Machine learning for condensed matter physics

Machine learning assisted sorting of active microswimmers

Active matter systems, being in a non-equilibrium state, exhibit complex behaviors, such as self-organization, giving rise to emergent phenomena. There are many examples of active particles with biological origins, including bacteria and spermatozoa, or with artificial origins, such as self-propelled swimmers and Janus particles. The ability to manipulate active particles is vital for their effective application, e.g., separating motile spermatozoa from nonmotile and dead ones, to increase fertilization chance. In this study, we proposed a mechanism-an apparatus-to sort and demix active particles based on their motility values (Péclet number). Initially, using Brownian simulations, we demonstrated the feasibility of sorting self-propelled particles. Following this, we employed machine learning methods, supplemented with data from comprehensive simulations that we conducted for this study, to model the complex behavior of active particles. This enabled us to sort them based on their Péclet number. Finally, we evaluated the performance of the developed models and showed their effectiveness in demixing and sorting the active particles. Our findings can find applications in various fields, including physics, biology, and biomedical science, where the sorting and manipulation of active particles play a pivotal role.

Variational tensor neural networks for deep learning

Deep neural networks (NNs) encounter scalability limitations when confronted with a vast array of neurons, thereby constraining their achievable network depth. To address this challenge, we propose an integration of tensor networks (TN) into NN frameworks, combined with a variational DMRG-inspired training technique. This in turn, results in a scalable tensor neural network (TNN) architecture capable of efficient training over a large parameter space. Our variational algorithm utilizes a local gradient-descent technique, enabling manual or automatic computation of tensor gradients, facilitating design of hybrid TNN models with combined dense and tensor layers. Our training algorithm further provides insight on the entanglement structure of the tensorized trainable weights and correlation among the model parameters. We validate the accuracy and efficiency of our method by designing TNN models and providing benchmark results for linear and non-linear regressions, data classification and image recognition on MNIST handwritten digits.

Discovery of novel materials through machine learning

Experimental exploration of new materials relies heavily on a laborious trial-and-error approach. In addition to substantial time and resource requirements, traditional experiments and computational modelling are typically limited in finding target materials within the enormous chemical space. Therefore, creating innovative techniques to expedite material discovery becomes essential. Recently, machine learning (ML) has emerged as a valuable tool for material discovery, garnering significant attention due to its remarkable advancements in prediction accuracy and time efficiency. This rapidly developing computational technique accelerates the search and optimization process and enables the prediction of material properties at a minimal computational cost, thereby facilitating the discovery of novel materials. We provide a comprehensive overview of recent studies on discovering new materials by predicting materials and their properties using ML techniques. Beginning with an introduction of the fundamental principles of ML methods, we subsequently examine the current research landscape on the applications of ML in predicting material properties that lead to the discovery of novel materials. Finally, we discuss challenges in employing ML within materials science, propose potential solutions, and outline future research directions.

Integrating machine learning and the finite element method for assessing stiffness degradation in photovoltaic modules

This study introduces a novel machine learning (ML) method utilizing a stacked auto-encoder network to predict stiffness degradation in photovoltaic (PV) modules with pre-existing cracks. The input data for the training process was derived from numerical simulations, ensuring a comprehensive representation of module behavior under various conditions. The findings highlight the robust predictive capability of the model, as evidenced by its impressive R2value of 0.961 and notably low root mean square error (RMSE) of 4.02%. These metrics significantly outperform those of other conventional methods, including the artificial neural network with R2of 0.905 and RMSE of 9.43%, the space vector machine with R2of 0.827 and RMSE of 17.93%, and the random forest (RF) with R2of 0.899 and RMSE of 11.02%. Moreover, the findings suggest that the predictive dynamics of degradation are affected by the varying weight functions of different input parameters, such as climate temperature (CT), grain size (GS), material effort, and pre-crack size, as the degradation level changes. Furthermore, a geometric analysis reveals model deficiencies where significant overestimations correlate with thicker glass components, while pronounced underestimations are predominantly associated with thinner layers of polycrystalline silicon wafer and Ethylene Vinyl Acetate in the module. As a case study, it demonstrated that to maintain a constant degradation level between 1.30 and 1.32 in a PV module with components featuring consistent geometric attributes, the input parameters must be kept within specific ranges: CT ranging from 33 °C to 57 °C, GS ranging from 36 to 81μm, material effort ranging from 0.74 to 0.81, and pre-crack size ranging from 24 to 32μm. Therefore, this underscores that the ML model not only predicts degradation but also delineates the parameter space required to achieve a consistent output value.

