In Algebraic Statistics, M.A. Cueto, J. Morton and B. Sturmfels introduced a statistical model, the Restricted Boltzmann Machine, which introduced the Hadamard product of two or more vectors of an affine or projective space, i.e., the componentwise product of their entries, forcing Algebraic Geometry to enter. The Hadamard product X⋆Y of two subvarieties X,Y⊂Pn is defined as the Zariski closure of the Hadamard product of its elements. Recently, D. Antolini and A. Oneto introduced and studied the definition of Hadamard rank, and we prove some results on it. Moreover, we prove some theorems on the dimension and shape of the Hadamard powers of X. The aim is to describe the images of the Hadamard products without taking the Zariski closure. We also discuss several scenarios describing the case in which some of the data, i.e., the variety X, is wrong or it is not possible to recover it.

Restricted Boltzmann machine (RBM) is a typical stochastic neural network with wide applications in image generation and recognition. Hardware implementation of RBMs using emerging devices (e.g., memristors) is promising for high energy efficiency, but it remains challenging because the devices with distinctly different switching characteristics are required to implement the stochastic neurons and deterministic synapses of the RBM, respectively. Here, an all-ferroelectric RBM, which consists of neurons based on nanoscale ferroelectric field-effect transistors (FeFETs) exhibiting probabilistic switching behavior and synapses based on microscale ferroelectric tunnel junctions (FTJs) showing deterministic switching behavior, is demonstrated by simulation. It is shown that the all-ferroelectric RBM is capable of effectively capturing and processing essential features of input images, achieving high image reconstruction performance. Moreover, the all-ferroelectric RBM's feature extraction capability is further enhanced by optimizing the device parameters of the FeFET and the mapping coefficients in the pulse scheme. Subsequently, the optimized all-ferroelectric RBM is integrated with an FTJ-based artificial neural network for image recognition. This integrated system achieves an accuracy of ∼91.8%, which almost reaches the ∼92% benchmark of a software model, demonstrating the effectiveness of the all-ferroelectric RBM as a feature extractor. This study presents a novel hardware-based approach for developing RBMs by using ferroelectric memristors with distinct switching characteristics.

Next-generation neuromorphic systems require hardware platforms that seamlessly integrate sensing, memory, and computation. Here, we present a light-programmable optoelectronic memristor based on an ITO/IGZO/W structure, capable of emulating a broad spectrum of synaptic and neuronal functions under purely optical stimulation through the transparent ITO top electrode. The device exhibits short-term plasticity, including excitatory postsynaptic current (EPSC), paired-pulse facilitation (PPF), and spike-dependent learning behaviors (SADP, SWDP, SNDP). It also replicates nociceptive responses such as threshold activation, no adaptation, relaxation, and sensitization. Pavlovian associative learning is demonstrated using optical stimuli, showing acquisition, extinction, and recovery behaviors driven by persistent photoconductivity. Furthermore, a 4-bit optical pulse-driven reservoir computing architecture achieves 97.005% MNIST classification accuracy through a convolutional neural network readout. A light-induced stochastic activation function, extracted from threshold-switching behavior, is applied in a Restricted Boltzmann Machine to model probabilistic neurons, reaching 96.35% image reconstruction accuracy. Postforming optical modulation enables light-intensity-dependent trap/detrap dynamics and fine-tuning of the conductive filament. These results highlight the proposed IGZO-based optoelectronic memristor as a versatile and energy-efficient platform for multifunctional neuromorphic computing, combining sensory, deterministic, and probabilistic intelligence in a single reconfigurable device.

Business English translation plays a crucial role in promoting international trade and technological exchange, but traditional translation methods suffer from issues such as low efficiency and insufficient accuracy. To address these challenges, this paper proposes a business English translation model based on Weighted Multi objective Deep Belief Network (WM-DBN), combined with artificial intelligence (AI) and fifth generation mobile communication (5G) technology, aiming to improve translation quality and shorten modeling time. This model constructs a deep network structure by stacking multiple Restricted Boltzmann Machines (RBMs) and optimizes it using a strategy that combines unsupervised layer by layer pretraining with supervised fine-tuning to better capture the complex dependencies between texts. This study not only effectively improves the quality and efficiency of business English translation, providing strong support for information flow and communication between multinational enterprises, but also provides new technical ideas for translation work in other fields.

