Information Entropy Based Crystal Structure Prediction of Chemically Disordered Alloys via Graph Convolutional Neural Networks

Abstract: The phase prediction of chemically disordered alloys poses a significant computational challenge due to the combinatorial complexity of such materials. The high-throughput compositional exploration of chemically disordered alloys, including high-entropy alloys, requires an approach to efficiently explore the potential energy landscape of such complex materials. Additionally, a metric to quantify the potential energy landscape explored for phase prediction of the compositions needs to be defined. We propose an information-theoretic approach to phase prediction in chemically disordered alloys in the present work. We demonstrate the applicability of alchemical Monte Carlo sampling using an efficient Graph Convolutional Neural Network-Based machine learning model. We additionally demonstrate the applicability and limitations of the Bond Disproportion Vector (BDV) as a low-computational-cost descriptor and benchmark it against the state-of-the-art Smooth Overlap of Atomic Positions (SOAP) descriptor. We show the applicability of an information entropy-based metric for the phase prediction of binary (CoNi, MoW, FeNi and TaW), ternary (CoCrNi, CrFeNi), quaternary (CoCrFeNi) and quinary ($\mathrm{Al_x(CoCrFeNi)_{1-x}}$) alloys. Information entropy-based phase prediction can be applicable in challenging cases where conventional approaches are not feasible.

• The paper presents a combined GCNN and alchemical Monte Carlo framework that utilizes information entropy to robustly predict phase stability in disordered alloys.
• It compares descriptor-based models (SOAP versus BDV), showing BDV’s smoother convergence and scalability in complex, multicomponent systems.
• The study validates its approach on binary to quinary alloys, demonstrating consistent phase predictions aligned with ab initio and experimental results.

Information-Entropy-Based Framework for Crystal Structure Prediction in Chemically Disordered Alloys

Problem Context and Motivations

Predicting the crystal structures and phase stability of chemically complex, disordered alloys—particularly high-entropy alloys (HEAs)—is a core challenge in computational materials science. The astronomical combinatorial possibilities in atomic arrangements for multicomponent systems render exhaustive first-principles statistical sampling infeasible. Classical thermodynamic approaches such as CALPHAD lack transferability to unexplored compositional domains or structures, while accurate free energy and entropy calculations quickly become computationally prohibitive. Machine learning, and in particular graph-based neural networks (GNNs), have demonstrated significant success for structural and energetic property prediction across disordered alloy landscapes, but rigorous frameworks for robust, information-theoretically motivated phase prediction remain underdeveloped. The present work introduces such a framework, integrating alchemical Monte Carlo sampling, GCNN-based energy prediction, and information entropy metrics to address phase prediction in chemically disordered alloys.

Descriptor Engineering and Featurization Strategies

Atomic environment representation is the primary determinant of ML model fidelity in capturing local energetics in disordered systems. This paper benchmarks two classes of descriptors:

• Smooth Overlap of Atomic Positions (SOAP): Encodes local atomic environments as rotationally invariant power spectra derived from expansions in radial and angular basis functions, enabling rich representations of both bond lengths and angles. However, descriptor dimensionality and associated model complexity scale unfavourably with the number of chemical species, limiting its efficacy in high-throughput settings for multicomponent alloys.
• Bond Disproportion Vector (BDV): Proposed herein as a low-cost, physically interpretable descriptor, BDV quantifies statistical enrichments or depletions of first-shell bond types relative to the fully random (ideal solid solution) baseline. BDV is compact—scaling linearly with the number of bond types—and more resistant to the curse of dimensionality. It is, however, agnostic to geometric details.

For both classes, featurization uses Term Frequency-Inverse Document Frequency (TF-IDF) weighting, conceptually repurposed from NLP. This approach highlights rare yet impactful coordination motifs, downweighting ubiquitous patterns—crucial for efficiently informing ML models on large, intrinsically imbalanced atomistic datasets.

Graph Convolutional Neural Network Approach

GCNNs model each atomistic configuration as a graph where nodes correspond to atoms (featurized by BDV or SOAP), and edges reflect local connectivity. The network architecture consists of stacked GraphConv layers with LeakyReLU activation and a terminal regression layer predicting atomic energy contributions. The model is trained with AdamW on mean squared error loss, and employs standard techniques (e.g., gradient clipping, mini-batching) for stability and scalability.

