra of generated structures and the average reference spectrum. B. Average normalized spectral error for four amorphous systems and four methodologies: reverse Monte Carlo (RMC) initialized with random (rnd) or liquid (liq) structures, and GLASS (unconditional or conditional). Error bars indicate the standard deviation across spectral modalities and independent replicas. C. Schematic comparing the spectral error (orange) and reference diversity (blue) of a-Si across spectroscopic modalities. The difference

\Delta ED=\mathrm{error}-\mathrm{diversity}

indicates whether generated spectra fall within the natural variability of the reference ensemble, and is represented with the color of the circle. D. Performance of RMC, unconditional, and conditional generation across densities and spectroscopic modalities for amorphous silicon (a-Si). Colors indicate

\Delta ED

values, with red circles corresponding to worse reconstruction. E. Performance of conditional generation of Cu-Zr metallic glass across out-of-distribution compositions. F. Spectra of generated structures (black) obtained when different reference spectroscopic modalities (orange) are used as guidance (columns). Reference spectra of the training structures used for the score model are shown in blue. G. Average

\Delta ED

of generated spectra for a-Si when conditioning on different spectroscopic observables. Benchmark datasets include amorphous carbon (a-C), amorphous silicon (a-Si), amorphous silica (SiO 2 ), Cu-Zr metallic glass, and Zr-Ni-Al alloys spanning variations in density, cooling rate, and composition (see Sec. 1.3 and Figs. 9 – 13 ).

Figure 3: Generating amorphous-crystalline mixtures in silicon via PDF-guided denoising. A. Pair distribution functions

g(r)

of mixed crystalline-amorphous configurations as a function of increasing crystallinity (colors). B. Dihedral-angle distributions of generated structures from different crystallinity (colors), which are not used in GLASS and provide partial validation on the structures. C. Crystallinity of generated structures correctly capture the trends of reference values despite underpredictions. Each point represents the mean over independent denoising runs and error bars indicate the standard deviation. D. Representative generated structures at different crystallinity levels. Atoms are colored according to local environment: cubic diamond (blue), hexagonal diamond (orange), and other environments (gray). E. GLASS-generated structures (red) reproduce the experimental

g(r)

(black) used as target, including subtle, yet important variations at medium-range (inset). F. Crystallinity in samples generated (gen) conditioned on the experimental PDF are consistently above zero. In contrast, the crystallinity of samples from unconditional generation or conditioned on PDFs from melt-quench reference configurations is consistently zero. The box and whiskers depict the interquartile range and range of the distribution, respectively, and the central line depicts the median.

Figure 4: Generating structures for liquid–liquid phase transition (LLPT) in sulfur from experimental data. A. Pressure-temperature phase diagram of sulfur illustrating the LLPT between low-density liquid (LDL) and high-density liquid (HDL). Colored points correspond to the experimental results used as guidance, with colors matching panels B - G . Black curves indicate the experimentally reported phase boundaries. B. Experimental (solid) and generated (dashed)

g(r)

across the LLPT. C. Signatures of the LLPT at the medium-range of generated

g(r)

. The peak near 4.45 Å associated with S 8 rings is suppressed, while a feature near 4.0 Å shifts toward

\sim 4.15

Å upon entering the HDL phase. Structural descriptors extracted from reconstructed configurations: D , fraction of intact S 8 rings, E , fraction of chain terminations (S 1 ), and F , atomistic information entropy. Box-and-whisker points represent the interquartile range, range, and median of the statistics of independently generated structures. G. Example of generated configurations at pressures of 0.11, 0.17, 0.36, and 0.56 GPa. S 8 rings are removed for clarity. Small triangular motifs (arrows) arise from artifacts in the experimental

g(r)

at very short distances (

\sim 1.2

Å).