This work resolves the crucial problem of secure data transmission in IoT-aided healthcare devices, where confidential patient data is vulnerable to breaches and cyberattacks. To resolve these complexities, this work proposes a novel secure data transmission system that combines blockchain technology, Multiobjective Weighted Restricted Boltzmann Machine (MW-RBM) for feature extraction, Magnified Feeding-based American Zebra Optimization (MFAZO) for weight optimization, and a Multiscale Stacked Residual-Gated Recurrent Unit (MSRes-GRU) for attack detection. The novelty of this work lies in combining residual GRU and blockchain for secure IoT healthcare data transmission, guaranteeing both transparency and attack detection. The improvement is displayed in weight optimization through MFAZO, which refines the feature extraction task and boosts the accuracy of the technique in attack detection. The designed approach involves gathering attack detection data, performing feature extraction utilizing MW-RBM with optimized weights and identifying IoT node attacks via the MSRes-GRU technique’s multiscale layers and the residual connections. The Homomorphic Polynomial Encryption (HPE) is further employed to secure the healthcare data during transmission. Lastly, the performance of the model is determined with conventional models. The accuracy of the designed MSRes-GRU is 96.22%, which is higher than the existing models such as DNN (85.65%), LSTM (80.71%), SVM (89.99%), and GRU (94.14%). The key results demonstrate the technique’s high detection accuracy and robust performance in recognizing the IoT-based attacks while guaranteeing effective, secure and transparent data transmission via blockchain. This research contributes to improving the secure and scalable IoT-enabled healthcare devices, providing a reliable model for trustworthy healthcare applications that preserve data integrity and privacy.

Riboswitches are structured allosteric RNA molecules that change conformation upon metabolite binding, triggering a regulatory response. Here we focus on the de novo design of riboswitch-like aptamers, the core part of the riboswitch undergoing structural changes. We use Restricted Boltzmann machines (RBM) to learn generative models from homologous sequence data. We first verify, on four different riboswitch families, that RBM-generated sequences correctly capture the conservation, covariation and diversity of natural aptamers. The RBM model is then used to design new SAM-I riboswitch aptamers. To experimentally validate the properties of the structural switch in designed molecules, we resort to chemical probing (SHAPE and DMS), and develop a tailored analysis pipeline adequate for high-throughput tests of diverse sequences. We probe a total of 476 RBM-designed and 201 natural sequences. Designed molecules with high RBM scores, with 20% to 40% divergence from any natural sequence, display ≈ 30% success rate of responding to SAM with a structural switch similar to their natural counterparts. We show how the capability of the designed molecules to switch conformation is connected to fine energetic features of their structural components.

Non-Hermitian (NH) quantum systems have emerged as a powerful framework for describing open quantum systems, nonequilibrium dynamics, and engineered quantum optical materials. However, solving the ground-state properties of NH systems is challenging due to the exponential scaling of the Hilbert space and exotic phenomena such as the emergence of exceptional points. Another challenge arises from the limitations of traditional methods like exact diagonalization (ED). For the past decade, neural networks (NNs) have shown promise in approximating many-body wavefunctions, yet their application to NH systems remains largely unexplored. In this paper, we explore different NN architectures to investigate the ground-state properties of a parity-time-symmetric, one-dimensional NH, transverse field Ising model with a complex spectrum by employing a recurrent neural network (RNN), a restricted Boltzmann machine (RBM), and a multilayer perceptron (MLP). We construct the NN-based many-body wavefunctions and validate our approach by recovering the ground-state properties of the model for small system sizes, finding excellent agreement with ED. Furthermore, for larger system sizes, we demonstrate that the RNN outperforms both the RBM and MLP. However, we show that the accuracy of the RBM and MLP can be significantly improved through transfer learning, allowing them to perform comparably to the RNN for larger system sizes. These results highlight the potential of neural-network-based approaches—particularly for accurately capturing the low-energy physics of NH quantum systems in case of both weak and strong non-Hermiticity.

Abstract Bearings are critical components of mechanical equipment, and predicting their remaining useful life (RUL) is important in industry. This paper proposes a RUL prediction method based on the assessment of a bearing’s health status. Features from the time, frequency, and time-frequency domains of the bearing’s vibration signal are extracted to construct a feature set. A multibranch encoder and restricted Boltzmann machine are used to improve the stacked autoencoder to reduce dimensionality. Local weights and health-sample means are introduced into the Mahalanobis distance to improve the health index. Subsequently, the weighted convolutional Euclidean distance serves as the distance metric in K-means clustering to achieve a more accurate health status assessment and provide historical data for RUL prediction. An improved self-attention (ISA) mechanism is proposed by incorporating depthwise separable convolutions and residual-like connections into self-attention mechanisms, enhancing the global and local dependencies of the temporal convolutional network (TCN). Thus, a more accurate RUL prediction is achieved. Comparative and ablation experiments confirm that the proposed ISA-TCN achieves superior predictive accuracy. Generalization experiments further demonstrate its strong adaptability, while anti-noise experiments demonstrate its strong robustness to uncertainties. Finally, experiments on multi-output RUL predictions validate the model’s effectiveness. This approach offers valuable insights for RUL prediction of rotating machinery under real-world scenarios involving variable operating conditions, noise interference, and multi-device environments.