• Training Dynamics: SOAP-based models exhibit oscillatory loss convergence patterns, interpreted as consequences of high curvature (large Hessian eigenvalues) in the parameter loss landscape generated by complex, correlated features. BDV-based models, by contrast, display smoother, monotonic convergence due to a more regular and lower-dimensional loss surface.
• Extensibility: The graph-based formalism is modular; alternative GNN operators (e.g., GCNConv, SAGEConv, GATConv) and edge-based physical features can be seamlessly incorporated.

Alchemical Monte Carlo Sampling and Energy Landscape Exploration

Configurational sampling is performed via the Genetic Algorithm-based Atomistic Sampling Protocol (GAASP), employing alchemical (elemental identity) swaps with Metropolis filtering to bias the population toward thermodynamically favourable states. At each sampling step, the GCNN provides rapid energy evaluation. This hybrid protocol enables efficient, high-throughput exploration of the potential energy landscape (PEL) for candidate structures (BCC, FCC, etc.), generating statistically meaningful energy distributions indispensable for subsequent entropy analysis.

Information Entropy and Phase Prediction Methodology

The core methodological advance is the application of information-theoretic entropy—specifically Shannon entropy difference between sampled PELs of distinct candidate structures—as a surrogate phase-selection criterion. Formally, given candidate PELs $P$ and $Q$ (e.g., BCC vs. FCC), their relative KL-divergence to a hypothetical equilibrium (Boltzmann-distributed) reference ensemble is shown to reduce to the difference in Shannon entropies:

$$\Delta_{P,Q} = D_{KL}(P\,||\,Eq) - D_{KL}(Q\,||\,Eq) = H(Q) - H(P)$$

where $P$, $Q$ are sampled energy distributions, and $H(\cdot)$ denotes Shannon entropy. This approach sidesteps the infeasibility of direct equilibrium ensemble sampling and explicit free energy calculation.

• Phase Selection Principle: The crystal structure whose sampled energy distribution exhibits higher Shannon entropy at the target temperature is predicted to be thermodynamically preferred—a proxy for lower KL-divergence and thus closer information-theoretic alignment with the true equilibrium structure. This approach leverages the computational tractability of energy evaluation by GCNN.
• Practical Implementation: Prediction results for various binary (CoNi, MoW, FeNi, TaW), ternary (CoCrNi, CrFeNi), quaternary (CoCrFeNi), and quinary (Alx(CoCrFeNi)1-x) alloys demonstrate the utility of this metric. For example, at 300 K, the FCC phase is favoured (higher entropy) in CoNi, FeNi, CoCrNi, CrFeNi, and CoCrFeNi, while the BCC phase is favoured in MoW and TaW, consistent with ab initio and experimental studies.

Performance Benchmarks: Descriptor Suitability and Model Accuracy

• Numerical Outcomes: SOAP-based GCNNs outperform BDV-based models in binary alloys, resolving atom-type segregation in energetic predictions. However, as system complexity increases (from ternary to quinary), the predictive gap narrows; BDV-based models approach SOAP performance for both MSE and $R^2$, indicating that low-dimensional, chemically meaningful descriptors suffice in sufficiently disordered, high-complexity systems.
• Computational Efficiency: Descriptor overhead remains manageable for BDV across alloy complexity, supporting large-scale, high-throughput screening even for extreme composition spaces—critical for AI-driven materials design pipelines.

Implications and Future Directions

This information-entropy-based GCNN/GAASP framework establishes a scalable paradigm for crystal structure and phase stability prediction in chemically complex alloys where conventional approaches are intractable. By directly connecting atomistic configuration sampling, efficient ML-based energy prediction, and rigorous information-theoretic metrics, the methodology enables:

• Rapid, physically interpretative ranking of candidate structures even in ultra-high compositional spaces (e.g., HEAs, medium-entropy alloys).
• A modular basis for incorporating additional physical constraints or descriptors, edge-based features, or domain-adapted graph neural architectures.
• Theoretical extension to temperature- or pressure-dependent phase diagrams if PEL sampling is adapted to varying thermodynamic conditions.

Possible future work includes rigorous uncertainty quantification in entropy-based phase prediction, integration of ab initio-generated reference ensembles for enhancing model transferability, and direct coupling with experiment via active learning protocols.

Conclusion

The presented framework—comprising alchemical Monte Carlo sampling, GCNN-based energy regression, and information entropy-based phase selection—delivers a computationally tractable, physically motivated method for crystal structure prediction in chemically disordered alloys. It demonstrates descriptor- and model-agnostic scalability, provides strong numerical evidence for phase identification consistent with known thermodynamics, and suggests broad applicability for accelerating the computational discovery and optimization of complex alloys.