To investigate the orientation of ellipsoidal nanoparticles in turbulence, a Physics-Informed Neural Network (PINN) framework for fractional-order problems was constructed, in which the fluid was described by a fractional-order fluid constitutive model in a periodically sinking channel. By introducing a fractional derivative into the constitutive relation, the model combines the nonlocal and memory effects of complex turbulence, making the effective rheology more accurate than classical integer-order closures. In addition, the second-order moment tensor equation derived from the Fokker-Planck equation is used to characterize the rotational diffusion behavior of ellipsoidal nanoparticles, and the variation of nanoparticle concentration in the channel is considered. The results show that the PINN based on the fractional constitutive model can predict the flow field. Based on the reconstructed flow field, the data-free neural network is used to predict the orientation and concentration distributions of nanoparticles in turbulent flow. The parameter analysis shows that as the rotating Péclet number increases, the shear effect controls rotating diffusion more strongly, resulting in ellipsoidal nanoparticles aligning more quickly and more clearly along the mainstream direction. Alignment near the wall precedes alignment at the channel center, usually accompanied by an overshoot. A smaller lateral Péclet number leads to a steep directional gradient, coupled with wall migration, resulting in higher alignment near the wall and greater dispersion at the center. In general, this study emphasizes the effectiveness of combining the fractional fluid constitutive model with PINN to solve the transport and orientation of multi-scale nanoparticles in turbulence.

DP-SGD is a standard tool for training machine learning models with differential privacy, but the Gaussian noise required to protect individual records can destabilize optimization and significantly reduce accuracy, especially under tight privacy budgets and small sampling rates. We propose Fractional-DP-SGD, a simple plug-in modification that preserves the core DP-SGD privacy mechanism—per-example gradient clipping and Gaussian noise injection—while changing only the update dynamics. Instead of updating parameters directly with the current noisy gradient, Fractional-DP-SGD applies a finite-window, normalized fractional-memory filter to the sequence of previously released DP gradients. This introduces a power-law temporal aggregation that behaves like a long-memory smoother, shaping the injected noise over time and potentially reducing the effective variance of parameter updates while keeping privacy accounting unchanged. We provide a formal privacy guarantee using the post-processing property of differential privacy together with standard Rényi DP accounting under subsampling, and we analyze the additional computational and memory costs introduced by the finite window. Finally, we outline an extensive experimental protocol comparing Fractional-DP-SGD against DP-SGD and widely used DP optimizers such as DP-Adam, DP-Adagrad, and DP-FTRL across multiple datasets and models. Our evaluation includes systematic ablations over fractional order, window length, kernel family, clipping norm, and noise multiplier to isolate when and why fractional memory improves stability and utility at matched privacy.

Generative modeling of dinosaur imagery is challenging due to large inter-species morphological variation, complex textures (scales, skin folds, armour, horns), and limited curated datasets. This paper investigates diffusion-based image generation for dinosaur images using a Denoising Diffusion Probabilistic Models (DDPM) and introduces fractional-order image analysis as a principled tool for evaluating the quality of generated images. Experiments are conducted on a 15-species dinosaur image dataset derived from the Jurassic Park franchise, consisting of two thousand images spanning diverse body structures, postures, and visual styles.

A DDPM model is trained to generate dinosaur images from Gaussian noise using a U-Net–based denoiser. To assess generation quality beyond conventional pixel-wise or perceptual metrics, we employ fractional calculus–based texture and structure measures. Fractional-order gradients, fractional Laplacian energy, and fractional spectral statistics are computed on both real and generated images across a range of fractional orders. These metrics enable continuous interpolation between fine-scale texture sensitivity and coarse morphological structure, providing a richer characterization of generative fidelity than integer-order operators.

Quantitative analysis reveals that fractional-order metrics are highly sensitive to differences in textural realism, edge continuity, and long-range spatial correlations between real and synthetic images. An optimal fractional order is observed at which the statistical distributions of generated images most closely match those of real dinosaur images, particularly for species with distinctive skin patterns and skeletal outlines. Visual inspection further confirms that fractional-order analysis highlights improvements in fine-scale detail preservation as diffusion training progresses.

The results demonstrate that fractional calculus provides an effective and interpretable framework for evaluating diffusion-generated images, especially for complex, texture-rich objects such as dinosaurs. This work highlights the synergy between diffusion models and fractional-order image analysis, offering new directions for generative modelling assessment in data-limited scientific and educational image processing domains.

Image-based dinosaur classification presents unique challenges due to limited data, large intra-class shape variability, and strong pose and illustration-style differences. Conventional integer-order image operators often fail to capture both fine-scale texture and global morphological structure simultaneously, particularly in small-sample data science challenges. This work explores fractional-order (FO) image processing and pattern recognition as an alternative representation framework for dinosaur image analysis.