The growing reliance on satellites highlights the need to understand how space weather quantitatively impacts the reliability and efficiency of power subsystems. While it is well established that space weather disturbances can trigger anomalies in satellite operations, most existing studies lack integrated, data-driven approaches capable of capturing the complex, nonlinear interactions between space weather parameters and satellite health telemetry. This study addresses this gap by introducing a novel four-stage data driven workflow to examine the relationship between key space weather indicators (proton flux, AL index, galactic cosmic rays (GCR), and solar wind density) and the NN1_Voltage and TBS1_Current and temperature of the EgyptSat-1 satellite power subsystem (T1BS, T3BS). The workflow includes: (1) data preprocessing; (2) To handle the high dimensionality and complexity of the data, a two-stage non-linear feature selection approach was employed. In the first stage, an unsupervised Restricted Boltzmann Machine (RBM) was applied to extract a compact and structurally stable feature subset. This was followed by a supervised mutual information (MI) validation step to ensure maximum predictive relevance to the satellite target parameters (T1BS and T3BS). (3) six machine learning models namely: Convolutional Neural Networks (CNN), Long Short-Term Memory (LSTM) networks, Random Forest Regressor, Adaptive Boost, Gradient Boosting, and Voting Regressor, to capture dynamic system behaviours; and (4) anomaly detection and validation by correlating prediction residuals with space weather disturbances, using STL decomposition and Z-score for GCR and P10 anomaly detection, and coincidence rate analysis to assess temporal alignment. The Random Forest (RF) model exhibited strong predictive performance. For NN1_Voltage, the mean squared error (MSE) was 0.00147 (95% CI: 0.00120-0.00184), the root mean squared error (RMSE) was 0.038 (95% CI: 0.0346-0.0428), the mean absolute error (MAE) was 0.028 (95% CI: 0.026-0.031), and the mean absolute percentage error (MAPE) was 0.09% (95% CI: 0.08-0.10%). For TBS1_Current, RF achieved an MSE of 0.0405 (95% CI: 0.0319-0.0489), RMSE of 0.201 (95% CI: 0.179-0.221), MAE of 0.153 (95% CI: 0.136-0.169), and MAPE of 2.4% (95% CI: 2.1-2.6%). Furthermore, analysis of detected anomalies revealed temporal coincidence rates of 31% with GCR disturbances and 27% with P10 proton events. Statistical validation using chi-squared and Fisher's exact tests yielded significant p-values (e.g., p = 2.83 × 10⁻³ for GCR; p = 5.63 × 10⁻⁷ for P10), suggesting a potential relationship worth further investigation. This analysis is particularly relevant for assessing unexplained satellite failures such as the loss of EgyptSat-1 and contributes to improved resilience and monitoring strategies for future missions. While the proposed workflow shows strong predictive performance, its validation is currently limited to a single satellite dataset, highlighting the need for broader cross-mission testing. This study not only enhances our understanding of space weather impacts on satellite power systems but also demonstrates the potential of machine learning in improving anomaly detection and resilience of satellites operating in challenging space environments.

The steady-state visual evoked potential (SSVEP), a non-invasive EEG modality, is a prominent approach for brain–computer interfaces (BCIs) due to its high signal-to-noise ratio and minimal user training. However, its practical utility is often hampered by susceptibility to noise, artifacts, and concurrent brain activities, complicating signal decoding. To address this, we propose a novel hybrid deep learning model that integrates a multi-channel restricted Boltzmann machine (RBM) with a convolutional neural network (CNN). The framework comprises two main modules: a feature extraction module and a classification module. The former employs a multi-channel RBM to unsupervisedly learn latent feature representations from multi-channel EEG data, effectively capturing inter-channel correlations to enhance feature discriminability. The latter leverages convolutional operations to further extract spatiotemporal features, constructing a deep discriminative model for the automatic recognition of SSVEP signals. Comprehensive evaluations on multiple public datasets demonstrate that our proposed method achieves competitive performance compared to various benchmarks, particularly exhibiting superior effectiveness and robustness in short-time window scenarios.