Figure 5: Generative reconstruction of medium-density amorphous (MDA) ice. A. The generation trajectory starts from crystalline hexagonal ice (I

h

) and converges to an amorphous MDA structure while passing through random intermediate states, as also shown in B ,

g_{\mathrm{OO}}(r)

. Oxygen atoms are shown in red and hydrogen atoms in white. C. Comparison of

g_{\mathrm{OO}}(r)

across amorphous ice, with reference low-density amorphous ice (LDA, blue), liquid water (LIQ, cyan), and high-density amorphous ice (HDA, purple), and generated MDA (pink). The inset highlights the trends of the second coordination shell near

\sim 4.5

Å. D. Comparison of calculated structure factors

S(Q)

of reference and generated structures and experimental measurements (black). E. Ring-size distributions of the hydrogen-bond network for LIQ, LDA, MDA, and HDA. The reconstructed MDA shows fewer six-membered rings relative to LDA and broader ring distribution resembling the liquid state.

Figure 6: Computational cost of spectroscopic feature extraction and gradient evaluation. Wall-time for computing spectroscopic observables and their gradients as a function of system size (number of atoms). All timings correspond to wall-clock time measured on CPU hardware under default multi-threaded execution. Dashed lines denote the cost of feature extraction for the PDF, ADF, and XRD observables, while solid lines denote the corresponding gradient calculations used during conditional denoising. Gradients for PDF and ADF are computed using differentiable formulations with costs comparable to their forward evaluations, whereas differentiable XRD gradients are moderately more expensive. For comparison, the cost of computing XRD gradients using finite-difference approximations is also shown, which is orders of magnitude more expensive than the differentiable implementation. All benchmarks were performed on the same underlying atomic configurations while varying the number of atoms (300–1800) to evaluate scaling with system size.

Figure 7: Averaged EXAFS spectra for Si across densities and structural environments in the validation set. Averaged EXAFS spectra

\chi(k)

for Si at densities

\rho=1.5

,

2.5

, and

3.5

g/cm 3 are shown, comparing neural network predictions against reference spectra. For each density, spectra are averaged over all atoms in the validation set belonging to a given structural class (amorphous, liquid, or randomly placed Si). Dashed lines denote the reference spectra; solid lines indicate the model predictions. The averages are computed over 582 validation atom-wise spectra at

1.5

g/cm 3 , 644 atom-wise spectra at

2.5

g/cm 3 , and 690 validation atom-wise-spectra at

3.5

g/cm 3 , with atoms randomly sampled from amorphous and liquid configurations generated by LAMMPS molecular dynamics simulations, as well as randomly placed Si structures. All predicted spectra are generated using a single model trained jointly across multiple densities and structural environments, including amorphous, liquid, and randomly initialized configurations.

Figure 8: Atom-wise EXAFS spectra for Si at

\rho=2.5

g/cm 3 . Atom-resolved EXAFS spectra predicted by the neural network model compared against reference spectra for validation atoms. Each panel shows the EXAFS signal

\chi(k)

for an individual atom, with dashed lines indicating the reference spectrum and solid lines indicating the model prediction. Panels are ranked by per-atom spectral error, quantified as the mean absolute error over the full

k

-range (0–14 Å -1 ), computed across a validation set of 644 atom-wise spectra randomly sampled from amorphous and liquid configurations (obtained from LAMMPS molecular dynamics simulations), as well as randomly placed Si structures (which also serve as the initial configurations for denoising). The top two rows show the ten best-fitted atoms, while the bottom two rows show the ten worst-fitted atoms. Line color encodes the structural environment of the atom (blue: amorphous; orange: liquid; green: random). Notably, all best-predicted configurations originate from amorphous structures, reflecting their higher structural similarity to the training distribution, whereas all worst-predicted configurations correspond to randomly generated Si structures.

Figure 9: Variability of reference spectroscopic signals with density in amorphous carbon and silicon. A. Amorphous carbon (a-C) and B. amorphous silicon (a-Si). Representative spectra are shown for three densities (1.5, 2.5, and 3.5 g/cm 3 ), selected from the five densities used in the dataset, with 2.0 and 3.0 g/cm 3 omitted for clarity. Solid lines denote the mean spectra averaged over 10 independent MD runs, while shaded regions indicate the corresponding standard deviations.

