Classical Numerical Methods Research Articles

Developing of neural network computing methods for solving inverse elasticity problems

This paper examines the use of neural network methods to solve inverse problems in the mechanics of elastic materials. The aim is to design physics-informed neural networks that can predict the parameters of structural components, and the physical properties of materials based on a specified displacement distribution. A key feature of the specified neural networks is the integration of differential equations and boundary conditions into the loss function calculation. This approach ensures that the error in approximating unknown functions has a direct impact on optimizing the network's weights. As a result, the resulting neural network approximations of unknown functions comply with the differential equations and boundary conditions. To test the capabilities of the designed neural networks, inverse problems involving the bending of plates and beams have been solved, focusing on determining one or two unknown parameters. Comparison of predicted and exact values demonstrates high accuracy of the constructed neural network models, with a relative prediction error of less than 3 % across all cases. Unlike analytical methods for solving inverse problems, the primary advantage of physics-informed neural networks is their flexibility when addressing both linear and nonlinear problems. For instance, the same network architecture can be employed to solve various boundary-value problems without modification. Compared to classical numerical methods, the parallelization capability of neural networks is inherently supported by modern software libraries. Therefore, the application of physics-informed neural networks for solving inverse elasticity problems of plates and beams is effective, as evidenced by the achieved relative errors and the computational robustness of the method. In practice, the proposed solution can be used for relevant calculations during the design of structural elements. The developed software code can also be integrated into automated design systems or computer algebra systems

Utilizing deep learning algorithms for the resolution of partial differential equations

Partial differential equations (PDEs) are mathematical equations that are used to model physical phenomena around us, such as fluid dynamics, electrodynamics, general relativity, electrostatics, and diffusion. However, solving these equations can be challenging due to the problem known as the dimensionality curse, which makes classical numerical methods less effective. To solve this problem, we propose a deep learning approach called deep Galerkin algorithm (DGA). This technique involves training a neural network to approximate a solution by satisfying the difference operator, boundary conditions and an initial condition. DGA alleviates the curse of dimensionality through deep learning, a meshless approach, residue-based loss minimisation and efficient use of data. We will test this approach for the transport equation, the wave equation, the Sine-Gordon equation and the Klein-Gordon equation.

Neural network-augmented differentiable finite element method for boundary value problems

Classical numerical methods such as finite element method (FEM) face limitations due to their low efficiency when addressing large-scale problems. As a novel paradigm, the physics-informed neural network (PINN) has demonstrated significant potential to solve partial differential equations. However, conventional PINNs utilize meshless control at discrete sampling points, which limits their ability to effectively handle complex boundaries. Moreover, catastrophic failure may occur in the deep energy method (DEM, a specific type of PINN). To handle these challenges, this study proposes a Neural Network-augmented Differentiable Finite Element Method (NNDFEM) by combining PINN and finite element approximation. In NNDFEM, the neural network backend solely predicts nodal variables. Derivatives and complex boundary conditions can be well handled by the finite element frontend. The governing equation over the domain, Dirichlet, and Neumann boundary conditions are directly enforced on the finite element frontend. Thus, losses of boundary conditions in PINN are rendered unnecessary. The overfitting problem in DEM is also significantly mitigated. Fully connected neural network (FCNN), modified FCNN, and graph-convolutional network are tested as backends. NNDFEM circumvents nodal force calculation and matrix assembly in FEM. Functional losses of linear elasticity, finite strain nonlinear elasticity, heat conduction, and flow in porous media are validated. A systematic exploration unveils the role of 3D finite element mesh. For large-scale problems, a multi-fidelity learning strategy is employed. Thus, the three-dimensional case with over three million degrees of freedom trains well in two minutes. Benefiting from the fast inference of the neural network backend, the forward pass is 8,550 times faster than FEM.