Neuromodulation via Krotov-Hopfield improves accuracy and robustness of restricted Boltzmann machines

Extracting offline retail shopping patterns: a restricted Boltzmann machines approach to customer segmentation and cross-selling

ABSTRACT The traditional cybersecurity solution cannot offer the necessary protection against all forms of cyberattacks for management systems since cyberattacks are of a scattered nature. Thus, an efficient security method is suggested in this work to prevent data loss from the information system using optimization and deep learning. The major aim of this work is to offer suitable recommendations for developing a cybersecurity decision support model for information systems with risk analysis and cybersecurity models. In the information system, the security threats are eliminated using this model. So, the authentication, prediction of the risk levels, and the mitigation of the security threats are regarded as the major objectives of this research work. From the CICIDS 2017 KNN dataset, UNSW‐NB15 dataset, and NSL‐KDD dataset, the data is collected. The collected data is given to the executed Deep Security Network (DSecNet). In the DSecNet, the Restricted Boltzmann Machine (RBM) is utilized to retrieve the features and then the extracted features from the RBM are fused with optimized weights, which are selected using the Enhanced Sandpiper Optimization Algorithm (ESOA). The obtained weighted features are further provided to the Deep Capsule Neural Network (DeepCapsNet). The implemented DSecNet is used for verifying the authentication of the user, detecting the intrusion, and determining the associated risk levels caused by the attacks on the information system. Once the threats are detected, the mitigation of the threat based on their risk level is carried out. The effectiveness of the proposed model is proven by the validation process. From the simulation outcomes, the accuracy rate of the developed model is 97.11% in the NSL‐KDD dataset and 96.70% in the UNSW‐NB15 dataset based on intrusion detection performance. Therefore, the efficacy of the developed security system is higher for the performance of the authentication, intrusion detection, and risk prediction, which elevates the security of the cyber networks.

INTRODUCTIONMachine learning models leverage baseline data to create artificial intelligence (AI)–generated digital twins (DTs)—individualized predictions of each participant's clinical outcomes if they had received placebo. Incorporating DTs may increase statistical power or reduce required sample sizes in Phase 2 or 3 trials, and therefore improve efficiency in Alzheimer's disease (AD) trials. Here we demonstrate these properties using data from an AD Phase 2 clinical trial (AWARE, NCT02880956).METHODSA conditional restricted Boltzmann machine (CRBM) model was trained on historical clinical trials and observational data from 6736 unique subjects after data harmonization to generate DTs of participants from the AWARE trial. The AWARE trial enrolled 453 subjects with mild cognitive impairment (MCI) or mild AD. DTs were assessed as prognostic covariates to evaluate gains in variance and sample size reduction.RESULTSPositive partial correlation coefficients were found between DTs and change score from baseline in key cognitive assessments ranging from 0.30 to 0.39 at Week 96 in the AWARE trial. These correlations were consistent with validation results from three independent trials, which ranged from 0.30 to 0.46. Total residual variance was reduced by ~9% to 15% with DTs. While maintaining statistical power, DTs could reduce total sample size by ~9% to 15%, and control arm sample size by 17% to 26% in future AD trials.DISCUSSIONEfficiency was improved in AD clinical trials using machine learning models to generate prognostic DTs by including them in statistical analysis modeling. This methodology aligns with regulatory guidance and represents an application of machine learning models suitable for the analysis of pivotal trial data. Validated DTs have the potential to improve clinical development efficiency in AD and in other neurological indications.HighlightsDigital twins (DTs) were generated by artificial intelligence (AI) models trained on historical datasets.Use of digital twin (DT) as a covariate in the analysis model can reduce treatment effect variability.By coupling DT with the analysis model, trial sample size can be reduced.DT technology was accepted by the U.S. Food and Drug Administration and European Medicines Agency for applications in clinical trials.

Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: One is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three, and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.

