Uncertainty quantification (UQ) plays a pivotal role in reduction of uncertainties during both optimization and decision making processes. It can be applied to solve a variety of real-world applications in science and engineering. Bayesian approximation and ensemble learning techniques are two most widely-used UQ methods in the literature. In this regard, researchers have proposed different UQ methods and examined their performance in a variety of applications such as computer vision (e.g., self-driving cars and object detection), image processing (e.g., image restoration), medical image analysis (e.g., medical image classification and segmentation), natural language processing (e.g., text classification, social media texts and recidivism risk-scoring), bioinformatics, etc. This study reviews recent advances in UQ methods used in deep learning. Moreover, we also investigate the application of these methods in reinforcement learning (RL). Then, we outline a few important applications of UQ methods. Finally, we briefly highlight the fundamental research challenges faced by UQ methods and discuss the future research directions in this field.

We investigate the effects of the Gauss-Bonnet (GB) gravitational trace anomaly on the circular motion of test particles around black holes (BHs) and its implications for quasi-periodic oscillations (QPOs) in various theoretical models. Beginning with the equations of motion, we study the effects on effective potential, angular momentum, specific energy, and the innermost stable circular orbit (ISCO) induced by the anomaly parameter $\alpha$. The fundamental frequencies are calculated. Moreover, we examine several QPO models, including PR, RP, WD, TD, and ER2-ER4, and study the relationship between the upper and lower QPO frequencies as well as the corresponding resonance radii for frequency ratios of 1:1, 3:2, 4:3, and 5:4. Our results show that increasing $\alpha$ leads to deviations from the Schwarzschild case in both upper and lower QPO frequencies correlations and QPO orbital radii, with model-dependent trends. Further, we constrain the BH parameters using the observational data using MCMC analysis. Finally, we calculate the upper and lower QPO frequencies for a few BH candidates on the basis of the RP model using the constrained parameter values and find a good agreement with the observed results.

Our study explores gravitational baryogenesis in the context of f(R, Lm, T) gravity, where R denotes the Ricci scalar, Lm represents the Lagrangian density of the matter field, and T stands for the metric contraction of T_{mu nu}. We focus on a linear model: f(R, Lm, T) = alpha R + beta Lm + gamma T, and examine the parameter constraints for a successful baryon asymmetry generation in four different eras of the cosmos under the assumption of a power-law cosmic expansion. The computed baryon-to-entropy ratio is found to be consistent with the observed order of asymmetry ratio, 9.42 x 10^-11. Furthermore, the study is extended to the generalized framework of gravitational baryogenesis, where the outcome shows strong agreement with the current observational data. Our findings indicate that the f(R, Lm, T) framework provides a compatible theoretical foundation for producing the observed matter imbalance of the cosmos, thereby emphasizing its potential significance in early-universe cosmology.

The thermodynamics of black holes provides a profound link between gravity, quantum theory and statistical mechanics. It serves as a useful tool for testing theories beyond Einstein's gravity. In this work of ours, we investigate the newly found restricted phase space thermodynamics (RPST) of charged static and charged rotating black holes in $f(R)$ gravity. Unlike the extended phase space (EPST) approach, RPST keeps the cosmological constant fixed and introduces the central charge $C$ along with its conjugate chemical potential $\mu$, thereby allowing the black hole mass to be consistently interpreted as internal energy. Within this framework, we derive the relevant thermodynamic quantities and analyse the temperature-entropy $(T-S)$ and Helmholtz free energy-temperature $(F-T)$ behaviours. Our results reveal characteristic features of first-order phase transitions through non-monotonic $T-S$ curves along with the swallow-tail structures in $F-T$ plots, while second-order transitions appear at critical points. To further validate these findings, we employ the formalism of geometrothermodynamics (GTD), which provides a Legendre-invariant geometric description of thermodynamic geometry. We demonstrate that the curvature singularities of the GTD scalar curvature coincides exactly with that of the divergences in the specific heat capacity curves, thereby establishing a geometric correspondence for phase transitions. This study facilitates the first systematic exploration of RPST within $f(R)$ gravity and highlights the universality of RPST in capturing black hole criticality in modified gravity theories.