Numerical implementation of solving a boundary value problem with parameter for Fredholm integro-differential equation

The boundary value problem with parameter for Fredholm integro-differential equation with degenerate kernel is investigated in this paper. The aim of the paper is to establish the solvability conditions, to construct analytical and numerical solutions of the considered problem. The basis for achieving the goal is the ideas of Dzhumabayev parameterization method, classical numerical methods of solving Cauchy problems and numerical integration techniques. A problem with parameters is obtained by introducing an additional parameter and a new unknown function. A system of equations with respect to parameters is compiled according to the initial data of the considered equation and boundary conditions. The unknown function is found as a solution of the Cauchy problem for the ordinary differential equation. The equivalence of the original problem and the problem with parameters, the conditions of unique solvability are established and the formula for finding an analytical solution is obtained. Test examples of finding analytical and approximate solutions of the original problem are given.

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

Large-Deformation Modeling of Surface Instability and Ground Collapse during Tunnel Excavation by Material Point Method

Recent rapid urbanization has led to an increase in tunnel construction, escalating the prevalence of ground collapses. Ground collapses, characterized by large deformation and strain-softening, pose a significant challenge for classical numerical theories and simulation methods. Consequently, a numerical framework combining the material point method (MPM) and strain-softening Drucker–Prager plasticity is introduced in this study to more accurately describe the evolution process and failure mechanism of the subgrade during tunnel excavation. The proposed numerical framework was validated against an analytic solution employing a typical ‘dry bottom’ dam model with solid non-linearity and large deformation; some of the results are also compared with those of the SPH method and centrifugal modeling tests to verify the validity of the MPM method in this paper. The validated model was used in this study to conduct a comprehensive analysis of surface instability and ground collapse under varying soil conditions. This included factors such as strata thickness, cohesion, internal friction angle, and a quantitative description of the development of longitudinal subsidence of the surface. The aim was to clarify deformation responses, failure patterns, and excavation mechanisms, providing insights for underground tunneling practices.

A percolation model for numerical simulations of 2D non-gravity impregnation in porous media

The aim of this work is to propose a new percolation model for numerical simulation of impregnation in porous media. Indeed, the impregnation phenomena concern many engineering fields, and could affect mechanical behavior of the medium, specifically when it is coupled with other physics such as heat or chemistry. The understanding of this phenomenon requires sophisticated numerical models that take into account the coupling between different physics. Classical numerical methods present known problems such as high computational costs, spurious oscillations and are not well adapted to such a complex coupling problem. To avoid these computational difficulties, Self-organized Gradient Percolation (SGP) model could be a good compromise. SGP is a numerical model based on a probabilistic approach, namely gradient percolation, already introduced in the 1D case, [1]. In this work we explore the non-reactive case. Instead of using finite element methods to solve Richards’ equation, we model the propagation of the saturation in the medium by the evolution of gradient percolation clusters. We hence delimitate the “wet” zone of the porous media to which we associate macroscopic saturation using interpolation techniques. The obtained results are free from spurious oscillation, and computational cost is drastically reduced compared with the classical approaches based on the finite element method. We believe that the 2D-SGP method can be enhanced by a futur 3D extension, but also to include the chemical and thermo-mechanical aspects.

Comparative analysis of filtering methods for measurement data from complex well configurations

This article presents a comparative analysis of various filtering methods for synthetic measurements that simulate data from well test analysis (WTA). The main objective of this work is to identify the most effective filtering methods for noisy WTA data, with the aim of preserving useful information and facilitating the subsequent interpretation of the results. The initial dataset consisted of 200 synthetic pressure drawdown (PDD) and pressure buildup (PBU) curves with varying levels of artificially introduced noise. Both classical filtering methods (Kalman filter, Savitzky–Golay filter, one-dimensional Gaussian filtering) and numerical methods based on neural networks (autoencoders) and machine learning (support vector machines) were considered for data filtering. The comparative analysis demonstrated that the performance of different filtering methods depends on the type of curve (PDD or PBU) and the well characteristics. The best results in terms of signal-to-noise ratio (SNR) and root mean square error (RMSE) were achieved using modern autoencoder-based methods. The conclusion is that the choice of an optimal filtering method requires a detailed analysis of the specific problem and the characteristics of the input data. A combination of different filtering methods is proposed to improve the quality of processing and interpretation of WTA data for complex well designs. The obtained results have practical significance, as they can simplify the segmentation of PDD and PBU curves, which is necessary for the correct identification of various operating periods of the well during the investigation process.