We propose a fractional-order computer vision pipeline combining three complementary components. First, fractional-order gradient and edge operators are applied to dinosaur images to capture multi-scale contour and texture information associated with skeletal structure, armour, horns, wings, and body outlines. Second, fractional spectral descriptors, derived from fractional Fourier and multi-resolution wavelet statistics, model long-range spatial correlations and anisotropic shape patterns. Third, fractional-order pattern recognition tools, including fractional diffusion embeddings and fractional-order feature statistics, are used for exploratory data analysis, visualization, and classification. The approach is evaluated on two datasets: (i) a five-class dinosaur dataset with limited samples per class, and (ii) a 40-class carnivorous–herbivorous dinosaur dataset with train/validation splits. Fractional features are coupled with classical machine-learning classifiers, avoiding data-hungry deep models.

Experimental analysis shows that fractional-order representations provide improved class separability as compared to integer-order baselines, particularly for visually similar dinosaur species. Fractional diffusion embeddings reveal smooth morphological manifolds that align with anatomical traits such as body mass, posture, and feeding type. In the carnivorous–herbivorous dinosaur classification task, FO features demonstrate robust discrimination despite large inter-species variation and small sample sizes.

This study demonstrates that fractional-order image processing and pattern recognition offer an effective, interpretable, and data-efficient framework for dinosaur image analysis. By introducing a continuous order parameter, fractional methods naturally capture multi-scale shape and texture variations, making them well suited for paleontological and educational vision tasks with limited image data.

Face analysis systems commonly rely on integer-order image operators that emphasize either local texture or global structure, limiting their ability to represent the continuous, multi-scale nature of facial appearance. This paper presents a fractional-order (FO) image processing and statistical learning framework for face analysis and evaluates its effectiveness on two large-scale tasks: age estimation and gender classification. Experiments are conducted on the UTKFace dataset and a second large gender-classification benchmark, demonstrating that fractional calculus provides a principled and flexible representation for modeling gradual facial variations across scales.

The proposed framework integrates three complementary FO components. First, variable-order fractional gradients are computed on perceptually meaningful colour channels, where the fractional order adapts spatially to local structural complexity, enabling a smooth transition between fine texture sensitivity and coarse shape information. Second, fractional spectral representations, including fractional Fourier and multi-resolution wavelet statistics, are employed to capture long-range spatial correlations and anisotropic facial patterns that are poorly represented by conventional integer-order operators. Third, fractional diffusion–based embeddings are constructed on feature similarity graphs to support exploratory data analysis, preserving non-Gaussian and anomalous diffusion characteristics in the resulting low-dimensional manifolds. The extracted FO statistics are combined with standard machine-learning regression and classification models for age prediction and gender recognition.

Experimental results on UTKFace show that fractional-order features improve age estimation accuracy across the full lifespan (0–116 years), with particularly notable error reductions in early childhood and older age groups where texture–shape transitions are subtle. For gender classification, the proposed representation demonstrates increased robustness to illumination, pose, and image quality variations compared with integer-order baselines. Exploratory analyses further reveal smooth manifolds aligned with age progression while avoiding over-segmentation caused by dataset artifacts. Although race annotations are available in UTKFace, they are not used for prediction.

Understanding and forecasting coastline retreat under storm forcing is critical for coastal resilience planning. This study presents a multi-site, data-driven framework integrating fractal shoreline metrics, ERA5-derived storm parameters, and machine learning models to predict monthly shoreline retreat across three morphologically distinct Turkish coasts (Istanbul Karaburun, Karasu, and Sinop Boyabat). Satellite imagery (Sentinel-2, Landsat-8) was processed to extract historical shorelines, followed by computation of box-counting (D_box) and boundary-method (D_bm) fractal dimensions. Storm indicators—including storm index, high-wind hours, significant wave height exceedance durations, and wave energy metrics—were derived from ERA5 reanalysis. To address data sparsity, a block-bootstrap data augmentation strategy generated 300 synthetic years, yielding a final dataset of 10,815 monthly observations.

Five machine learning models were trained and evaluated: Random Forest, GRU, MLP, Ridge, and SVR. Results demonstrate that the Random Forest model achieved the highest performance (R² = 0.9866, MAE = 0.0113 m/month), followed by the GRU model (R² = 0.9552). Feature importance analysis revealed that shoreline-specific characteristics, fractal metrics, and storm intensity indicators are the dominant predictors. Cross-site generalization tests show that a single unified model can effectively predict retreat patterns across morphologically diverse coastlines.

The findings highlight: (1) the strong predictive capacity of storm indices and fractal shoreline complexity; (2) the suitability of tree-based and recurrent neural models for coastal change prediction; and (3) the viability of bootstrap-based synthetic augmentation for long-term coastal datasets. This framework provides a scalable and transferable methodology for early-warning systems and climate-adaptation strategies. Future work will explore scenario-based forecasting under extreme storm thresholds and SHAP-based interpretability to enhance model transparency.

