Conventional Numerical Methods Research Articles

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.

Optimization of the generalized covariance estimator in noncausal processes

This paper investigates the performance of routinely used optimization algorithms in application to the Generalized Covariance estimator (GCov) for univariate and multivariate mixed causal and noncausal models. The GCov is a semi-parametric estimator with an objective function based on nonlinear autocovariances to identify causal and noncausal orders. When the number and type of nonlinear autocovariances included in the objective function are insufficient/inadequate, or the error density is too close to the Gaussian, identification issues can arise. These issues result in local minima in the objective function, which correspond to parameter values associated with incorrect causal and noncausal orders. Then, depending on the starting point and the optimization algorithm employed, the algorithm can converge to a local minimum. The paper proposes the Simulated Annealing (SA) optimization algorithm as an alternative to conventional numerical optimization methods. The results demonstrate that SA performs well in its application to mixed causal and noncausal models, successfully eliminating the effects of local minima. The proposed approach is illustrated by an empirical study of a bivariate series of commodity prices.

A new method for solving multidimensional heat conduction problems

In this study, we propose an efficient boundary-type method named the half boundary method (HBM) for solving multidimensional heat conduction equations. Unlike conventional numerical methods, the core idea behind the HBM is to concentrate variables along half of the boundary and employ them to represent node variables within the solution domain. This approach streamlines the solving process by dealing with smaller-order matrices, substantially reducing storage requirements. We derived the fundamental equations of the HBM for solving multidimensional heat conduction differential equations. Furthermore, by introducing the Cayley–Hamilton formula, we optimize the program to enhance computational efficiency. Additionally, we explored heat conduction problems in solution domains differing from rectangular regions. Notably, we pioneer the application of the HBM to address three-dimensional (3-D) problems for the first time, ensuring its accuracy through rigorous validation.

Neural network approach for cholera dynamics: Integrating deterministic and stochastic insights

In this article, a comprehensive qualitative analysis framework is presented that integrates stochastic and deterministic models to explore the transmission dynamics of Cholera disease. The significance of the reproduction number in understanding the spread of the disease and in formulating effective control strategies is emphasized. The sensitivity analysis of model parameters is conducted to identify those that critically influence transmission. By converting the deterministic model into a stochastic form, Brownian motion and stochastic parameters are incorporated to capture the inherent randomness in disease dynamics. This approach is further enriched by the application of neural networks (NN) to simulate and validate the complex interactions involved in Cholera transmission. The analysis extends to the global positive solution, stochastic reproduction number, and virus-free stochastic dynamics, providing an improved understanding of the disease's behavior. The critical conditions necessary for the extinction and long-term elimination of the virus are also addressed, highlighting the importance of controlling its spread. The stationary distribution of the disease is examined through the lens of a regular Markov process, enhancing the understanding of the stochastic model's dynamics. The results verified by neural networks, compared with those obtained from conventional numerical methods, underscore the accuracy and versatility of neural networks in epidemiological research, offering new perspectives and methodologies for tackling this global health challenge.

Thermal performance and optimization of an integrated collector–storage solar air heater based on lap joint-type flat micro-heat pipe arrays

This study seeks to optimize the performance of an integrated collector–storage solar air heater (ICSSAH) based on lap joint-type (LJT) flat micro-heat pipe arrays (FMHPAs) and latent thermal storage through numerical simulation. A novel dynamic heat transfer model is developed by incorporating the heat transfer correlation describing the thermal characteristics of phase change material (PCM) in narrow and elongated confined spaces within plate fins and validated using experimental data to address the issue of deviation caused by the fixed thermal resistance of the PCM in conventional numerical simulation methods. The validated model is used to simulate and predict the dynamic performance of LJT-FMHPA-ICSSAH under different heat transfer structures. Several optimization parameters, including the air gap thickness, the optical properties of the collector and glass cover plates, and fins structure, are considered. Simulation results indicate that the heat lost to the external environment through the glass cover plate accounted for over 70 % of the total heat dissipation. The optimal combination included a 50 mm gas layer; copper-based blue titanium coating; ultra-white glass; and fin spacing, thickness, height, and length of 5, 0.2, 60, and 195 mm, respectively. Under autumn/winter operating conditions, the optimized thermal performance is 1.13–1.35 times higher than the performance prior to optimization.