Adaptive Meta-heuristic Framework for Real-time Dynamic Obstacle Avoidance in Redundant Robot Manipulators

Robotic manipulator faces a challenge in navigating dynamic environments while ensuring collision-free trajectories, especially for redundant manipulators. Inverse kinematics involves finding joint angles to reach a specific point in 3D space. The shift from classical analytical and numerical methods to optimization heuristic algorithms is driven by the increasing complexity of robotic systems and the demand for more versatile and adaptive solutions. Meta-heuristic algorithms offer a transformative approach by framing the inverse kinematics problem as an optimization challenge, providing a flexible and robust means to navigate complex solution spaces. Metaheuristic algorithms, known for their ability to explore high-dimensional search spaces and avoid local optima, offer robust solutions for these challenges. They enhance computational efficiency, enabling real-time decision-making and obstacle avoidance, making them ideal for complex robotic applications. These characteristics of the metaheuristic algorithms can used in developing an integrated framework that offers complete solution to robot manipulators. This research article presents a generalized framework leveraging meta-heuristic algorithms to address dynamic obstacle avoidance in redundant manipulators. The framework uses meta-heuristic algorithms as the inverse kinematics solver, 3D trajectory planner, and obstacle avoidance mechanism, encompassing both static and dynamic obstacles. The proposed framework is generalized and gives the user to select the type of robot manipulator, with any number of links with any custom trajectory within the workspace of the robot manipulator. Also, any metaheuristic algorithm can be used in the proposed framework. The proposed framework is implemented in MATLAB’s app designer for simulation with six different meta-heuristic algorithms. The effectiveness of the framework was evaluated in terms of its capability to generate 3d path, its ability to follow generated trajectory, while seamlessly adapting to dynamically changing environments. Through simulation, the framework showcased robust performance in navigating workspaces with moving obstacles, ensuring collision-free motion for redundant manipulators.

A deep learning operator-based numerical scheme method for solving 1D wave equations

Abstract In this paper, we introduce the deep numerical technique DeepNM, which is designed for solving one-dimensional (1D) hyperbolic conservation laws, particularly wave equations. By creatively integrating traditional numerical schemes with deep learning techniques, the method yields improvements over conventional approaches. Specifically, we compare this approach against two established classical numerical methods: the discontinuous Galerkin method (DG) and the Lax–Wendroff correction method (LWC). While maintaining a comparable level of accuracy, DeepNM significantly improves computational speed, surpassing conventional numerical methods in this aspect by more than tenfold, and reducing storage requirements by over 1000 times. Furthermore, DeepNM facilitates the utilization of higher-order numerical schemes and allows for an increased number of grid points, thereby enhancing precision. In contrast to the more prevalent PINN method, DeepNM optimally combines the strengths of conventional mathematical techniques with deep learning, resulting in heightened accuracy and expedited computations for solving partial differential equations. Notably, DeepNM introduces a novel research paradigm for numerical equation-solving that can be seamlessly integrated with various traditional numerical methods.

