Amorphous DPR: Diffusion Modeling for Amorphous Materials

• Amorphous DPR is a diffusion-probabilistic representation framework that applies DDPMs to generate atomic configurations in glasses and disordered solids, achieving up to 1000x speed improvement over MD simulations.
• It utilizes an E(3)-equivariant GNN to condition the generation process on physical parameters, ensuring accurate short- and medium-range structural fidelity.
• Validated against mechanical and structural metrics, amorphous DPR offers a promising inverse design route for accelerating simulations of complex disordered materials.

Amorphous DPR refers to "diffusion-probabilistic representation" (DPR) frameworks specifically designed for the generative modeling and simulation of amorphous materials. The key innovation is the application of denoising diffusion probabilistic models (DDPMs) to directly generate atomic configurations of glasses and disordered solids, enabling efficient, accurate sampling of physical structures at scale. This approach has achieved order-of-magnitude improvements in computational efficiency and sampling diversity relative to classical simulation methods for amorphous matter, including complex silica glasses and metallic glasses (Yang et al., 7 Jul 2025).

1. Theoretical Foundations of Amorphous Diffusion-Probabilistic Representation

Amorphous DPR leverages the DDPM paradigm for modeling the generation of atomic coordinates $x \in \mathbb{R}^{3N}$ as a stochastic denoising process. The method defines a forward Markovian process where Gaussian noise is iteratively added to the atomic positions:

$$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$

with a controlled noise schedule, typically linear: $\sigma_t = \sigma_{\max}(1 - t/T)$, often with $\sigma_{\max} \sim 1.0$ Å. The reverse (generative) process employs a learned model $p_\theta(x_{t-1}|x_t)$, which denoises $x_t$ towards realistic atomic configurations conditioned on material composition, volume, and process parameters.

The training objective is the noise prediction (score-matching) loss:

$$\min_\theta\, \mathbb{E}_{t, x_0, \varepsilon} \left\|\varepsilon_\theta(x_t, t) - \varepsilon\right\|^2,\qquad x_t = x_0 + \sigma_t \varepsilon,~\varepsilon \sim \mathcal{N}(0,I)$$

This construction enables learning a distribution over atomic configurations consistent with the statistics of observed melt–quench structures, without requiring explicit physical simulation at each sample (Yang et al., 7 Jul 2025).

2. Network Architecture and Conditioning

The denoising model, $\varepsilon_\theta(\cdot)$, is implemented as an E(3)-equivariant graph neural network (GNN), extending architectures such as NequIP for direct atomic structure generation. Essential architectural features include:

• Input: Noised atomic coordinates $x_t$ and per-atom species
• Species embeddings: 8-dimensional
• Three equivariant convolutional layers (64 scalar, 32 vector channels per layer)
• Edge features: Spherical harmonics up to rank-2, radial expansion via two-layer MLP
• Local neighborhoods up to 12 neighbors and 5 Å cutoff distance
• Output: Per-atom noise vector in $\mathbb{R}^{3N}$
• Optional conditioning: Physical parameters, e.g., cooling rate $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$0, included via radial basis expansion and MLP embedding added to the GNN hidden state at each layer

This design ensures E(3)-equivariance, necessary for modeling atomic systems with rotational, translational, and permutational invariance (Yang et al., 7 Jul 2025).

3. Training and Sampling Protocol

The training dataset consists of atomic structures obtained from molecular dynamics (MD) melt–quench trajectories, over a range of processing conditions and compositions (e.g., 3000-atom SiO$$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$1 glasses at $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$2 K/ps; Cu$$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$3Zr$$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$4 metallic glass). For each sample, noisy versions $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$5 are generated for uniformly sampled $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$6.

The generative sampling strategy is as follows:

1. Initialize $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$7 in the simulation cell.
2. For $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$8 down to 1, repeatedly:
   • Use the GNN to predict $$q(x_t | x_{t-1}) = \mathcal{N}(x_{t-1}, \sigma_t^2 I), \quad t = 1, \ldots, T$$9.
   • Denoise: $\sigma_t = \sigma_{\max}(1 - t/T)$0.
   • Sample $\sigma_t = \sigma_{\max}(1 - t/T)$1, set $\sigma_t = \sigma_{\max}(1 - t/T)$2.
3. Optionally, apply a short molecular dynamics refinement ($\sigma_t = \sigma_{\max}(1 - t/T)$325 ps NVT, 25 ps NPT) to correct rare outlier bonding environments.

