The Poisson equation is a fundamental equation of mathematical physics that describes the potential distribution in static fields. Solving the Poisson equation on a grid is computationally intensive and can be challenging for large grids. In recent years, quantum computing has emerged as a potential approach to solving the Poisson equation more efficiently. This article uses quantum algorithms, particularly the Harrow–Hassidim–Lloyd (HHL) algorithm, to solve the 2D Poisson equation. This algorithm can solve systems of equations faster than classical algorithms when the matrix A is sparse. The main idea is to use a quantum algorithm to transform the state vector encoding the solution of a system of equations into a superposition of states corresponding to the significant components of this solution. This superposition is measured to obtain the solution of the system of equations. The article also presents the materials and methods used to solve the Poisson equation using the HHL algorithm and provides a quantum circuit diagram. The results demonstrate the low error rate of the quantum algorithm when solving the Poisson equation.

Quantum machine learning (QML) has emerged as a promising approach for enhancing image classification by exploiting quantum computational principles such as superposition and entanglement. However, practical applications on complex datasets like CIFAR-100 remain limited due to the low expressivity of shallow circuits and challenges in circuit optimization. In this study, we propose HQCNN–REGA—a novel hybrid quantum–classical convolutional neural network architecture that integrates data re-uploading and genetic algorithm optimization for improved performance. The data re-uploading mechanism allows classical inputs to be encoded multiple times into quantum states, enhancing the model’s capacity to learn complex visual features. In parallel, a genetic algorithm is employed to evolve the quantum circuit architecture by optimizing gate sequences, entanglement patterns, and layer configurations. This combination enables automatic discovery of efficient parameterized quantum circuits without manual tuning. Experiments on the MNIST and CIFAR-100 datasets demonstrate state-of-the-art performance for quantum models, with HQCNN–REGA outperforming existing quantum neural networks and approaching the accuracy of advanced classical architectures. In particular, we compare our model with classical convolutional baselines such as ResNet-18 to validate its effectiveness in real-world image classification tasks. Our results demonstrate the feasibility of scalable, high-performing quantum–classical systems and offer a viable path toward practical deployment of QML in computer vision applications, especially on noisy intermediate-scale quantum (NISQ) hardware.

This paper presents a quantum algorithm for solving the one-dimensional heat equation with Dirichlet boundary conditions. The algorithm utilizes discretization techniques and employs quantum gates to emulate the heat propagation operator. Central to the algorithm is the Trotter–Suzuki decomposition, enabling the simulation of the time evolution of the temperature distribution. The initial temperature distribution is encoded into quantum states, and the evolution of these states is driven by quantum gates tailored to mimic the heat propagation process. As per the literature, quantum algorithms exhibit an exponential computational speedup with increasing qubit counts, albeit facing challenges such as exponential growth in relative error and cost functions. This study addresses these challenges by assessing the potential impact of quantum simulations on heat conduction modeling. Simulation outcomes across various quantum devices, including simulators and real quantum computers, demonstrate a decrease in the relative error with an increasing number of qubits. Notably, simulators like the simulator_statevector exhibit lower relative errors compared to the ibmq_qasm_simulator and ibm_osaka. The proposed approach underscores the broader applicability of quantum computing in physical systems modeling, particularly in advancing heat conductivity analysis methods. Through its innovative approach, this study contributes to enhancing modeling accuracy and efficiency in heat conduction simulations across diverse domains.

This paper presents the work of predicting oil production using machine learning methods. As a machine learning method, a multiple linear regression algorithm with polynomial properties was implemented. Regression algorithms are suitable and workable methods for predicting oil production based on a data-driven approach. The synthetic dataset was obtained using the Buckley-Leverett mathematical model, which is used to calculate hydrodynamics and determine the saturation distribution in oil production problems. Various combinations of parameters of the oil production problem were chosen, where porosity, viscosity of the oil phase and absolute permeability of the rock were taken as input parameters for machine learning. And the value of the oil recovery factor was chosen as the output parameter. More than 400 thousand synthetic data were used to test multiple regression algorithms. To estimate the quality of regression algorithms, the mean square error metrics and the coefficient of determination were used. It was found that linear regression does not cover all patterns in the data due to underfitting. Various degrees of polynomial regression were deployed and tested, and it was also found that for our synthetic data, the quadratic polynomial model trains quite well and perfectly predicts the value of the oil recovery factor. To solve the overfitting problem, L1 regularization known as the Lasso regression method was applied. For the quadratic polynomial regression model, the coefficient of determination was 0.96, which is a pretty good result for the test data. Thus, it is assumed that the data-driven machine learning methods discussed in the paper can be useful for predicting the oil recovery factor using practical data from oil fields at the stages of production

Newton’s method has been widely used in simulation multiphase, multicomponent flow in porous media. In addition, to solve systems of linear equations in such problems, the generalized minimal residual method (GMRES) is often used. This paper analyzed the one-dimensional problem of multicomponent fluid flow in a porous medium and solved the system of the algebraic equation with the Newton-GMRES method. We calculated the linear equations with the GMRES, the GMRES with restarts after every m steps—GMRES (m) and preconditioned with Incomplete Lower-Upper factorization, where the factors L and U have the same sparsity pattern as the original matrix—the ILU(0)-GMRES algorithms, respectively, and compared the computation time and convergence. In the course of the research, the influence of the preconditioner and restarts of the GMRES (m) algorithm on the computation time was revealed; in particular, they were able to speed up the program.

The problem of oil displacement was solved using neural networks and machine learning classifiers. The Buckley-Leverett model is selected, which describes the process of oil displacement by water. It consists of the equation of continuity of oil, water phases and Darcy’s law. The challenge is to optimize the oil displacement problem. Optimization will be performed at three levels: vectorization of calculations; implementation of classical algorithms; implementation of the algorithm using neural networks. A feature of the method proposed in the work is the identification of the method with high accuracy and the smallest errors, comparing the results of machine learning classifiers and types of neural networks. The research paper is also one of the first papers in which a comparison was made with machine learning classifiers and neural and recurrent neural networks. The classification was carried out according to three classification algorithms, such as decision tree, support vector machine (SVM) and gradient boosting. As a result of the study, the Gradient Boosting classifier and the neural network showed high accuracy, respectively 99.99 % and 97.4 %. The recurrent neural network trained faster than the others. The SVM classifier has the lowest accuracy score. To achieve this goal, a dataset was created containing over 67,000 data for class 10. These data are important for the problems of oil displacement in porous media. The proposed methodology provides a simple and elegant way to instill oil knowledge into machine learning algorithms. This removes two of the most significant drawbacks of machine learning algorithms: the need for large datasets and the robustness of extrapolation. The presented principles can be generalized in countless ways in the future and should lead to a new class of algorithms for solving both forward and inverse oil problems