The Loss Landscape of Powder X-Ray Diffraction-Based Structure Optimization Is Too Rough for Gradient Descent

TL;DR

The study investigates reconstructing crystal structures from powder XRD using gradient-based optimization, revealing that standard XRD-similarity objectives produce rugged, non-convex landscapes that trap optimizers. By constraining optimization to a ground-truth crystal family, recovery improves and symmetry emerges as a beneficial inductive bias, though non-convexity can persist along symmetry axes. The authors compare XRD-based optimization with energy-relaxation approaches, finding energy landscapes generally smoother and more reliable, suggesting multi-objective strategies that couple diffraction fidelity with physical-energy guidance. Overall, the work highlights the critical role of symmetry awareness in navigating the inverse mapping from diffraction to structure and motivates symmetry-guided generative or hybrid methods.

Abstract

Solving crystal structures from powder X-ray diffraction (XRD) is a central challenge in materials characterization. In this work, we study the powder XRD-to-structure mapping using gradient descent optimization, with the goal of recovering the correct structure from moderately distorted initial states based solely on XRD similarity. We show that commonly used XRD similarity metrics result in a highly non-convex landscape, complicating direct optimization. Constraining the optimization to the ground-truth crystal family significantly improves recovery, yielding higher match rates and increased mutual information and correlation scores between structural similarity and XRD similarity. Nevertheless, the landscape may remain non-convex along certain symmetry axes. These findings suggest that symmetry-aware inductive biases could play a meaningful role in helping learning models navigate the inverse mapping from diffraction to structure.

Figures (10)

• Figure 1: Results of XRD-based optimization under two types of structural noise. Crystal structures were optimized with respect to XRD similarity metrics using the snap method riesel2024crystal, which struggles to recover the correct structure under both lattice and coordinate perturbations. The plots show match rates computed with StructureMatcher ($\text{ltol}=0.1, \ \text{stol}=0.2, \ \text{angle\_{tol}}= 5^\circ$) under random lattice (a) and coordinate (b) perturbations. Error bars represent 95% Jeffreys binomial credible intervals brown2001interval. For lattice distortions, incorporating symmetry constraints significantly improves robustness, even at high noise levels.
• Figure 2: 2D landscape of XRD cosine similarity (CS) loss as a function of lattice parameters $a$ and $c$ of U2Ti structure, illustrating the presence of multiple local minima. (a) Cosine similarity loss topographic map showing non-convex behavior with several local minima. (b) XRD patterns for the structures corresponding to the marked local minima: all exhibit reasonably high cosine similarity to the ground truth pattern despite having different lattice parameters.
• Figure 3: 2D landscape of XRD cosine similarity (CS) loss as a function of lattice parameters, with simulated optimization paths for XRD-based gradient descent (GD). Unconstrained GD, unconstrained GD with a constrained initialization, and fully constrained GD. Unconstrained GD converges to some local minima, even with constrained initialization, whereas constrained GD reaches the ground truth. (a) Lattice parameters $a$ and $b$ of cubic Au2S are perturbed. (b) Lattice parameters $a$ and $\gamma$ of monoclinic Na3MnCoNiO6 are perturbed.
• Figure 4: Average Minimum Distance (AMD) vs. XRD Similarity Metrics. (a) Cosine similarity. (b) Entropy similarity. (c) Mean squared error (MSE). (d) Cosine similarity with symmetry constraints applied during optimization. All panels compare structures obtained from XRD-based optimization following lattice distortions of 0.1. In this case, using entropy similarity as the optimization objective yields higher mutual information (MI) than cosine similarity or MSE. However, this trend is not consistent across all noise types and levels (see Table \ref{['tab:mutual_info']}). Applying symmetry constraints improves MI as well as linear and Spearman correlations.
• Figure 5: Lattice and XRD patterns of Na3MnCoNiO6. Each row shows the unit cell relative to the ground truth and corresponding XRD pattern. From top to bottom: ground truth; distorted lattice structure with 0.1 noise level; result of XRD-based GD optimization without constraints; and result of XRD-based GD optimization with symmetry-based constraints. For each, the cosine similarity to the ground truth pattern and the structure match status according to StructureMatcher are reported.