In many scientific and engineering fields, such as hydrogeology, petroleum engineering, geotechnical research, and developing renewable energy solutions, fluid flow modeling in porous media is essential. In these areas, optimizing extraction techniques, forecasting environmental effects, and guaranteeing structural safety all depend on an understanding of the behavior of single-phase flows—fluids passing through connected pore spaces in rocks or soils. Darcy’s law, which results in an elliptic partial differential equation controlling the pressure field, is usually the mathematical basis for such modeling. Analytical solutions to these partial differential equations are seldom accessible due to the complexity and variability in natural porous formations, which makes the employment of numerical techniques necessary. To approximate subsurface flow solutions, traditional methods like the finite difference method, two-point flux approximation, and multi-point flux approximation have been employed extensively. Accuracy, stability, and computing economy are trade-offs for each, though. Deep learning techniques, in particular convolutional neural networks, physics-informed neural networks, and neural operators such as the Fourier neural operator, have become strong substitutes or enhancers of conventional solvers in recent years. These models have the potential to generalize across various permeability configurations and greatly speed up simulations. The purpose of this study is to examine and contrast the mentioned deep learning and numerical approaches to the problem of pressure distribution in single-phase Darcy flow, considering a 2D domain with mixed boundary conditions, localized sources, and sinks, and both homogeneous and heterogeneous permeability fields. The result of this study shows that the two-point flux approximation method is one of the best regarding computational speed and accuracy and the Fourier neural operator has potential to speed up more accurate methods like multi-point flux approximation. Different permeability field types only impacted each methods’ accuracy while computational time remained unchanged. This work aims to illustrate the advantages and disadvantages of each method and support the continuous development of effective solutions for porous medium flow problems by assessing solution accuracy and computing performance over a range of permeability situations.

The article investigates the accuracy of nonparametric univariate density estimation methods applied to various Gaussian mixture models. A comprehensive comparative analysis is performed for four popular estimation approaches: adaptive kernel density estimation, projection pursuit, log-spline estimation, and wavelet-based estimation. The study is extended with modified versions of these methods, where the sample is first clustered using the EM algorithm based on Gaussian mixture components prior to density estimation. Estimation accuracy is quantitatively evaluated using MAE and MAPE criteria, with simulation experiments conducted over 100,000 replications for various sample sizes. The results show that estimation accuracy strongly depends on the density structure, sample size, and degree of component overlap. Clustering before density estimation significantly improves accuracy for multimodal and asymmetric densities. Although no formal statistical tests are conducted, the performance improvement is validated through non-overlapping confidence intervals obtained from 100,000 simulation replications. In addition, several decision-making systems are compared for automatically selecting the most appropriate estimation method based on the sample’s statistical features. Among the tested systems, kernel discriminant analysis yielded the lowest error rates, while neural networks and hybrid methods showed competitive but more variable performance depending on the evaluation criterion. The findings highlight the importance of using structurally adaptive estimators and automation of method selection in nonparametric statistics. The article concludes with recommendations for method selection based on sample characteristics and outlines future research directions, including extensions to multivariate settings and real-time decision-making systems.