The digital revolution has significantly impacted financial transactions, leading to a notable increase in credit card usage. However, this convenience comes with a trade-off: a substantial rise in fraudulent activities. Traditional machine learning methods for fraud detection often struggle to capture the inherent interconnectedness within financial data. This paper proposes a novel approach for credit card fraud detection that leverages Graph Neural Networks (GNNs) with attention mechanisms applied to heterogeneous graph representations of financial data. Unlike homogeneous graphs, heterogeneous graphs capture intricate relationships between various entities in the financial ecosystem, such as cardholders, merchants, and transactions, providing a richer and more comprehensive data representation for fraud analysis. To address the inherent class imbalance in fraud data, where genuine transactions significantly outnumber fraudulent ones, the proposed approach integrates an autoencoder. This autoencoder, trained on genuine transactions, learns a latent representation and flags deviations during reconstruction as potential fraud. This research investigates two key questions: (1) How effectively can a GNN with an attention mechanism detect and prevent credit card fraud when applied to a heterogeneous graph? (2) How does the efficacy of the autoencoder with attention approach compare to traditional methods? The results are promising, demonstrating that the proposed model outperforms benchmark algorithms such as Graph Sage and FI-GRL, achieving a superior AUC-PR of 0.89 and an F1-score of 0.81. This research significantly advances fraud detection systems and the overall security of financial transactions by leveraging GNNs with attention mechanisms and addressing class imbalance through an autoencoder.

This study develops and evaluates a novel hybridWavelet SARIMA Transformer, WST framework to forecast using monthly rainfall across five meteorological subdivisions of Northeast India over the 1971 to 2023 period. The approach employs the Maximal Overlap Discrete Wavelet Transform, MODWT with four wavelet families such as, Haar, Daubechies, Symlet, Coiflet etc. to achieve shift invariant, multiresolution decomposition of the rainfall series. Linear and seasonal components are modeled using Seasonal ARIMA, SARIMA, while nonlinear components are modeled by a Transformer network, and forecasts are reconstructed via inverse MODWT. Comprehensive validation using an 80 is to 20 train test split and multiple performance indices such as, RMSE, MAE, SMAPE, Willmotts d, Skill Score, Percent Bias, Explained Variance, and Legates McCabes E1 demonstrates the superiority of the Haar-based hybrid model, WHST. Across all subdivisions, WHST consistently achieved lower forecast errors, stronger agreement with observed rainfall, and unbiased predictions compared with stand alone SARIMA, stand-alone Transformer, and two-stage wavelet hybrids. Residual adequacy was confirmed through the Ljung Box test, while Taylor diagrams provided an inte- grated assessment of correlation, variance fidelity, and RMSE, further reinforcing the robustness of the proposed approach. The results highlight the effectiveness of integrating multiresolution signal decomposition with complementary linear and deep learning models for hydroclimatic forecasting. Beyond rainfall, the proposed WST framework offers a scalable methodology for forecasting complex environmental time series, with direct implications for flood risk management, water resources planning, and climate adaptation strategies in data-sparse and climate-sensitive regions.

This paper proposes a novel graph-based framework for robust and interpretable multiclass fault diagnosis in rotating machinery. The method integrates entropy-optimized signal segmentation, time-frequency feature extraction, and graph-theoretic modeling to transform vibration signals into structured representations suitable for classification. Graph metrics, such as average shortest path length, modularity, and spectral gap, are computed and combined with local features to capture global and segment-level fault characteristics. The proposed method achieves high diagnostic accuracy when evaluated on two benchmark datasets, the CWRU bearing dataset (under 0-3 HP loads) and the SU gearbox and bearing datasets (under different speed-load configurations). Classification scores reach up to 99.8% accuracy on Case Western Reserve University (CWRU) and 100% accuracy on the Southeast University datasets using a logistic regression classifier. Furthermore, the model exhibits strong noise resilience, maintaining over 95.4% accuracy at high noise levels (standard deviation = 0.5), and demonstrates excellent cross-domain transferability with up to 99.7% F1-score in load-transfer scenarios. Compared to traditional techniques, this approach requires no deep learning architecture, enabling lower complexity while ensuring interpretability. The results confirm the method's scalability, reliability, and potential for real-time deployment in industrial diagnostics.