Enhanced accuracy through machine learning-based simultaneous evaluation: a case study of RBS analysis of multinary materials

We address the high accuracy and precision demands for analyzing large in situ or in operando spectral data sets. A dual-input artificial neural network (ANN) algorithm enables the compositional and depth-sensitive analysis of multinary materials by simultaneously evaluating spectra collected under multiple experimental conditions. To validate the developed algorithm, a case study was conducted analyzing complex Rutherford backscattering spectrometry (RBS) spectra collected in two scattering geometries. The dual-input ANN analysis excelled in providing a systematic analysis and precise results, showcasing its robustness in handling complex data and minimizing user bias. A comprehensive comparison with human supervision analysis and conventional single-input ANN analysis revealed a reduced susceptibility of the dual-input ANN analysis to inaccurately known setup parameters, a common challenge in material characterization. The developed multi-input approach can be extended to a wide range of analytical techniques, in which the combined analysis of measurements performed under different experimental conditions is beneficial for disentangling details of the material properties.

Predicting superconducting transition temperature through advanced machine learning and innovative feature engineering

Superconductivity is a remarkable phenomenon in condensed matter physics, which comprises a fascinating array of properties expected to revolutionize energy-related technologies and pertinent fundamental research. However, the field faces the challenge of achieving superconductivity at room temperature. In recent years, Artificial Intelligence (AI) approaches have emerged as a promising tool for predicting such properties as transition temperature (Tc) to enable the rapid screening of large databases to discover new superconducting materials. This study employs the SuperCon dataset as the largest superconducting materials dataset. Then, we perform various data pre-processing steps to derive the clean DataG dataset, containing 13,022 compounds. In another stage of the study, we apply the novel CatBoost algorithm to predict the transition temperatures of novel superconducting materials. In addition, we developed a package called Jabir, which generates 322 atomic descriptors. We also designed an innovative hybrid method called the Soraya package to select the most critical features from the feature space. These yield R2 and RMSE values (0.952 and 6.45 K, respectively) superior to those previously reported in the literature. Finally, as a novel contribution to the field, a web application was designed for predicting and determining the Tc values of superconducting materials.

Machine learning for structure-property mapping of Ising models: Scalability and limitations

We present a scalable machine learning (ML) framework for predicting intensive properties and particularly classifying phases of Ising models. Scalability and transferability are central to the unprecedented computational efficiency of ML methods. In general, linear-scaling computation can be achieved through the divide-and-conquer approach, and the locality of physical properties is key to partitioning the system into subdomains that can be solved separately. Based on the locality assumption, ML model is developed for the prediction of intensive properties of a finite-size block. Predictions of large-scale systems can then be obtained by averaging results of the ML model from randomly sampled blocks of the system. We show that the applicability of this approach depends on whether the block-size of the ML model is greater than the characteristic length scale of the system. In particular, in the case of phase identification across a critical point, the accuracy of the ML prediction is limited by the diverging correlation length. We obtain an intriguing scaling relation between the prediction accuracy and the ratio of ML block size over the spin-spin correlation length. Implications for practical applications are also discussed. While the two-dimensional Ising model is used to demonstrate the proposed approach, the ML framework can be generalized to other many-body or condensed-matter systems.

ScGAN: a generative adversarial network to predict hypothetical superconductors

Despite having been discovered more than three decades ago, high temperature superconductors (HTSs) lack both an explanation for their mechanisms and a systematic way to search for them. To aid this search, this project proposes ScGAN, a generative adversarial network (GAN) to efficiently predict new superconductors. ScGAN was trained on compounds in Open Quantum Materials Database and then transfer learned onto the SuperCon database or a subset of it. Once trained, the GAN was used to predict superconducting candidates, and approximately 70% of them were determined to be superconducting by a classification model-a 23-fold increase in discovery rate compared to manual search methods. Furthermore, more than 99% of predictions were novel materials, demonstrating that ScGAN was able to potentially predict completely new superconductors, including several promising HTS candidates. This project presents a novel, efficient way to search for new superconductors, which may be used in technological applications or provide insight into the unsolved problem of high temperature superconductivity.

Localization of quantum walks with classical randomness: Comparison between manual methods and supervised machine learning

A transition of quantum walk induced by classical randomness changes the probability distribution of the walker from a two-peak structure to a single-peak one when the random parameter exceeds a critical value. We first establish the generality of the localization by showing its emergence in the presence of random rotation or translation. The transition point can be located manually by examining the probability distribution, momentum of inertia, and inverse participation ratio. As a comparison, we implement three supervised machine learning methods, the support vector machine (SVM), multilayer perceptron neural network, and convolutional neural network with the same data and show they are able to identify the transition. While the SVM sometimes underestimates the exponents compared to the manual methods, the two neural-network methods show more deviations for the case with random translation due to the fluctuating probability distributions. Our work illustrates potentials and challenges facing machine learning of physical systems with mixed quantum and classical probabilities.