Fractional calculus (FC), with its roots dating back to 1965 and pioneered by Leibniz and L'Hospital, is increasingly attracting attention in the applications of machine learning, a rising star in artificial intelligence (AI). Machine learning (ML), particularly in artificial neural network (ANN) applications, captures an internal memory dynamic by utilizing patterns in datasets. Fractional differential calculus, fractional derivatives, and integrals are used in models developed to internalize this dynamic and make learning algorithms more effective. According to studies in the literature, the Caputo definition of the fractional derivative is used in artificial neural networks called Fractional-order Artificial Neural Networks (FAANs). Here, we observe that activation functions based on fractional derivatives stand out. These activation functions are more flexible than their classical counterparts due to the need to carry adjustable parameters, and they increase the learning capacity and accuracy of the network. Based on this, this study will introduce the activation functions found in the literature at the intersection of FC and ANNs. This study shall discuss the convergence results in Banach spaces by considering the binary and multiple components of the relevant activation functions actively used in ANN operators. New activation functions obtained from the combination of these very general activation functions with more specific ones will be examined. Convergence analysis will be discussed using Banach space-valued Caputo fractional derivative and Caputo fractional Taylor formulas. Finally, a mathematical analysis of these activation functions—recently added to the literature—will be performed. The work closes with a nuanced examination of existing challenges and identifies several fruitful directions for future exploration in uniting fractional calculus with neural network methodologies.

The knee joint is a complex biotribological system. Its health relies on the interaction between articular cartilage and synovial fluid. Specifically, cartilage acts as a poroelastic structure, while synovial fluid exhibits complex non-Newtonian behavior. To elucidate the lubrication mechanisms, it is essential to investigate the fluid-structure coupling between them. This study presents a computational framework using Physics-Informed Neural Networks (PINNs) to solve the fractional squeeze-film flow problem in a knee joint model. The model characterizes the synovial fluid as a fractional Maxwell fluid, capturing its viscoelastic memory effects. At the same time, the cartilage is a porous elastic layer, and the femoral component is a rigid sphere. By directly embedding the governing equations into the neural network's loss function, this complex flow behavior can be effectively solved. The results indicate that the permeability of the cartilage layer is a critical parameter regulating joint lubrication performance. High permeability accelerates the exudation of synovial fluid from the loaded region, leading to a rapid decline in film thickness and a significant reduction in fluid load-supporting capacity. Meanwhile, an increase in the fractional order and a decrease in the relaxation time also weaken the stability and load-bearing performance of the fluid film, further accelerating lubrication failure. Collectively, these mechanisms exacerbate cartilage wear and elevate the risk of osteoarthritis. This study provides novel insights into the lubrication failure of diseased cartilage. It also highlights the potential of PINNs for tackling complex biomechanical problems.

In this study, we investigate the integration of fractional calculus and Riemannian geometry within the framework of machine learning. We propose a fractional-order learning model that operates on the Symmetric Positive Definite (SPD) manifold, a curved geometric space naturally suited for representing covariance-based data. Each data instance is modeled as an SPD matrix, ensuring that the learning dynamics respect the manifold’s intrinsic geometry. The UCI Human Activity Recognition (HAR) dataset was used to demonstrate the method, in which multichannel sensor recordings were transformed into SPD covariance representations. The learning algorithm extends classical gradient descent by incorporating a Caputo-type fractional derivative, introducing a controllable memory effect that enables updates to depend on both the current and historical gradients. Experiments performed for fractional orders ν=1.0, 1.2, and 1.5 show that increasing the fractional order yields smoother convergence trajectories, mitigates oscillations, and improves stability on the SPD manifold. This fractional–geometric coupling yields a regularized, memory-aware learning paradigm that enhances generalization capability, particularly for small or noisy datasets. The results suggest that fractional dynamics can act as an implicit temporal regularizer for Riemannian optimization, offering a principled approach to stabilize manifold-based learning algorithms. Overall, the study introduces a coherent framework that bridges fractional calculus, information geometry, and modern machine learning, and highlights the potential of fractional-order geometric learning as a unifying paradigm that links mathematical theory, computational modeling, and real-world sensor data analysis.

A nonlinear mathematical model is proposed to examine the role of plants with varying abilities to absorb atmospheric carbon dioxide (COâ‚‚) and its broader ecological impacts. Different plant species exhibit distinct COâ‚‚ absorption capacities, which can significantly influence the overall reduction in atmospheric COâ‚‚ levels. The study aims to analyze how disparities in plant absorption capacities affect atmospheric COâ‚‚ concentrations, with a particular focus on the influence of plant growth and harvesting rates. The model is formulated using a discrete difference operator, facilitating a numerical exploration of the system's dynamics. A hybrid computational framework is employed, namely, the Discrete Numerical Iterative Method integrated with the Levenberg–Marquardt neural network algorithm (DNIM-LM). Artificial intelligence techniques are used to assess model performance, including training progress, error distribution, regression accuracy, and overall fitness. The dataset is partitioned into 70% for training, 15% for validation, and 15% for testing. The results reveal that plant species with higher COâ‚‚ absorption capacities lead to more rapid decreases in atmospheric COâ‚‚ as their growth rates increase. Conversely, higher harvesting rate coefficients are associated with increased atmospheric COâ‚‚ concentrations. The study concludes that differences in plant absorption abilities significantly shape the dynamic behavior of atmospheric COâ‚‚ reduction. These findings underscore the critical role of plant growth and harvesting practices in regulating COâ‚‚ levels, offering valuable insights for ecosystem management and carbon sequestration strategies.