Bars are fundamental structures in disc galaxies, although their role in galaxy evolution is still not fully known. This study investigates the effect of the presence of bars on the environmental dependence of disc galaxies' properties using the volume-limited sample from Mapping Nearby Galaxies at APO (MaNGA) survey. The disc galaxies with and without bars samples were obtained using the Galaxy Zoo 2 project then assigned into isolated and non-isolated sub-samples. These sub-samples were used to compare the stellar mass, star formation rate, $g-r$ colour, concentration index and gas phase metallicity, and their relationships between isolated and non-isolated environments. Then these are used to investigate if there is an existence of any difference between galaxies with and without bars. A one-to-one correspondence between isolated and non-isolated galaxy properties was observed, and a strong dependence on the environment for properties of unbarred galaxies was observed when compared to barred. The stellar mass against star formation rate, $g-r$ colour against concentration index and stellar mass against gas phase metallicity of unbarred galaxies strongly depend on environment while for barred these relations weakly depend on environment. The study concludes that bars in disc galaxies decrease the dependence of analysed properties and their relations on the environment.

In this work, we derive a novel black hole solution surrounded by a Dehnen-(1,4,0) type dark matter halo by embedding a Schwarzschild black hole within the halo, constituting a composite dark matter-black hole system. The thermodynamics of the resulting effective black hole spacetime is then studied with particular attention to the influence of the dark matter parameters on various thermodynamic properties. We examine the specific heat and free energy to assess the thermodynamic stability of the model. Furthermore, the null geodesics and the effective potential of light rays are studied to investigate how the dark matter parameters affect these geodesics and the radii of circular orbits. The stability of circular null geodesics is analyzed using dynamical systems and Lyapunov exponents, which represents the dynamical behaviour of the circular photon orbits. Finally, we studied if the circular geodesics exhibit chaotic behaviour using the chaos-bound condition.

Purpose: This paper aims to enhance bearing fault diagnosis in industrial machinery by introducing a novel method that combines Graph Attention Network (GAT) and Long Short-Term Memory (LSTM) networks. This approach captures both spatial and temporal dependencies within sensor data, improving the accuracy of bearing fault detection under various conditions. Methodology: The proposed method converts time series sensor data into graph representations. GAT captures spatial relationships between components, while LSTM models temporal patterns. The model is validated using the Case Western Reserve University (CWRU) Bearing Dataset, which includes data under different horsepower levels and both normal and faulty conditions. Its performance is compared with methods such as K-Nearest Neighbors (KNN), Local Outlier Factor (LOF), Isolation Forest (IForest) and GNN-based method for bearing fault detection (GNNBFD). Findings: The model achieved outstanding results, with precision, recall, and F1-scores reaching 100\% across various testing conditions. It not only identifies faults accurately but also generalizes effectively across different operational scenarios, outperforming traditional methods. Originality: This research presents a unique combination of GAT and LSTM for fault detection, overcoming the limitations of traditional time series methods by capturing complex spatial-temporal dependencies. Its superior performance demonstrates significant potential for predictive maintenance in industrial applications.