Data-driven stochastic simulation leading to the allometric scaling laws in complex systems

We propose a data-driven stochastic method that allows the simulation of a complex system's long-term evolution. Given a large amount of historical data on trajectories in a multi-dimensional phase space, our method simulates the time evolution of a system based on a random selection of partial trajectories in the data without detailed knowledge of the system dynamics. We apply this method to a large data set of time evolution of approximately one million business firms for a quarter century. Accordingly, from simulations starting from arbitrary initial conditions, we obtain a stationary distribution in three-dimensional log-size phase space, which satisfies the allometric scaling laws of three variables. Furthermore, universal distributions of fluctuation around the scaling relations are consistent with the empirical data.

Thermodynamics of the Ising Model Encoded in Restricted Boltzmann Machines

The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.

Searching for the ground state of complex spin-ice systems using deep learning techniques

Searching for the ground state of a given system is one of the most fundamental and classical questions in scientific research fields. However, when the system is complex and large, it often becomes an intractable problem; there is essentially no possibility of finding a global energy minimum state with reasonable computational resources. Recently, a novel method based on deep learning techniques was devised as an innovative optimization method to estimate the ground state. We apply this method to one of the most complicated spin-ice systems, aperiodic Penrose P3 patterns. From the results, we discover new configurations of topologically induced emergent frustrated spins, different from those previously known. Additionally, a candidate of the ground state for a still unexplored type of Penrose P3 spin-ice system is first proposed through this study. We anticipate that the capabilities of the deep learning techniques will not only improve our understanding on the physical properties of artificial spin-ice systems, but also bring about significant advances in a wide range of scientific research fields requiring computational approaches for optimization.

Importance of feature construction in machine learning for phase transitions

Machine learning is an important tool in the study of the phase behavior from molecular simulations. In this work we use un-supervised machine learning methods to study the phase behavior of two off-lattice models, a binary Lennard-Jones (LJ) mixture and the Widom-Rowlinson (WR) mixture. We find that the choice of the feature vector is crucial to the ability of the algorithm to predict a phase transition. We consider two feature vectors, one where the elements are distances of the particles of a given species from a probe (distance-based feature) and one where the elements are +1 if there is an excess of particles of the same species within a cut-off distance and -1 otherwise (affinity-based feature). We use principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) methods to investigate the phase behavior at a critical composition. We find that the choice of the feature vector is key to the success of unsupervised machine learning algorithm in predicting the phase behavior, and the sophistication of the machine learning algorithm is of secondary importance. In the case of the LJ mixture both feature vectors are adequate to accurately predict the critical point, but in the case of the WR mixture the affinity-based feature vector provides accurate estimates of the critical point, but the distance-based feature vector does not provide a clear signature of the phase transition. The study suggests that physical insight in choice of input features is an important aspect of implementing machine learning methods.

Hierarchical classical metastability in an open quantum East model

We study in detail an open quantum generalization of a classical kinetically constrained model-the East model-known to exhibit slow glassy dynamics stemming from a complex hierarchy of metastable states with distinct lifetimes. Using the recently introduced theory of classical metastability for open quantum systems, we show that the driven open quantum East model features a hierarchy of classical metastabilities at low temperature and weak driving field. We find that the effective long-time description of its dynamics not only is classical, but shares many properties with the classical East model, such as obeying an effective detailed balance condition and lacking static interactions between excitations, but with this occurring within a modified set of metastable phases which are coherent, and with an effective temperature that is dependent on the coherent drive.

Deep Learning Analysis of Polaritonic Wave Images

Deep learning (DL) is an emerging analysis tool across the sciences and engineering. Encouraged by the successes of DL in revealing quantitative trends in massive imaging data, we applied this approach to nanoscale deeply subdiffractional images of propagating polaritonic waves in complex materials. Utilizing the convolutional neural network (CNN), we developed a practical protocol for the rapid regression of images that quantifies the wavelength and the quality factor of polaritonic waves. Using simulated near-field images as training data, the CNN can be made to simultaneously extract polaritonic characteristics and material parameters in a time scale that is at least 3 orders of magnitude faster than common fitting/processing procedures. The CNN-based analysis was validated by examining the experimental near-field images of charge-transfer plasmon polaritons at graphene/α-RuCl₃ interfaces. Our work provides a general framework for extracting quantitative information from images generated with a variety of scanning probe methods.