Figure 10: Variability of reference spectroscopic signals with cooling rates in amorphous silica (a-SiO 2 ). Representative spectra are shown for three cooling rates (1, 10, and 100 K ps -1 shown in blue, orange, and green, respectively) from the reference dataset generated using LAMMPS molecular dynamics (MD) simulations. Solid lines denote the mean spectra averaged over 10 independent MD runs, while the shaded regions indicate the corresponding standard deviations. The spectra are highly similar across the different cooling rates, reflecting the relatively weak structural sensitivity to cooling rate in these simulations, with the small system size further contributing to the overlapping spectral signals.

Figure 11: Variability of reference spectroscopic signals with composition in amorphous Cu–Zr alloys. Representative spectra are shown for three compositions (Cu 20 Zr 80 , Cu 50 Zr 50 , and Cu 80 Zr 20 ), selected from the seven compositions included in the dataset. Solid lines denote the mean spectra averaged over 10 independent MD runs, while shaded regions indicate the corresponding standard deviations.

Figure 12: Variability of reference spectroscopic signals with composition in amorphous Zr–Ni–Al alloys. Representative spectra are shown for four compositions (Zr 1 Ni 1 Al 1 , Zr 1 Ni 1 Al 3 , Zr 1 Ni 3 Al 1 , and Zr 3 Ni 1 Al 1 ), selected from 25 total compositions generated by varying the stoichiometric ratio of each element from 1 to 3. Compositions that are simple multiples of one another (e.g., 1:1:1, 2:2:2, and 3:3:3) are considered equivalent and therefore counted only once. Solid lines denote the mean spectra averaged over 10 independent MD runs, while shaded regions indicate the corresponding standard deviations.

Figure 13: Comparison of spectral reconstruction errors across amorphous materials, including SiO 2 . Average (normalized) spectral error of reverse Monte Carlo (RMC) and our generative model in reproducing the structure of five characteristic amorphous materials: a-C, a-Si, SiO 2 , CuZr, and ZrNiAl. Error bars represent the standard deviation of errors aggregated across all spectral modalities and structural variations in the reference datasets, including 5 different densities for a-C and a-Si, 3 different cooling rates for SiO 2 , and different alloy compositions for CuZr (7 compositions) and ZrNiAl (25 compositions). For SiO 2 , only unconditional denoising results are shown because the unconditional model already achieves consistently low errors with small variance across both spectral observables and cooling rates. This indicates that the spectroscopic signals themselves do not strongly distinguish between different cooling rates, and therefore additional conditional denoising provides little benefit in this case.

Figure 14: Effect of initialization on Reverse Monte Carlo (RMC) refinement for amorphous carbon (a-C). A. Heatmap summary of RMC performance across spectroscopic observables for different initialization strategies. For each condition, ten independent starting configurations were generated, and each was refined using ten independent RMC runs. Two evaluation modes are shown: best run , which selects the run with the lowest PDF error from the best-fitting sample, and mean , which averages over all runs and samples. Because the selection criterion is based on the PDF guidance signal, the best-run strategy does not necessarily yield lower errors across other spectroscopic observables, reflecting occasional overfitting to the PDF. The dominant effect instead arises from the initialization: starting from liquid configurations consistently improves agreement across spectra compared with random initializations. B. Representative spectra at

\rho=2.5

g/cm 3 illustrating this behavior. Solid lines show the mean spectrum, and shaded regions indicate the standard deviation across ensembles (multiple RMC runs and samples for the refined structures, and multiple configurations for the reference). Liquid-initialized refinements produce spectra that more closely match the reference across the observables than those obtained from random starting structures.

