Machine learning identifies scale-free properties in disordered materials

Disordered systems, encompassing periodic, quasiperiodic, and various disordered structures, offer vast design freedom for signal processing but pose challenges for deterministic design. Traditional approaches using mathematical microstructural descriptors are time-consuming and limited by the complexity of higher-order descriptors. This paper proposes a deep learning approach leveraging the power of convolutional neural networks (CNNs) to circumvent these limitations. The authors aim to establish a data-driven relationship between disordered structures and wave behaviors, enabling efficient prediction and design of materials with desired wave interaction properties. This deep learning framework is expected to overcome the challenges associated with the vast design space inherent in disordered systems and provide a more efficient and versatile method for material design.

Previous research has explored disordered structures and their wave behaviors through mathematical microstructural descriptors like n-point probability, percolation, or cluster functions. These descriptors capture specific aspects of structural patterns, enabling classification and revealing origins of wave behaviors. Inverse design methods, such as stochastic, genetic, and topological optimizations, have been developed to generate disordered structures with target wave properties. However, these approaches are hindered by the large design freedom in disordered systems, making them time-consuming and problem-specific. Most studies focus on lower-order descriptors due to the complexity of higher-order ones. Recent advancements in deep learning have shown promise in handling complex physics problems, including crystal classification, topological order identification, phase transition analysis, optical device designs, and image reconstruction. The application of deep learning to disordered systems is expected to facilitate the exploration of complex wave behaviors.

The authors utilize two deep convolutional neural networks (CNNs): a disorder-to-localization (D2L) CNN and a localization-to-disorder (L2D) CNN. Disordered structures are represented as multicolor images, where the x and y displacements of atomic sites are projected onto separate color channels. Wave localization is quantified by the normalized mode area (inverse of the inverse participation ratio). The D2L CNN is trained to predict the mode area from the input image representing the disordered structure. The training data consists of a large dataset of randomly deformed lattices and their corresponding localization properties, incorporating both collective and individual atomic site deformations. The mean absolute percentage error (MAPE) is used as the cost function. The L2D CNN is trained in conjunction with the pre-trained D2L CNN (forming an L2D2L network) to generate disordered structures from target localization profiles. The L2D2L network uses the same MAPE cost function. Dropout and L2 regularization techniques are employed to prevent overfitting. The networks are implemented using Google TensorFlow. The deformation of lattices in the dataset is described by equations involving collective and individual displacements of atomic sites (equation 5 in the paper).

The D2L CNN achieves a high test accuracy (≈94.80%) in predicting localization properties. The L2D CNN, trained using the L2D2L network, exhibits a high test accuracy (≈94.21%) in generating disordered structures corresponding to target localization profiles. Crucially, the ML-generated disordered structures demonstrate scale invariance, following a power-law distribution, unlike the normal distribution of the seed structures. This scale invariance results in a two-to-four order-of-magnitude improvement in robustness to accidental defects compared to conventionally disordered structures. Analysis reveals heavy-tailed distributions and the presence of hub atoms, which are characteristic of scale-free networks and explain the observed robustness to accidental attacks and sensitivity to targeted attacks. Further analysis shows a strong correlation between the scale-free properties of the ML-generated structures and the weight distribution in the L2D CNN's fully connected layer, indicating the influence of network architecture on the generated material properties. This scale-free behavior is robust across different test accuracies, suggesting it's a fundamental property arising from the network architecture rather than simply a consequence of high accuracy.

The findings demonstrate that machine learning can identify and generate disordered materials with specific wave localization properties and scale-free characteristics. The scale-free nature of the ML-generated structures, with their heavy-tailed distributions and hub atoms, translates to significantly improved robustness against accidental errors and increased sensitivity to targeted modulations. This aligns with the behavior of scale-free networks observed in other fields. The strong correlation between network architecture and material properties suggests that the design of CNNs can be tailored to generate materials with specific scale-free properties. This study opens new possibilities for designing materials with tailored wave interactions.

This research successfully demonstrates the use of machine learning for identifying and generating disordered materials exhibiting scale-free wave properties. The developed CNNs effectively predict localization and generate structures with heavy-tailed distributions, leading to enhanced robustness to defects. The study highlights the critical role of neural network architecture in shaping the generated material's characteristics. This work paves the way for designing advanced wave devices leveraging the robustness and tunability of scale-free materials.

The study focuses on two-dimensional lattices. The generalizability to three-dimensional systems and other types of disorder requires further investigation. The accuracy of the L2D CNN might be limited by the extrapolation beyond the training data range, particularly in the strong localization regime. The perfect scale-free property is only observed in the infinite-size limit, and this study is limited by the finite size of the lattice.