This yields valid atomic configurations faithfully sampling the amorphous structure distribution. For system sizes of 3000 atoms, this procedure is up to $\sigma_t = \sigma_{\max}(1 - t/T)$4 times faster than MD melt–quench simulations at experimental cooling rates (Yang et al., 7 Jul 2025).

4. Validation and Physical Metrics

Validation of amorphous DPR-generated structures encompasses multiple levels:

• Short-range order: Partial pair distribution functions (PDFs) $\sigma_t = \sigma_{\max}(1 - t/T)$5 for relevant atomic pairs, with peak positions and intensities matching MD and experiment.
• Medium-range order: Ring-size distributions computed via algorithms such as RINGS, verifying the preservation of medium-range connectivity.
• Mechanical properties: Elastic moduli ($\sigma_t = \sigma_{\max}(1 - t/T)$6, $\sigma_t = \sigma_{\max}(1 - t/T)$7, $\sigma_t = \sigma_{\max}(1 - t/T)$8, $\sigma_t = \sigma_{\max}(1 - t/T)$9) extracted from the stiffness tensor $\sigma_{\max} \sim 1.0$0, with distributions indistinguishable from MD ground truth.
• Diversity and novelty: Information-theoretic metrics such as differential entropy and environment overlap (QUESTS), confirming that generated samples exhibit comparable structural diversity to those from MD, without mode collapse.

For metallic glasses, generated polyhedral distributions (e.g., Voronoi indices $\sigma_{\max} \sim 1.0$1) and stress–strain responses match their reference MD and experimental counterparts, including accurate reproduction of radical features such as “3–3” pair peaks and medium-range shoulders in pair distribution functions (Yang et al., 7 Jul 2025).

5. Conditional and Generalized Generation

Amorphous DPR models are capable of conditional structure generation. Most notably, the model can be conditioned on cooling rate $\sigma_{\max} \sim 1.0$2 across several orders of magnitude (e.g., $\sigma_{\max} \sim 1.0$3–$\sigma_{\max} \sim 1.0$4 K/ps), allowing direct sampling of amorphous structures at process conditions unattainable with classical MD due to prohibitive timescales.

Key observations include:

• Structural evolution with cooling rate: Mean information entropy and bond angle statistics are correctly reproduced as functions of $\sigma_{\max} \sim 1.0$5.
• Mechanical behavior: Fracture testing on generated structures reveals accurate ductile-to-brittle transitions as system size and cooling rate vary.
• Mesoporous silica: By initializing positions outside a predefined exclusion region and applying the diffusion process, correct non-bridging oxygen densities are achieved, demonstrating the model's ability to generate complex topological motifs.

Extension to new material classes (e.g., metallic glasses with experimental atomic electron tomography data) confirms the method's generality (Yang et al., 7 Jul 2025).

6. Computational Efficiency and Limitations

The amortized cost of generative structure sampling with amorphous DPR is nearly independent of system size and cooling rate, scaling as $\sigma_{\max} \sim 1.0$6 with the number of atoms due to the local message passing in the GNN. For 3000 atoms, DPR can generate atomic structures in $\sigma_{\max} \sim 1.0$7 CPU-hours, versus $\sigma_{\max} \sim 1.0$8 CPU-hours for MD at $\sigma_{\max} \sim 1.0$9 K/ps.

Limitations of current implementations include:

• Occasional need for post-generation MD refinement to remove unphysical environments.
• Underestimation of yield strengths in mechanical testing, which may indicate model sensitivity or missing physical constraints.
• Density is fixed during sampling; variable-density generative models remain an open research direction.

These constraints delimit the direct applicability of DPR and suggest future extensions toward fully unsupervised, variable-density amorphous sampling and enhanced physical fidelity (Yang et al., 7 Jul 2025).

7. Impact and Prospective Developments

The emergence of amorphous DPR methodologies has provided a concrete roadmap for inverse design and simulation of amorphous and glassy materials directly from data. By reproducing short- and medium-range order, mechanical properties, and processing–structure–property relationships with high fidelity, and doing so at multiple orders of magnitude lower computational cost, these models are positioned to significantly accelerate the study of complex disordered materials.

Potential future advances include the integration of experimental spectroscopic or diffraction data for direct inverse reconstruction, online adaptation to new compositions or processing regimes, and integration of variable-density modeling for true ab initio amorphous material design (Yang et al., 7 Jul 2025).