In this work, we study the thermodynamic topology of a static, a charged static and a charged, rotating black hole in $f(R)$ gravity. For charged static black holes, we work in two different ensembles: fixed charge$(q)$ ensemble and fixed potential$(\phi)$ ensemble. For charged, rotating black hole, four different types of ensembles are considered: fixed $(q, J)$, fixed $(\phi, J)$, fixed $(q,\Omega)$ and fixed $(\phi,\Omega)$ ensemble, where $J$ and $\Omega$ denotes the angular momentum and the angular frequency respectively. Using the generalized off-shell free energy method, where the black holes are treated as topological defects in their thermodynamic spaces, we investigate the local and global topology of these black holes via the computation of winding numbers at these defects. For static black hole we work in three model. We find that the topological charge for a static black hole is always $-1$ regardless of the values of the thermodynamic parameters and the choice of $f(R)$ model. For a charged static black hole, in the fixed charge ensemble, the topological charge is found to be zero. Contrastingly, in the fixed $\phi$ ensemble, the topological charge is found to be $-1.$ For charged static black holes, in both the ensembles, the topological charge is observed to be independent of the thermodynamic parameters. For charged, rotating black hole, in fixed $(q, J)$ ensemble, the topological charge is found to be $1.$ In $(\phi, J)$ ensemble, we find the topological charge to be $1.$ In case of fixed $(q,\Omega)$ ensemble, the topological charge is $1$ or $0$ depending on the value of the scalar curvature($R$). In fixed $(\Omega,\phi)$ ensemble, the topological charge is $-1,0$ or $1$ depending on the values of $R,\Omega$ and $\phi.$

Understanding data and reaching valid conclusions are of paramount importance in the present era of big data. Machine learning and probability theory methods have widespread application for this purpose in different fields. One critically important yet less explored aspect is how data and model uncertainties are captured and analyzed. Proper quantification of uncertainty provides valuable information for optimal decision making. This paper reviewed related studies conducted in the last 30 years (from 1991 to 2020) in handling uncertainties in medical data using probability theory and machine learning techniques. Medical data is more prone to uncertainty due to the presence of noise in the data. So, it is very important to have clean medical data without any noise to get accurate diagnosis. The sources of noise in the medical data need to be known to address this issue. Based on the medical data obtained by the physician, diagnosis of disease, and treatment plan are prescribed. Hence, the uncertainty is growing in healthcare and there is limited knowledge to address these problems. We have little knowledge about the optimal treatment methods as there are many sources of uncertainty in medical science. Our findings indicate that there are few challenges to be addressed in handling the uncertainty in medical raw data and new models. In this work, we have summarized various methods employed to overcome this problem. Nowadays, application of novel deep learning techniques to deal such uncertainties have significantly increased.

Galactic rotation curve is a powerful indicator of the state of the gravitational field within a galaxy. The flatness of these curves indicates the presence of dark matter in galaxies and their clusters. In this paper, we focus on the possibility of explaining the rotation curves of spiral galaxies without postulating the existence of dark matter in the framework of $f(\mathcal{R},T)$ gravity, where the gravitational Lagrangian is written by an arbitrary function of $\mathcal{R}$, the Ricci scalar and of $T$, the trace of energy-momentum tensor $T_{\mu\nu}$. We derive the gravitational field equations in this gravity theory for the static spherically symmetric spacetime and solve the equations for metric coefficients using a specific model that has minimal coupling between matter and geometry. The orbital motion of a massive test particle moving in a stable circular orbit is considered and the behavior of its tangential velocity with the help of the considered model is studied. We compare the theoretical result predicted by the model with observations of a sample of nineteen galaxies by generating and fitting rotation curves for the test particle to check the viability of the model. It is observed that the model could almost successfully explain the galactic dynamics of these galaxies without the need of dark matter at large distances from the galactic center.

The Frolov black hole (BH) is a charged extension of the Hayward BH, having regularity at the central point $r = 0$ and an asymptotically Schwarzschild form for large values of $r$. Such a BH is parameterized by a length scale parameter, $\alpha_0$. In this paper, we analyze the thermodynamic properties, null and timelike geodesics, and shadows of a Frolov BH immersed in a quintessence field. Our results indicate that the smaller BH is locally thermodynamically stable yet globally unstable at all horizon radii. Neither the quintessence parameter nor the other model parameters like the charge $q$ and length scale parameter $\alpha_0$ change this global instability. We extend the study of the null and timelike geodesics to the vicinity of the BH by analyzing how the geodesic motion depends on the model parameters. Finally, we analyze the shadow of the BH system and find that the shadow radii are sensitively dependent on model parameters. In contrast, the influence of the quintessence parameter itself on the size of the shadow is found to be rather weak.