Figure 15: Effect of initialization on Reverse Monte Carlo (RMC) refinement for amorphous silicon (a-Si). A. Heatmap summary of RMC performance across spectroscopic observables for different initialization strategies. As in the a-C case, ten independent starting configurations were generated for each condition, and each was refined with ten independent RMC runs. Unlike a-C, neither random nor liquid initializations yield consistently good agreement with the reference structures across the observables. This behavior arises because the initial PDFs of both random and liquid configurations differ substantially from the target amorphous structure, making it difficult for RMC to reach the correct structural basin during refinement. B. Representative spectra illustrating this limitation. Both random and liquid starts remain noticeably offset from the reference spectra across multiple observables, highlighting the strong dependence of RMC on the quality of the initial configuration.

Figure 16: Comparison of different structural states for A. C and B. Si at 1.5 g/cm 3 . Pair distribution functions (PDFs) are shown for three representative starting configurations: randomly placed atoms, liquid configurations, and amorphous structures. The liquid and amorphous configurations were obtained from LAMMPS molecular dynamics simulations. Solid lines denote the mean PDFs averaged over multiple configurations, while shaded regions indicate the corresponding standard deviations. For a-C, the liquid structure closely resembles the amorphous state, whereas random configurations differ substantially. This structural similarity explains why reverse Monte Carlo (RMC) refinements converge more reliably when initialized from liquid configurations. In contrast, for a-Si the liquid structure remains structurally distinct from the amorphous network, making liquid initialization less advantageous and highlighting the stronger dependence of the reconstruction on the choice of starting configuration.

Figure 17: PDF-guided RMC loss as a function of Monte Carlo atom steps. A total of

10^{8}

(104-million) atom-steps were performed. The figure shows the evolution of multiple spectroscopic losses for amorphous Si at 1.5 g/cm 3 , one of the most challenging densities to recover. The RMC error corresponds to the PDF loss minimized during refinement using the FULLRMC package, and is distinct from the PDF curves reported elsewhere, which are computed using our differentiable framework. Within the

10^{8}

steps, the first half corresponds to the exploration stage, followed by a refinement stage. The simulation is initialized from a liquid-derived structure, which is already closer to the target than random initialization.

Figure 18: Reconstruction performance across cooling rates and alloy compositions. A. Performance of the unconditional generative model for amorphous silica (a-SiO 2 ) across different cooling rates. B–D. Reconstruction performance for Cu–Zr metallic glass across compositions

x

in Cu x Zr 1-x . B. Unconditional generative models with progressively larger model sizes. C. Two-step denoising strategy in which structures are first denoised using the unconditional model and then refined by conditional guidance using the total pair correlation function

g(r)

. D. Two-step denoising with guidance from partial pair distribution functions (PDF). Rows correspond to spectroscopic observables. Circle color encodes the reconstruction score

\Delta ED

, where progressively red circles indicate larger values and thus worse agreement with the reference spectra. Columns outlined in black denote conditions included in the training dataset.

Figure 19

Figure 19: Cross-composition reconstruction performance for amorphous Zr–Ni–Al alloys. A. Performance of the unconditional generative model across compositions

x,y,z

in Zr x Ni y Al z . B. Performance of the two-step denoising strategy, where randomly initialized structures are first denoised using the unconditional model and then refined through conditional guidance using the total pair distribution function

g(r)

. Rows correspond to spectroscopic observables, while columns represent alloy compositions generated by varying

x,y,z\in\{1,2,3\}

with equivalent compositions (e.g., 111, 222, 333) treated as identical and therefore omitted. Circle color encodes the reconstruction score

\Delta ED

, where progressively red circles indicate larger values and thus worse agreement with the reference spectra. Columns outlined in black denote compositions included in the training dataset.

Figure 20: Cross-density reconstruction performance for amorphous carbon and silicon. A. Unconditional generative model performance for amorphous carbon (a-C). B. Best two-stage denoising strategy for a-C, where randomly placed atoms are first denoised with the unconditional model to recover training-like amorphous behavior, followed by PDF-guided conditional denoising at lower noise to reach the target density. C. Unconditional generative model performance for amorphous silicon (a-Si). D. Direct PDF-guided denoising starting from the initial random structure, without the unconditional pre-denoising step, resulting in consistently poorer reconstructions. E. Two-stage denoising strategy applied to a-Si, analogous to panel B . Rows correspond to spectroscopic observables, and columns indicate the target density used during denoising. Columns highlighted in black denote densities included in the training dataset. Circle color encodes the reconstruction score

\Delta ED

, where progressively red circles indicate larger values and thus worse agreement with the reference spectra.