Estimation of Marshall–Olkin Extended Generalized Extreme Value Distribution Parameters under Progressive Type-II Censoring by Using a Genetic Algorithm

In this article, we consider the statistical analysis of the parameter estimation of the Marshall–Olkin extended generalized extreme value under liner normalization distribution (MO-GEVL) within the context of progressively type-II censored data. The progressively type-II censored data are considered for three specific distribution patterns: fixed, discrete uniform, and binomial random removal. The challenge lies in the computation of maximum likelihood estimations (MLEs), as there is no straightforward analytical solution. The classical numerical methods are considered inadequate for solving the complex MLE equation system, leading to the necessity of employing artificial intelligence algorithms. This article utilizes the genetic algorithm (GA) to overcome this difficulty. This article considers parameter estimation through both maximum likelihood and Bayesian methods. For the MLE, the confidence intervals of the parameters are calculated using the Fisher information matrix. In the Bayesian estimation, the Lindley approximation is applied, considering LINEX loss functions and square error loss, suitable for both non-informative and informative contexts. The effectiveness and applicability of these proposed methods are demonstrated through numerical simulations and practical real-data examples.

Semi-analytic PINN methods for boundary layer problems in a rectangular domain

Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

Physics-informed neural network for diffusive wave model

The diffusive wave model (DWM), a nonlinear second-order simplified form of the shallow water equation, has been widely used in hydraulic, hydrologic and irrigation engineering. Solution of the forward problem of the DWM can be utilized to predict evolution in water levels and discharge. Solution of its inverse problem allows for the identification of crucial parameters (such as Manning coefficient, rainfall intensity, etc.) based on observations. This paper applied the physics-informed neural network (PINN) with novel improvements to solve the DWM for both forward and inverse problems. In the forward problem, compared to traditional numerical methods, PINN was able to predict the evolution at any location. In the inverse problem, PINN provided a simple and efficient solution process. In order to overcome the gradient explosion in the training process caused by the characteristics of the DWM, the stop-gradient technique was adopted to train the neural network. To improve the estimation of DWM parameters, the concept of time division was developed, and a new network structure was proposed. To verify the effectiveness of PINN and its improved algorithm for DWM, seven examples were simulated. The PINN solutions for forward problems were compared with the results obtained by classical numerical methods, while the correct rainfall pattern was identified for the inverse problem.

Rock mass substitute geometry theory (SGT) and 3D centroid sliding pyramid (CSP) method

Discontinuous analysis methods are typically used for the stability analysis of fractured rock masses with multiple sets of joints. Among various discontinuous methods, the key block theory (KBT) requires less data input, has a higher computational efficiency, and is suitable for wide-ranging engineering applications. Classical discontinuous numerical methods typically require complete boundary information for rock block units. However, current information-gathering methods can only capture information on rock surfaces. The proposed rock mass substitute geometry theory (SGT) overcomes this limitation by using local geometric information to equivalently represent the original geometric body simplifying subsequent analyses. A 3D centroid sliding pyramid (CSP) method is introduced for motion state analysis of a rock block that uses only the collinear faces of a block’s free surfaces. A comparison of CSP with KBT proves that CSP not only addresses the issue of collecting only surface information of rock masses but it significantly minimizes the computation scale. Additionally, a 3D discontinuous analysis program was developed, encompassing the entire process of rock mass information gathering, 3D discontinuous modeling, and 3D CSP analysis. The effectiveness and efficiency of the method were validated using stochastically generated models, while its practical engineering applicability was demonstrated at an engineering site.

Intelligent optimal control of nonlinear diabetic population dynamics system using a genetic algorithm

Diabetes is a chronic disease affecting millions of people worldwide. Several studies have been carried out to control the diabetes problem, involving both linear and non-linear models. However, the complexity of linear models makes it impossible to describe the diabetic population dynamic in depth. To capture more detail about this dynamic, non-linear terms were introduced into the mathematical models, resulting in more complicated models strongly consistent with reality (capable of re-producing observable data). The most commonly used methods for control estimation are Pantryagain’s maximum principle and Gumel’s numerical method. However, these methods lead to a costly strategy regarding material and human resources; in addition, diabetologists cannot use the formulas implemented by the proposed controls. In this paper, the authors propose a straightforward and well-performing strategy based on non-linear models and genetic algorithms (GA) that consists of three steps: 1) discretization of the considered non-linear model using classical numerical methods (trapezoidal rule and Euler–Cauchy algorithm); 2) estimation of the optimal control, in several points, based on GA with appropriate fitness function and suitable genetic operators (mutation, crossover, and selection); 3) construction of the optimal control using an interpolation model (splines). The results show that the use of the GA for non-linear models was successfully solved, resulting in a control approach that shows a significant decrease in the number of diabetes cases and diabetics with complications. Remarkably, this result is achieved using less than 70% of available resources.