$\mu-\tau$ reflection symmetry is an attractive flavour symmetry in lepton mixing, which accommodates maximal values of atmospheric mixing angle ($\theta_{23}=\pi/4$) and Dirac CP phase ($\delta=\pi/2/3\pi/2$). Another significance of this symmetry is that it does not constrain $\theta_{13}$ to be zero. As the recent results from $T2K$ and $NO\nu A$ experiments indicate a near-maximal value of the Dirac CP phase, the role of $\mu-\tau$ reflection symmetry becomes more prominent. In this work, we study RG running effects as a perturbation to the $\mu-\tau$ reflection symmetry. Assuming the symmetry to be preserved at the seesaw scale, we study the deviations of mass eigenvalues and lepton mixing parameters at the electroweak scale due to RG running. We derive the one-loop RGEs of the mass eigenvalues and mixing parameters and solve them numerically. Numerical analysis shows that the deviations from $\mu-\tau$ reflection symmetry are consistent with $3\sigma$ range of global oscillation data.

This study delves into the intricate properties of a Schwarzschild black hole enveloped by King dark matter in an isotropic configuration. The thermodynamic characteristics of this black hole are meticulously analyzed, and the dynamics of massive and massless particles in its vicinity are investigated. In examining the trajectories of massless particles, the shadow cast in the presence of King dark matter is explored, revealing virtual ranges for the corresponding parameters. For the dynamics of massive particles, the radius of the innermost stable circular orbit, angular momentum, energy, and angular velocity of a test particle within the King dark matter framework surrounding the black hole are calculated. The effect of King dark matter on the accretion disk energy flux, effective radiation temperature, differential luminosity, and spectral luminosity are then investigated. The stability of the photon sphere in the presence of King dark matter is also studied, and finally, the thermodynamic potentials of this black hole are examined from a topological perspective.

The links between the deformation parameter $\beta$ of the generalized uncertainty principle (GUP) to the two free parameters $\hat{\omega}$ and $\gamma$ of the running Newtonian coupling constant of the Asymptotic Safe gravity (ASG) program, has been conducted recently in [Phys.Rev.D 105 (2022) 12, 124054]. In this paper, we test these findings by calculating and examining the shadow and quasinormal modes of black holes and demonstrate that the approach provides a theoretical framework for exploring the interplay between quantum gravity and GUP. Our results confirm the consistency of ASG and GUP and offer new insights into the nature of black holes and their signatures. The implications of these findings for future studies in quantum gravity are also discussed.

Bianchi type III (BIII) metric is an interesting anisotropic model for studying cosmic anisotropy as it has an additional exponential term multiplied to a directional scale factor. Thus, the cosmological parameters obtained for this BIII metric with the conventional energy-momentum tensor within the framework of a modified gravity theory and the estimation of their values with the help of Hubble, Pantheon plus and other observational data may provide some new information in cosmic evolution. In this work, we have studied the BIII metric under the framework of $f(R,T)$ gravity theory and estimated the values of the cosmological parameters for three different models of this gravity theory by using the Bayesian technique. In our study, we found that all the models show consistent results with the current observations but show deviations in the early stage of the Universe. In one model we have found a sharp discontinuity in the matter-dominated phase of the Universe. Hence through this study, we have found that all the $f(R,T)$ gravity models may not be suitable for studying evolutions and early stages of the Universe in the BIII metric even though they show consistent results with the current observations.

We study the Restricted Phase Space Thermodynamics (RPST) of magnetically charged Anti de Sitter (AdS) black holes sourced by nonlinear electrodynamics(NED). The first law and the corresponding Euler relation are examined using the scaling properties. While the mass is homogeneous in the first order, the intensive variables are observed to follow zeroth order homogeneity. We use numerical and graphical techniques to find the critical points of the various thermodynamic quantities. By utilizing the re-scaling properties of the equation of states, we study the thermodynamic processes using different pairs of variables. From our analysis, we infer that although the RPS thermodynamics of NED-AdS black hole resembles those of RN-AdS, Kerr-AdS, Kerr-Sen-Ads black holes in most of its aspects, hinting at a possible universality, there exists one particular $\mu-C$ process that differs in its behaviour from its counterparts in earlier reported works.