Learning impurity spectral functions from density of states

Using numerical renormalization group calculation, we construct a dataset with 100 K samples, and train six different neural networks for the prediction of spectral functions from density of states (DOS) of the host material. We find that a combination of gated recurrent unit (GRU) network and bidirectional GRU (BiGRU) performances the best among all the six neural networks. The mean absolute error of the GRU + BiGRU network can reach 0.052 and 0.043 when this network is evaluated on the original dataset and two other independent datasets. The average time of spectral function predictions from machine learning is on the scale of 10^-5-10^-6that of traditional impurity solvers for Anderson impurity model. This investigation pave the way for the application of recurrent neural network and convolutional neural network in the prediction of spectral functions from DOSs in machine learning solvers of magnetic impurity problems.

Topics in the mathematical design of materials

We present a perspective on several current research directions relevant to the mathematical design of new materials. We discuss: (i) design problems for phase-transforming and shape-morphing materials, (ii) epitaxy as an approach of central importance in the design of advanced semiconductor materials, (iii) selected design problems in soft matter, (iv) mathematical problems in magnetic materials, (v) some open problems in liquid crystals and soft materials and (vi) mathematical problems on liquid crystal colloids. The presentation combines topics from soft and hard condensed matter, with specific focus on those design themes where mathematical approaches could possibly lead to exciting progress. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

Semi-supervised learning for the study of structural formation in colloidal systems via image recognition

The analysis of the structural formation of colloidal systems using machine learning techniques has recently attracted much attention. In many of these studies, local bond-order parameters (LBOPs) were employed as descriptors, where such LBOPs are suitable mainly for the detection of crystal structures. On the other hand, image-based convolutional neural networks (CNNs) are quite effective in detecting not only crystals but also random structures, and the author demonstrated their efficiency in a previous paper. However, in supervised learning, it is difficult to obtain a correct result when there is an unexpected new phase that was unknown when training the CNN. In this paper, we propose a hybrid scheme that consists of supervised and unsupervised learning techniques, employing two different approaches: image-based CNN and generalized LBOP. The proposed method was applied to two-dimensional colloidal systems, and its efficiency was demonstrated.

Automatic learning of topological phase boundaries

Topological phase transitions, which do not adhere to Landau's phenomenological model (i.e., a spontaneous symmetry breaking process and vanishing local order parameters), have been actively researched in condensed matter physics. Machine learning of topological phase transitions has generally proved difficult due to the global nature of the topological indices. Only recently has the method of diffusion maps been shown to be effective at identifying changes in topological order. However, previous diffusion map results required adjustments of two hyperparameters: a data length scale and the number of phase boundaries. In this article we introduce a heuristic that requires no such tuning. This heuristic allows computer programs to locate appropriate hyperparameters without user input. We demonstrate this method's efficacy by drawing remarkably accurate phase diagrams in three physical models: the Haldane model of graphene, a generalization of the Su-Schreiffer-Haeger model, and a model for a quantum ring with tunnel junctions. These diagrams are drawn, without human intervention, from a supplied range of model parameters.

Assessment of the Wolf method using the Stillinger-Lovett sum rules: From strong electrolytes to weakly charged colloidal dispersions

The Ewald method has been the cornerstone in molecular simulations for modeling electrostatic interactions of charge-stabilized many-body systems. In the late 1990s, Wolf and collaborators developed an alternative route to describe the long-range nature of electrostatic interactions; from a computational perspective, this method provides a more efficient and straightforward way to implement long-range electrostatic interactions than the Ewald method. Despite these advantages, the validity of the Wolf potential to account for the electrostatic contribution in charged fluids remains controversial. To alleviate this situation, in this contribution, we implement the Wolf summation method to both electrolyte solutions and charged colloids with moderate size and charge asymmetries in order to assess the accuracy and validity of the method. To this end, we verify that the proper selection of parameters within the Wolf method leads to results that are in good agreement with those obtained through the standard Ewald method and the theory of integral equations of simple liquids within the so-called hypernetted chain approximation. Furthermore, we show that the results obtained with the original Wolf method do satisfy the moment conditions described by the Stillinger-Lovett sum rules, which are directly related to the local electroneutrality condition and the electrostatic screening in the Debye-Hückel regime. Hence, the fact that the solution provided by the Wolf method satisfies the first and second moments of Stillinger-Lovett proves, for the first time, the reliability of the method to correctly incorporate the electrostatic contribution in charge-stabilized fluids. This makes the Wolf method a powerful alternative compared to more demanding computational approaches.