Generative Inversion of Spectroscopic Data for Amorphous Structure Elucidation

Abstract

Determining atomistic structures from characterization data is one of the most common yet intricate problems in materials science. Particularly in amorphous materials, proposing structures that balance realism and agreement with experiments requires expert guidance, good interatomic potentials, or both. Here, we introduce GLASS, a generative framework that inverts multi-modal spectroscopic measurements into realistic atomistic structures without knowledge of the potential energy surface. A score-based model learns a structural prior from low-fidelity data and samples out-of-distribution structures conditioned on differentiable spectral targets. Reconstructions using pair distribution functions (PDFs), X-ray absorption spectroscopy, and diffraction measurements quantify the complementarity between spectral modalities and demonstrate that PDFs is the most informative probe for our framework. We use GLASS to rationalize three contested experimental problems: paracrystallinity in amorphous silicon, a liquid-liquid phase transition in sulfur, and ball-milled amorphous ice. In each case, generated structures reproduce experimental measurements and reveal mechanisms inaccessible to diffraction analysis alone.

Figures (32)

• 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. \ref{['sec:mol-dyn-sims']} and Figs. \ref{['si:fig:Si_C_all_features']}--\ref{['si:fig:comparison']}).
• 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 $\sim4.15$ 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 ($\sim1.2$ 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.