Peridynamics simulation of TBM cutting process in soft and hard composite strata with a simplified loading scheme

In this study, we address the complex geological conditions encountered during tunnel boring machine (TBM) excavation, particularly the challenges posed by soft and hard composite strata. To overcome limitations of conventional numerical methods for analyzing rock fracture in such strata, we employ Peridynamics to simulate rock fragmentation induced by disc cutters. We propose a virtual disc cutter loading scheme that enhances computational efficiency by replacing traditional contact models. Through numerical examples, we demonstrate the efficacy of this loading scheme in simulating rock fragmentation within soft and hard composite strata. With this method, we conduct numerical studies on kerfs-assisted rock fragmentation in such geological conditions, finding that kerfs can help mitigate differences in cutter forces between soft and hard rocks, thus showing potential for improving TBM performance. Our findings offer insights into optimizing TBM operations in challenging geological environments.

A hybridized DDA-DDM for modeling jointed rock masses

The complexity of geo-materials’ behaviors surpass the capabilities of conventional numerical methods alone, particularly when realistically modeling rock structures. Coupling methods can enhance the realism of numerical modeling. Discontinuous deformation analysis (DDA) and displacement discontinuity method (DDM) are combined to model block displacement and crack propagation mechanisms in a blocky rock mass. DDA calculates stresses, strains, rotations, and displacements of the blocks, while DDM predicts crack propagation paths based on specified boundary conditions. The displacements result from DDA are transformed into stresses, followed by investigating crack propagation mechanisms within each block using Kelvin's solution. Boundary stresses are adjusted as cracks propagate, updating stress boundary conditions in DDA. This iterative process continues until crack propagation halts or a new block emerges. In this paper, the complete failure process of a rock slab with one pre-existing non-persistent joint located on the crest of a rock slope and compression test on the rock-like specimen with a single random crack with a 45° slope angle is simulated using the proposed numerical approach. The proposed numerical solution is compared with existing numerical results. This comparison confirms the accuracy and effectiveness of the proposed procedure, as the predicted crack propagation paths from the coupled numerical method align well with the corresponding numerical results from the rock-like samples.

Real-Time 3D Tracking of Multi-Particle in the Wide-Field Illumination Based on Deep Learning.

In diverse realms of research, such as holographic optical tweezer mechanical measurements, colloidal particle motion state examinations, cell tracking, and drug delivery, the localization and analysis of particle motion command paramount significance. Algorithms ranging from conventional numerical methods to advanced deep-learning networks mark substantial strides in the sphere of particle orientation analysis. However, the need for datasets has hindered the application of deep learning in particle tracking. In this work, we elucidated an efficacious methodology pivoted toward generating synthetic datasets conducive to this domain that resonates with robustness and precision when applied to real-world data of tracking 3D particles. We developed a 3D real-time particle positioning network based on the CenterNet network. After conducting experiments, our network has achieved a horizontal positioning error of 0.0478 μm and a z-axis positioning error of 0.1990 μm. It shows the capability to handle real-time tracking of particles, diverse in dimensions, near the focal plane with high precision. In addition, we have rendered all datasets cultivated during this investigation accessible.

A DC fault current fast-computing method of MMC-HVDC grid with short circuit protection equipment

The multi-terminal modular multi-level converter-based high voltage direct current (MMC-HVDC) grid with short circuit protection equipment (SCPE) is so complex that it is difficult to estimate its fault current and analyze the performance of SCPE by conventional time-domain numerical calculation method, it meets three big obstacles. This paper has made significant progress in overcoming these obstacles. 1). By applying the modern electrical circuit theory, a systematic formulation of the differential equation set for fault current calculation is developed to avoid a lot of complex and cumbersome matrix manual calculations. 2). A novel Y-Delta transformation in the s-domain is proposed to develop an eliminating virtual node approach for a complex MMC-HVDC grid, including the ring, radial, and hybrid topologies. 3). It is difficult to solve the equivalent circuit of MMC-HVDC grid with SCPE since SCPE is a time-variable-nonlinear circuit. A canonical voltage source model of SCPE is established to transform the time-variable-nonlinear circuit into a piecewise linear circuit. Based on the three significant progresses, a DC fault current fast-computing method of MMC-HVDC grid with SCPE is put forward to deal with all kinds of MMC-HVDC grids with several kinds of SCPEs. Then, the performance of several kinds of SCPE is analyzed and compared by this method. Consequently, the proposed DC fault current fast-computing method is a new powerful tool to estimate the fault current of MMC-HVDC grid and analyze the performance of SCPE.