Figure 21: Spectroscopic features of unconditional denoised Cu–Zr structures relative to reference signals. The score model was trained only on the Cu 50 Zr 50 composition and used without spectroscopic guidance to generate amorphous structures at Cu 30 Zr 70 and Cu 70 Zr 30 . Despite the model being trained on a single composition, the generated structures reproduce several qualitative structural features of the target systems, with notable overlap observed in the partial pair distribution functions (PDFs).

Figure 22: Spectroscopic features of unconditional denoised Cu–Zr structures from a multi-composition prior. The score model was trained on multiple Cu–Zr compositions (Cu 80 Zr 20 , Cu 60 Zr 40 , Cu 40 Zr 60 , and Cu 20 Zr 80 and used without spectroscopic guidance to generate amorphous structures at Cu 30 Zr 70 and Cu 70 Zr 30 . Despite the substantially larger training set, the generated spectra exhibit less overlap with the reference signals compared with the single-composition prior.

Figure 23: Spectroscopic features of

g(r)

-guided Cu–Zr structure generation. The score model was trained on Cu 50 Zr 50 and used to generate structures at Cu 30 Zr 70 and Cu 70 Zr 30 with total PDF

g(r)

guidance during denoising. In contrast to unconditional generation, the guided structures show stronger agreement with the reference signals, and the previously observed spectral overlaps disappear even when using the smallest prior model. Nevertheless, small variations in EXAFS remain noticeable even after the near-perfect fits.

Figure 24: Effect of prior–guidance weighting on conditional denoising. Heatmap summary of reconstruction performance as a function of the weighting parameter

w

, which controls the balance between the structural prior (score model) and the spectroscopic guidance term during denoising ( ). Smaller weights (moving left) correspond to stronger reliance on the score-model. Because the prior was trained on a limited dataset and only at 2.5 g/cm 3 , excessive reliance on the prior leads to poor agreement with the target spectra when reconstructing structures at 1.5 g/cm 3 . Conversely, larger weights (moving right) emphasize the guidance loss and can drive the optimization outside the physically reasonable configuration space learned by the prior, leading to unphysical structures and degraded agreement across other observables due to overfitting. Optimal performance occurs at intermediate weights, where the prior constrains the structural manifold while the guidance term steers the configuration toward agreement with the target spectrum. This figure presents the full weighting benchmark across multiple spectroscopic guidance signals (ADF, PDF, EXAFS, and XRD). The plots are identical to those shown in Fig. 2 B of the main text, except that the main text displays only the optimal weighting for each guidance signal, whereas here the full sweep over

w

is shown. All benchmarks were performed on amorphous Si, where the score model was trained exclusively on structures at 2.5 g/cm 3 and evaluated on reconstruction of amorphous structures at 1.5 g/cm 3 . Selection of

w

can be performed in a self-consistent way, where a good value of

w

minimizes reconstruction error for in-distribution validation data.

Figure 25: Structure of a-Si generated from experimental pair distribution function (PDF). Si-Si bonds are determined using a cutoff of 2.8 Å. Blue, orange, and white colors represent environments similar to cubic diamond, hexagonal diamond, and none, respectively, as determined by the polyhedral template matching method (Sec. 1.7 ).

Figure 26: Pair distribution functions (PDFs) for amorphous silicon. A. PDFs for generated a-Si directly from the prior model (unconditional) and conditioned to the experimental PDF (conditional). B. PDFs for generated a-Si conditioned on the reference PDF from melt-quench simulations.