Curtain grouting is widely used to reduce the permeability of dam foundations, yet forecasting cement intake remains challenging because flow pathways are governed by the three-dimensional connectivity of rock fractures. We develop a hybrid framework that explicitly embeds 3D fracture connectivity into data-driven prediction. A discrete fracture network (DFN) is constructed and traversed using depth-first search (DFS) from each grouting hole segment to capture both direct and multistep connections. Six connectivity descriptors are computed—the number of reachable fractures (N), average inclination (I), average dip angle (D), cumulative connected volume (V), average radius (r), and average width (w)—and combined with construction parameters as model inputs. Cement intake is predicted using an integrated model that combines a Restricted Boltzmann Machine (RBM)-pretrained multilayer perceptron with channel-wise squeeze-and-excitation (SE) attention, where key hyperparameters are optimized via a genetic algorithm (GA). Applied to a curtain-grouting project (448 segments), the connectivity-aware model improves agreement with observations over a no-connectivity baseline: the correlation coefficient (R) increases from 0.938 to 0.972, while mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE) decrease by 27.1%, 12.2%, and 32.8%, respectively. Ablation studies validate the contributions of each component, confirming that RBM pretraining enhances generalization, SE attention improves feature selection, and ensemble aggregation stabilizes predictions. Compared to four optimized baseline models (SVR, RF, ELM, and LSTM), the proposed integrated method achieves improvements of 3–34% in R and reductions of 22–55% in MAE, 31–66% in RMSE, and 2–48% in MAPE on the held-out test set. This model provides engineers with a simple, cost-effective tool for accurate predictions to support better decision-making in grouting.

Rapid advancement in banking technology not only enhances productivity and improves people's lives but also introduces significant risks. This work proposes an effective framework to predict financial risk in Supply Chain Management (SCM). At first, the data needed to perform the process is taken from various data sources. Further, the feature extraction process with the help of Principal Component Analysis (PCA) and Restricted Boltzmann Machine (RBM), and also the statistical features are extracted from the inputted data. Moreover, the features PCA, RBM, and statistical features are fed to the weighted fused features phase, whereas the weights are tuned by Revised Archimedes Optimisation (RAO). Then, the fused features are inputted into the Residual Convolution Gated Recurrent Unit (Res-CGRU) for the prediction process. Finally, the suggested Res-CGRU model tends to show a predicted outcome. The effectiveness of designed approach is compared with several baseline systems to confirm its superiority over others.

The classic paradigms for learning and memory recall focus on the strengths of synaptic couplings and how these can be modulated to encode memories. In this work, using analytical theory and computational inference schemes that use specialized restricted Boltzmann machines, we show that the dynamical steady state accessed due to a class of nonequilibrium detailed balance breaking dynamics is in fact similar to those accessed after the operation of a classic unsupervised scheme for improving memory recall, Hebbian unlearning, or “dreaming.” Our work suggests how nonequilibrium dynamics can provide an alternative route for controlling the memory encoding and retrieval properties of a variety of synthetic (neuromorphic) and biological systems.

MotivationMachine-generated or synthetic data is a valuable resource for training artificial intelligence algorithms, evaluating rare workflows, and sharing data under stricter data legislations. However, current statistical and deep learning methods struggle with large data volumes, are prone to hallucinating scenarios incompatible with reality, and seldom quantify privacy meaningfully.ResultsHere, we introduce Genomator, a logic solving approach (SAT solving), which efficiently produces private and realistic representations of the original data. We demonstrate the method on genomic data, which arguably is the most complex and private information. We benchmark Genomator against state-of-the-art methodologies (Markov generation, Wasserstein Generative Adversarial Network and Conditional Restricted Boltzmann Machines), demonstrating a 40%–530% accuracy improvement and 57%–172% higher privacy. Genomator is also 3–100 times more efficient, making it the only tested method that scales to whole genomes. We show the universal trade-off between privacy and accuracy, and use Genomator’s tuning capability to cater to all applications along the spectrum, from provable private representations of sensitive cohorts, to datasets with indistinguishable pharmacogenomic profiles. Demonstrating the production-scale generation of tuneable synthetic genomes hold great potential for balancing underrepresented populations in medical research and advancing global data exchange.Availability and implementationGenomator is available at https://github.com/csiro/genomator.

Abstract Next‐generation neuromorphic hardware must concurrently address computation, learning, and security demands. Here, a photonic‐driven neuromorphic cryptographic platform based on an ITO/IGZO/TaN memristive device is reported. Under dual‐wavelength optical stimuli (405 and 532 nm), the device emulates various synaptic plasticity behaviors, including spike‐amplitude‐dependent plasticity (SADP), spike‐number‐dependent plasticity (SNDP), and spike‐rate‐dependent plasticity (SRDP), enabling high‐accuracy reservoir computing (88.39%) on Fashion Modified National Institute of Standards and Technology Database (FMNIST). Light‐driven probabilistic learning using a Restricted Boltzmann Machine (RBM) achieved 95.06% image reconstruction accuracy via experimentally extracted sigmoid activation. Moreover, the device enables optical logic operations and generates robust physical unclonable functions by leveraging intrinsic material randomness and optical conductance modulation. This multifunctional platform offers a promising path toward secure, energy‐efficient, and reconfigurable neuromorphic systems integrating memory, computation, and hardware‐level encryption within a single device architecture.