Physics-Informed Neural Networks for Solving High-Index Differential-Algebraic Equation Systems Based on Radau Methods

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10−6, and the absolute errors for algebraic variables are maintained at 10−5. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

Potential of physics-informed neural networks for solving fluid flow problems with parametric boundary conditions

Fluid flows are present in various fields of science and engineering, so their mathematical description and modeling is of high practical importance. However, utilizing classical numerical methods to model fluid flows is often time consuming and a new simulation is needed for each modification of the domain, boundary conditions, or fluid properties. As a result, these methods have limited utility when it comes to conducting extensive parameter studies or optimizing fluid systems. By utilizing recently proposed physics-informed neural networks (PINNs), these limitations can be addressed. PINNs approximate the solution of a single or system of partial differential equations (PDEs) by artificial neural networks (ANNs). The residuals of the PDEs are used as the loss function of the ANN, while the boundary condition is imposed in a supervised manner. Hence, PDEs are solved by performing a nonconvex optimization during the training of the ANN instead of solving a system of equations. Although this relatively new method cannot yet compete with classical numerical methods in terms of accuracy for complex problems, this approach shows promising potential as it is mesh-free and suitable for parametric solution of PDE problems. This is achieved without relying on simulation data or measurement information. This study focuses on the impact of parametric boundary conditions, specifically a variable inlet velocity profile, on the flow calculations. For the first time, a physics-based penalty term to avoid the suboptimal solution along with an efficient way of imposing parametric boundary conditions within PINNs is presented.

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

Quantum computer based feature selection in machine learning

AbstractThe problem of selecting an appropriate number of features in supervised learning problems is investigated. Starting with common methods in machine learning, the feature selection task is treated as a quadratic unconstrained optimisation problem (QUBO), which can be tackled with classical numerical methods as well as within a quantum computing framework. The different results in small problem instances are compared. According to the results of the authors’ study, whether the QUBO method outperforms other feature selection methods depends on the data set. In an extension to a larger data set with 27 features, the authors compare the convergence behaviour of the QUBO methods via quantum computing with classical stochastic optimisation methods. Due to persisting error rates, the classical stochastic optimisation methods are still superior.

Triangular intuitionistic fuzzy linear system of equations with applications: an analytical approach

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two n × n crisp triangular intuitionistic fuzzy (TIF) systems of linear equations. In the original system, the coefficient matrix is considered as real crisp, while the unknown variable vector and right hand side vector are regarded as triangular intuitionistic fuzzy numbers. The Block Homotopy Modified Perturbation Method is found to be efficient and practical to solve n × n TIF linear systems as it only requires the non-singularity of the n × n TIF linear system's coefficient matrix, whereas the point Homotopy Perturbation Method and other classical numerical iterative methods typically require non-zero diagonal entries in the coefficient matrix. A set of theorems relevant to this study are presented and demonstrated. We solve an engineering application, i.e. a current flow circuit problem that is represented in terms of a triangular intuitionistic fuzzy environment, using the suggested method. The unknown current is then obtained as a triangle intuitionistic fuzzy number. The proposed semi-analytic method is used to solve some numerical test problems in order to validate their performance and efficiency in comparison to other existing techniques. The numerical results of the example are displayed on graphs with different degrees of uncertainty. The efficiency and accuracy of the proposed method are further demonstrated by comparisons to block Jacobi, Adomain Decomposition method, Successive Over-Relaxation method and the classical Gauss-Seidel numerical method.