Improved dynamic analysis for plane elasticity problems with a boundary meshfree method and distributed sources

In this paper, we present an improved dynamic analysis method to address plane elasticity problems. Distributed sources are employed in a boundary-free mesh-based approach known as Boundary Distributed Source (BDS). This method, which builds upon a modified Method of Fundamental Solutions (MFS), aims to eliminate singularities without the reliance on fictitious boundaries. Distributed sources are strategically positioned directly on the real boundary, centered over disks. The method's efficiency and simplicity are highlighted by its ability to operate without the need for precise data concerning adjacent points. The dual reciprocity method (DRM) is utilized to approximate the non-homogeneous inertial term. Furthermore, the present formulation investigates several factors that influence the convergence of the proposed method, including the number of collocation points, internal points, the radius of the circular disk, and the time step size. Based on the numerical results, it is evident that the method exhibits remarkable accuracy and efficiency when applied to plane elasticity problems including forced, harmonic, and free analyses. As compared to conventional numerical methods for dynamic analysis, the method outperforms in both accuracy and convergence rate.

Calculation of noise field in main transformer room of indoor substation based on thermal-acoustic coupling

In order to calculate the noise field distribution in the main transformer room of an indoor substation accurately, a thermal-acoustic coupling idea was introduced, and a noise field solving the algorithm based on thermal-acoustic coupling was proposed in view of the noise quality loss caused by the effect of ambient temperature changes. Firstly, the fluid-structure coupling model of the main transformer room is established, and the computational fluid dynamics method is used to calculate the flow field distribution and temperature boundary conditions under the action of the heat source; Then, extracting the rheological equation of the environmental boundary conditions in the flow field calculation, and use the difference function to couple the rheological equation to the sound grid to obtain the sound field coupling model under the condition of variable temperature; Finally, the modified parameter acoustic integral equation under the sound field coupling model is calculated based on the conventional Gauss numerical integration method, and the sound field of the main transformer room considering the effect of variable temperature is obtained. The algorithm has been successfully applied in the analysis of the noise field of Xi'an 110 kV Zhenxing Substation, and the error with the measured value is 2.083%.

Efficient Method for Numerical Calculations of Molecular Vibrational Frequencies by Exploiting Sparseness of Hessian Matrix.

Molecular vibrational frequency analysis plays an important role in theoretical and computational chemistry. However, in many cases, the analytical frequencies are unavailable, whereas frequency calculations using conventional numerical methods are very expensive. In this work, we propose an efficient method to numerically calculate the frequencies. Our main strategies are to exploit the sparseness of the Hessian matrix and to construct the N-fold two-variable potential energy surfaces to fit the parabola parameters, which are later used for the construction of Hessian matrices. A set of benchmark calculations is performed for typical molecules of different sizes and complexities using the proposed method. The obtained frequencies are compared to those calculated with the analytical methods and conventional numerical methods. It is shown that the results yielded with the new method are in very good agreement with corresponding accurate values (with a maximum error of ∼20 cm-1), while the required computation resource is largely reduced compared to that required by conventional numerical methods. For medium-sized molecules, the calculational scaling is lowered to O(N1.6) (this work) from that of O(N2) (conventional numerical methods). For even larger molecules, more computational savings can be achieved, and the scaling is estimated to be quasilinear with respect to the molecular size.

A Deep Learning Method for Dynamic Process Modeling of Real Landslides Based on Fourier Neural Operator

AbstractThe conventional numerical solvers for partial differential equations encounter a formidable challenge, as their computational efficiency and accuracy are heavily contingent on grid size. Recently, machine learning (ML) has exhibited substantial promise in addressing partial differential equations. Nevertheless, substantial hurdles persist in practical applications. In this work, we endeavor to establish a deep learning framework founded on the Fourier neural operator (FNO) for resolving the intricacies of simulating real landslide dynamic processes. Our findings demonstrate that the current FNO approach adeptly replicates landslide dynamic processes and boasts exceptional computational efficiency. Additionally, it is noteworthy that this data‐driven ML methodology can seamlessly incorporate data from other experimental sources or numerical simulation techniques. Consequently, this work underscores the significant potential of utilizing ML methodologies to supplant conventional numerical simulation methods.

Matrix formulation of the Gaussian expansion of coherent multiple beams in arbitrary dimensions.

Modeling the propagation of beams along laser beamlines is very challenging due to the multidimensional and multiscale configuration of the problem. Spatio-temporal couplings are particularly difficult to address with conventional numerical methods. Here we derive the Wigner function of a sum of Gaussian beams by calculating the multidimensional Fourier transform of the intercorrelation function of the fields. The matrix formulation allows for a simple propagation of the Wigner function in the framework of matrix optics. The relevancy of this approach is assessed by applying this model to one-dimensional and multidimensional configurations and by studying the influence of spatio-temporal couplings when considering propagation and dispersion by a diffraction grating.