Figure 27: Multi-modality validation comparing denoised structures guided by simulated melt-quench versus experimental ion-implantation PDFs. Spectra computed from the two denoised structures are compared across XRD, ND, EXAFS, XANES, PDF (Si–Si), and ADF (Si–Si–Si). The melt–quench–guided structure exhibits fully amorphous character with no detectable crystallinity, whereas the ion-implantation–guided structure shows signatures of partial crystallinity (see main text). This distinction is evident in the sharper and more pronounced peaks in XRD and ND for the ion-implantation case. In contrast, EXAFS and XANES remain largely similar between the two structures, indicating that local coordination environments and short-range physics are preserved despite differences in medium- and long-range order.

Figure 28: Training-set sufficiency analysis for the sulfur score model. A. Entropy-based learning curve obtained by randomly sampling fractions of the available configurations and computing the information entropy of the resulting environment distribution. Error bars denote the standard deviation across multiple sampling runs. The entropy rapidly saturates as additional environments are included, indicating that the structural diversity of the dataset is captured with relatively few configurations. B. Structural overlap score as a function of the number of configurations drawn from the two limiting structural manifolds of liquid sulfur: S 8 ring-dominated (r) and polymeric (p) networks. Training sets are constructed by progressively combining configurations from both manifolds (e.g., 1r, 1p, 1r+1p, 5r+5p, etc.). The overlap score quickly approaches unity once representative configurations from both structural states are included, demonstrating that only minimal manifold coverage is required to capture the relevant structural gradients governing the sulfur liquid–liquid transition.

Figure 29: Influence of prior model and initialization on sulfur structure generation A. X-ray diffraction (XRD) intensity profiles for structures generated using different score-model priors and initial configurations. B. Neutron diffraction (ND) spectra for the same structures. C. Total pair distribution function

g(r)

compared with the experimental reference at 0.11 GPa (gray band). D. Ring-size distribution showing the fraction of atoms participating in

n

-membered rings. Two different score-model priors are compared: an S-trained model (blue), trained on literature sulfur configurations containing both isolated S 8 rings and polymeric chains (Sec. 1.8 ), and a Si-trained model (orange), identical to the prior used in the paracrystalline silicon case. Two initialization strategies are used: S1 , randomly distributed sulfur monomers (solid lines), and S8 , randomly distributed S 8 rings (dashed lines).

Figure 30: Effect of guidance weighting on recovered structural correlations in MDA. A. Number-weighted total pair distribution functions

g(r)

obtained from denoising guided by either number-weighted or neutron-weighted

g(r)

. The resulting total

g(r)

profiles are nearly indistinguishable relative to the reference, indicating that both guidance schemes can reproduce the aggregate structural signal when evaluated in a number-weighted form. Corresponding number-weighted partial pair distribution functions (

g_{\mathrm{HH}}(r)

B. ,

g_{\mathrm{OH}}(r)

C. , and

g_{\mathrm{OO}}(r)

) D. reveal clear discrepancies between the two guidance schemes. Despite similar agreement in the total

g(r)

, guidance based on number-weighted

g(r)

fails to accurately recover the underlying partial correlations, particularly in the medium-range O–O network, due to the dominance of intramolecular O–H and H–H contributions. In contrast, neutron-weighted guidance better preserves physically relevant structural information.

Figure 31: Scaling behavior of unconditional and PDF-guided conditional denoising. A. GPU runtime as a function of system size for unconditional (prior-only), conditional (PDF-guided), and total denoising. All simulations were performed on a single NVIDIA RTX A6000 GPU using identical denoising hyperparameters. The computational cost increases systematically with the number of atoms, reflecting both the growing cost of graph construction and score evaluation at each denoising step. B. Comparison of the resulting pair distribution functions

g(r)

for systems ranging from 1,000 to 17,576 atoms against the reference structure. All runs used an identical denoising schedule of 512 steps for both part A and B . Despite the increase in system size, the agreement with the reference is consistently maintained without requiring additional denoising steps, indicating that a properly benchmarked and optimized set of hyperparameters enables size-transferable convergence.

Generative Inversion of Spectroscopic Data for Amorphous Structure Elucidation

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