A new modified deep learning technique based on physics-informed neural networks (PINNs) for the shock-induced coupled thermoelasticity analysis in a porous material

In this article, a new modified deep learning (DL) method based on physics-informed neural networks (PINNs) is proposed for analyzing generalized coupled thermoelasticity in a porous material under shock loadings using Lord–Shulman (LS) theory. The PINN-based method demonstrates remarkable capabilities in solving differential equations and identifying unknown parameters. It is employed to solve a system of coupled partial differential equations (PDEs) governing a porous half-space material, considering thermal and strain relaxation coefficients in the LS theory. The optimal structure of the PINN is investigated through sensitivity analyses. Two adaptive sampling techniques, residual-based adaptive refinement (RAR) and residual-based adaptive distribution (RAD), are employed to enhance solution quality within the optimized architecture. The proposed forward PINN utilizes known values of field variables at initial and boundary conditions. The efficiency and effectiveness of the proposed PINN approach are demonstrated through three distinct scenarios. Non-parametric statistical tests and L 2 relative error analysis validate the extraordinary potential of the proposed PINN-based method in accurately capturing the system behavior. The extrapolation results, represented as time history plots, showcase exceptional accuracy in this study, overcoming the limitations of conventional numerical methods in larger temporal domains.

Physics-Informed Neural Network (PINN) for Solving Frictional Contact Temperature and Inversely Evaluating Relevant Input Parameters

Ensuring precise prediction, monitoring, and control of frictional contact temperature is imperative for the design and operation of advanced equipment. Currently, the measurement of frictional contact temperature remains a formidable challenge, while the accuracy of simulation results from conventional numerical methods remains uncertain. In this study, a PINN model that incorporates physical information, such as partial differential equation (PDE) and boundary conditions, into neural networks is proposed to solve forward and inverse problems of frictional contact temperature. Compared to the traditional numerical calculation method, the preprocessing of the PINN is more convenient. Another noteworthy characteristic of the PINN is that it can combine data to obtain a more accurate temperature field and solve inverse problems to identify some unknown parameters. The experimental results substantiate that the PINN effectively resolves the forward problems of frictional contact temperature when provided with known input conditions. Additionally, the PINN demonstrates its ability to accurately predict the friction temperature field with an unknown input parameter, which is achieved by incorporating a limited quantity of easily measurable actual temperature data. The PINN can also be employed for the inverse identification of unknown parameters. Finally, the PINN exhibits potential in solving inverse problems associated with frictional contact temperature, even when multiple input parameters are unknown.

Real-time numerical differentiation of sampled data using adaptive input and state estimation

Real-time numerical differentiation plays a crucial role in many digital control algorithms, such as PID control, which requires numerical differentiation to implement derivative action. This paper proposes an algorithm for estimating the numerical derivative of a signal from noisy sampled data measurements. The method uses adaptive input estimation with adaptive state estimation (AIE/ASE), and thus it requires only minimal prior information about the signal and noise statistics. Furthermore, since the estimates of the derivative at step k provided by AIE/ASE depend only on data available up to step k, AIE/ASE is thus implementable in real time. The accuracy of AIE/ASE is compared numerically to several conventional numerical differentiation methods. Finally, AIE/ASE is applied to simulated vehicle position data, generated in the CarSim simulator software.

Physics-informed neural network for predicting hot-rolled steel temperatures during heating process

The heating process in hot-rolled steel manufacture is a key step for product quality. It is desired that the steel slabs can be heated to the target temperature in the furnace with good temperature uniformity but low energy consumption. Due to the extreme conditions inside the furnace, the internal temperatures of steel slabs cannot be directly monitored. The existing methods use key thermal parameters to numerically analyze and simulate the hot-rolled steel temperatures during the heating process. However, these conventional methods require substantially intensive calculations and thus cannot be applied to the real-time simulation of hot-rolled steel temperatures in the furnace. This paper presents a novel predictive model that is developed based on a physics-informed neural network for prediction of the hot-rolled steel temperatures. It embeds thermophysical information in the neural network, in order to efficiently predict the internal temperatures of steel slabs during the heating process. The temperature predictions derived from the physics-informed neural network-based model were compared with the results obtained from an experiment and the conventional numerical method. The comparisons generally indicate that the physics-informed neural network-based model yields accurate temperature predictions, but with greatly improved computational efficiency over the conventional numerical method.

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

The Boussinesq and Camassa–Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physics-informed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa–Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision time-splitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shapedsoliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method.

Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators.

We address the challenge of acoustic simulations in three-dimensional (3D) virtual rooms with parametric source positions, which have applications in virtual/augmented reality, game audio, and spatial computing. The wave equation can fully describe wave phenomena such as diffraction and interference. However, conventional numerical discretization methods are computationally expensive when simulating hundreds of source and receiver positions, making simulations with parametric source positions impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with parametric source positions, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 to 0.10 Pa. Notably, our method signifies a paradigm shift as-to our knowledge-no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains.