Nonlinear Problem Research Articles

Parameter Sensitivity Analysis for Diesel Spray Penetration Prediction Based on GA-BP Neural Network

Machine learning has started to be used in engine research to optimize combustion and predict fuel spray characteristics. This paper presents the development of a machine learning model using a Genetic Algorithm-Backpropagation (GA-BP) neural network to predict spray penetration. The GA-BP neural network was selected for its ability to optimize neural network weights and biases, thereby improving model convergence and avoiding local minima, which are common challenges in complex, non-linear problems such as spray prediction. The model was trained using experimental data from diesel injector spray tests, and its accuracy was evaluated through parametric sensitivity analysis, examining the influence of various input factors. A comparison between the machine learning model and the traditional empirical formulas of spray penetration revealed that the machine learning model achieved greater accuracy. In terms of the sensitivity to inputs, it is interesting to find that the cognition of machines is different from that of humans. When an input parameter does not have any functional relationship with other input parameters, the absence of this input parameter will lead to a significant decrease in the accuracy of the output result. The results demonstrate that the machine learning approach offers higher accuracy and better generalizability compared to traditional empirical methods. This study recommends the ways to get better results of penetration prediction with BP neural networks, which is efficient in training and utilizing Artificial Neural Networks (ANNs).

Adaptive Pseudospectral Successive Convex Optimization for Six-Degree-of-Freedom Powered Descent Guidance

Guidance for six-degrees-of-freedom powered descent is more challenging due to its stronger nonlinearity compared to three-degrees-of-freedom. The standard convex programming algorithm has been difficult to effectively address this problem. To enhance the performance of successive convex programming in terms of both optimality and accuracy, an adaptive pseudospectral successive convex optimization algorithm is proposed in this paper. First, it transforms the nonlinear optimal control problem into a convex subproblem by integrating global pseudospectral discretization with local linearization. Second, a parameter-adaptive successive convexification algorithm is proposed, which adaptively adjusts the trust region size based on the update rate of the optimal trajectory. Last, the global error and local error are accurately calculated based on the pseudospectral method. To tackle the issue of excessively large local errors caused by nonlinearity, an adaptive grid method is proposed to refine the mesh grid. Simulation results demonstrate the efficacy of the proposed pseudospectral convex optimization method and adaptive mesh grid method in reducing local errors and improving solution accuracy. Furthermore, the proposed parameter-adaptive successive convex optimization algorithm exhibits adaptability and optimality across different initial conditions, while also improving solution accuracy.

Weighted sum method based multi-objective optimal power flow considering various objectives: an application of whale optimization algorithm

Nowadays, multi-objective optimization plays a vital role in solving optimal power flow problems. Multi-objective optimal power flow (MOOPF) is a nonlinear optimization problem aimed at optimizing control variables while balancing multiple objective functions and satisfying both equality and inequality constraints and addresses this by integrating two more objectives into a single objective using a weighting factor. In this paper this weighted sum type multi-objective technique has been used to formulate the objective function. The whale optimization algorithm (WOA) has been used to reduce the cost, emission, losses, and voltage stability by considering various multi objectives like fuel cost along with emission, fuel cost with losses, fuel cost with voltage stability, fuel cost with voltage deviation and finally fuel cost with emission, losses, voltage deviation. In this paper, the IEEE 30 bus structure has been used to analyze the effect of WOA on the improvement of system performance. Obtained results with WOA have been compared with other optimization techniques like ensemble constraint handling technique with differential evolution (ECHT-DE), the superiority of feasible differential evolution (SF-DE), moth swarm algorithm (MSA), and moth-flame optimization (MFO), available in the literature.

A divergence-free and [formula omitted]-conforming embedded-hybridized DG method for the incompressible resistive MHD equations

We present a divergence-free and Hdiv-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and Hdiv-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems.

Adaptive reverse task offloading in edge computing for AI processes

Nowadays, we witness the proliferation of edge IoT devices, ranging from smart cameras to autonomous vehicles, with increasing computing capabilities, used to implement AI-based services in users’ proximity, right at the edge. As these services are often computationally demanding, the popular paradigm of offloading their tasks to nearby cloud servers has gained much traction and been studied extensively. In this work, we propose a new paradigm that departs from the above typical edge computing offloading idea. Namely, we argue that it is possible to leverage these end nodes to assist larger nodes (e.g., cloudlets) in executing AI tasks. Indeed, as more and more end nodes are deployed, they create an abundance of idle computing capacity, which, when aggregated and exploited in a systematic fashion, can be proved beneficial. We introduce the idea of reverse offloading and study a scenario where a powerful node splits an AI task into a group of subtasks and assigns them to a set of nearby edge IoT nodes. The goal of each node is to minimize the overall execution time, which is constrained by the slowest subtask, while adhering to predetermined energy consumption and AI performance constraints. This is a challenging MINLP (Mixed Integer Non-Linear Problem) optimization problem that we tackle with a novel approach through our newly introduced EAI-ARO (Edge AI-Adaptive Reverse Offloading) algorithm. Furthermore, a demonstration of the efficacy of our reverse offloading proposal using an edge computing testbed and a representative AI service is performed. The findings suggest that our method optimizes the system’s performance significantly when compared with a greedy and a baseline task offloading algorithm.

On some noncoercive nonlinear problems in unbounded domains

On some noncoercive nonlinear problems in unbounded domains

Research on Microgrid Optimal Scheduling Based on an Improved Honey Badger Algorithm

As global energy demands continue to grow and environmental protection pressures increase, microgrids have garnered widespread attention due to their ability to effectively integrate distributed energy sources, improve energy utilization efficiency, and enhance grid stability. Due to the complexity of internal structure, variety of energy sources, and uncertainty of load demand, the optimal scheduling problem of microgrids becomes extremely complicated. Traditional optimization methods often perform poorly in complex and dynamic microgrid environments, and it is assumed that the complexity is low or that more simplification is needed, which leads to poor convergence and local optimality when dealing with uncertainty and nonlinear problems, making intelligent optimization algorithms a crucial solution to this problem. To address the shortcomings of the traditional honey badger algorithm, such as the slow convergence speed and a tendency to fall into local optima in complex microgrid optimal scheduling problems, this paper proposes a multi-strategy improved honey badger algorithm. During the population initialization phase, a combined opposition-based learning strategy is introduced to enhance the algorithm’s exploration and exploitation capabilities. Additionally, the introduction of variable spiral factors and a linearly decreasing strategy for parameters improves the overall efficiency of the algorithm and reduces the risk of local optima. To further enhance population diversity, a hunger search strategy is employed, providing stronger adaptability and global search capabilities in varying environments. The improved honey badger algorithm is then applied to solve the multi-objective optimal scheduling problem in grid-connected microgrid modes. The simulation results indicate that the improved honey badger algorithm effectively enhances the economic and environmental benefits of microgrid operations, improving system operational stability.

Quasinormal mode expansion in thin elastic plates

Elastic wave manipulation using large arrays of resonators is driving the need for advanced simulation and optimization methods. To address this we introduce and explore a robust framework for wave control: quasinormal modes (QNMs). Specifically, we consider the problem for thin elastic plates, where the Green's function formalism is well known and readily exploited to solve multiple scattering problems. By studying the associated nonlinear eigenvalue problem we derive a dispersive QNM expansion, providing a reduced-order model for efficient forced response computations which reveals physical insight into the resonant mode excitation. Furthermore, we derive eigenvalue sensitivities with respect to resonator parameters and apply a gradient-based optimization to design quasi-bound states in the continuum and position eigenfrequencies precisely in the complex plane. Scattering simulations validate our approach in structures such as graded line arrays and quasicrystals. Drawing on QNM concepts from electromagnetism we demonstrate significant advances in elastic metamaterials, highlighting their potential for tailored wave manipulation. Published by the American Physical Society 2024

Determination of the Thermal Conductivity and Volumetric Heat Capacity of Substance from Heat Flux

The study of nonlinear problems related to heat transfer in a substance is of great practical important. Earlier, this paper’s authors proposed an effective algorithm for determining the volumetric heat capacity and thermal conductivity of a substance based on experimental observations of the dynamics of the temperature field in the object. In this paper, the problem of simultaneous identification of temperature-dependent volumetric heat capacity and thermal conductivity of the substance under study from the heat flux at the boundary of the domain is investigated. The consideration is based on the first boundary value problem for a one-dimensional unsteady heat equation. The coefficient inverse problem under consideration is reduced to a variational problem, which is solved by gradient methods based on the application of fast automatic differentiation. The uniqueness of the solution of the inverse problem is investigated.

Dynamic Description of soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

Physics-informed two-tier neural network for non-linear model order reduction

AbstractIn recent years, machine learning (ML) has had a great impact in the area of non-intrusive, non-linear model order reduction (MOR). However, the offline training phase often still entails high computational costs since it requires numerous, expensive, full-order solutions as the training data. Furthermore, in state-of-the-art methods, neural networks trained by a small amount of training data cannot be expected to generalize sufficiently well, and the training phase often ignores the underlying physical information when it is applied with MOR. Moreover, state-of-the-art MOR techniques that ensure an efficient online stage, such as hyper reduction techniques, are either intrusive or entail high offline computational costs. To resolve these challenges, inspired by recent developments in physics-informed and physics-reinforced neural networks, we propose a non-intrusive, physics-informed, two-tier deep network (TTDN) method. The proposed network, in which the first tier achieves the regression of the unknown quantity of interest and the second tier rebuilds the physical constitutive law between the unknown quantities of interest and derived quantities, is trained using pretraining and semi-supervised learning strategies. To illustrate the efficiency of the proposed approach, we perform numerical experiments on challenging non-linear and non-affine problems, including multi-scale mechanics problems.

Long-term load forecasting for smart grid

Abstract The load forecasting problem is a complicated non-linear problem connected with the weather, economy, and other complex factors. For electrical power systems, long-term load forecasting provides valuable information for scheduling maintenance, evaluating adequacy, and managing limited energy supplies. A future generating, transmission, and distribution facility's development and planning process begins with long-term demand forecasting. The development of advanced metering infrastructure (AMI) has greatly expanded the amount of real-time data collection on large-scale electricity consumption. The load forecasting techniques have changed significantly as a result of the real-time utilization of this vast amount of smart meter data. This study suggests numerous approaches for long-term load forecasting using smart-metered data from an actual distribution system on the NIT Patna campus. Data pre-processing is the process of converting unprocessed data into a suitable format by eliminating possible errors caused by lost or interrupted communications, the presence of noise or outliers, duplicate or incorrect data, etc. The load forecasting model is trained using historical load data and significant climatic variables discovered through correlation analysis. With a minimum MAPE and RMSE for every testing scenario, the proposed artificial neural network model yields the greatest forecasting performance for the used system data. The efficacy of the proposed technique has been through a comparison of the acquired results with various alternative load forecasting methods.

Studying a cylindrical moving boundary problem in ductal carcinoma in situ

In this paper, a mathematical model of ductal carcinoma in situ (DCIS) is considered as a moving boundary problem in cylindrical coordinates. Using the front-fixing technique, first, we transform the problem into a nonlinear parabolic problem on a fixed domain, and then some essential properties of the solution, such as the uniqueness of the solution, are shown. A numerical approach based on the finite difference method is established for solving the fixed domain problem. The solvability of the proposed finite difference method is proved, and the stability and convergence of the numerical solution are shown. For a test case problem, analytical and numerical results are compared to demonstrate the ability and effectiveness of the numerical method.

Wavelet-based collocation methods for solving multi-dimensional integro-differential systems

This paper presents a comprehensive investigation of wavelet-based collocation methods for solving multi-dimensional integro-differential systems. The research addresses the challenging problem of finding numerical solutions to integro-differential equations (IDEs) that are essential in various scientific and engineering applications. We propose novel numerical approximations for infinitely dimensional problems, including fractional Cattaneo-Rayleigh waves and nonlinear diffusion problems defined on unbounded domains. The study introduces a unified computational strategy that directly applies to IDEs without discretization, particularly focusing on the integer and fractional-order Cattaneo-Rayleigh model of thermoelasticity in one-dimensional zonal regions. Through numerical experiments, we demonstrate that our wavelet collocation approach achieves high efficiency and superior accuracy compared to traditional methods. The results show particularly strong performance in handling systems with differential physical singularities, functional physical singularities, and vector-valued physical singularities. Our approach provides a valuable toolbox for addressing complex modeling challenges in biology, mechanical signal processing, and digital communications where partial integro-differential equations naturally arise.

Disturbance Observer‐Based Neural Adaptive Command Filtered BacksStepping Funnel‐Like Control for the Chaotic PMSM With Asymmetric Prescribed Performance Constraints

ABSTRACTThis paper suggests a neural adaptive command filtered backstepping tracking control strategy for the chaotic permanent magnet synchronous motors with asymmetric prescribed performance constraints. Therefore, enable chaotic permanent magnet synchronous motors (PMSM) to obtain good robustness and better universality in practical industrial environments, and realizes more accurate control effect. The main challenge lies in devising a valid funnel‐like solution within the backstepping frame to handle the asymmetric performance constraints that traditional solutions cannot solve. To achieve this, a novel funnel‐like function is introduced, integrating a performance boundary function independent of initial output error, thereby transforming the system into an unbounded one. Additionally, the “explosion of complexity” with conventional backstepping is mitigated by introducing command filtering and constructing an error compensating system to reduce errors. By combining the theory of Lyapunov function and backstepping technique, the virtual controller and the real controller with adaptive law ensure the stability of the system. The disturbance observer and neural network solve the external disturbance and the uncertain nonlinear problem, respectively. Simulation comparisons confirm the robustness of the proposed control scheme and demonstrate its superiority over existing solutions.

Advancing nonlinear problem solutions: Newton-like iterative techniques for varied convergence orders using homotopy perturbation

Abstract Iterative methods are crucial in numerical analysis for approximating solutions to nonlinear problems encountered in various fields, such as biomathematics, thermodynamics, chemical engineering, and fluid mechanics. This paper introduces innovative iterative methods of convergence orders three and six to eight, developed using the homotopy perturbation technique (HPT). We address the limitations of traditional derivative-based methods by incorporating divided differences, resulting in derivative-free schemes with optimal convergence properties. We present four newly designed algorithms (HPM1, HPM2, HPM3, and HPM4) and provide a comprehensive convergence analysis. Numerical simulations and basins of attraction demonstrate the superior performance of the proposed methods compared to existing methods of the same order. The proposed methods exhibit faster convergence and larger basins of attraction, making them highly effective for solving nonlinear equations. Our new numerical methods rely on approximations and iterative processes, with each small step bringing us closer to the exact solution. Exploring these numerical methods encourages us to apply mathematics to solve challenging problems in physics, engineering, finance, and more.

A probabilistic integration of LSTM and Gaussian Process Regression for uncertainty-aware reservoir water level predictions

ABSTRACT Reservoir-level forecasting while being crucial for optimal operation, is challenged by complex physical processes and changing climate conditions. Machine learning approaches offer deterministic predictions but often neglect system physics and uncertainty. This article presents a probabilistic data-driven approach combining Long Short-Term Memory (LSTM) and Gaussian Process Regression (GPR) to provide both point forecasts and uncertainty estimates. The hybrid model leverages LSTM’s fitting capabilities with GPR’s robust Bayesian frameworks for uncertainty estimation in nonlinear problems, offering accurate predictions without extensive high-fidelity modeling, and avoiding frequent training and parameter optimization. Evaluation with real reservoir data from India shows the model’s superiority over the vanilla LSTM for both univariate and multivariate scenarios. The proposed model achieved a Nash Sutcliffe efficiency of 0.97 to 0.98, a mean biased error of -0.5634 to -1.0314 for 10-day forecasts, and a continuous ranked probability score of 5.80 and 1.87 for the Bhakra and Pong reservoirs, respectively.

Advancing Parameter Estimation in Differential Equations: A Hybrid Approach Integrating Levenberg–Marquardt and Luus–Jaakola Algorithms

This study presents an integrated approach that combines the Levenberg–Marquardt (LM) and Luus–Jaakola (LJ) algorithms to enhance parameter estimation for various applications. The LM algorithm, known for its precision in solving non-linear least squares problems, is effectively paired with the LJ algorithm, a robust stochastic optimization method, to improve accuracy and computational efficiency. This hybrid LM-LJ methodology is demonstrated through several case studies, including the optimization of MESH equations in distillation processes, modeling controlled diffusion in biopolymer films, and analyzing heat and mass transfer during the drying of cylindrical quince slices. By overcoming the convergence issues typical of gradient-based methods and performing global searches without initial parameter bounds, this approach effectively handles complex models and closely aligns simulation results with experimental data. These capabilities highlight the versatility of this approach in engineering and environmental modeling, significantly enhancing parameter estimation in systems governed by differential equations.

Joint Design of Altitude and Channel Statistics Based Energy Beamforming for UAV-Enabled Wireless Energy Transfer

In recent years, UAV-enabled wireless energy transfer (WET) has attracted significant attention for its ability to provide ground devices with efficient and stable power by flexibly navigating three-dimensional (3D) space and utilizing favorable line-of-sight (LoS) channels. At the same time, energy beamforming utilizing multiple antennas, in which energy beams are focused toward devices in desirable directions, has been highlighted as a key technology for substantially enhancing radio frequency (RF)-based WET efficiency. Despite its significant utility, energy beamforming has not been studied in the context of UAV-enabled WET system design. In this paper, we propose the joint design of UAV altitude and channel statistics based energy beamforming to minimize the overall charging time required for all energy-harvesting devices (EHDs) to meet their energy demands while reducing the additional resources and costs associated with channel estimation. Unlike previous works, in which only the LoS dominant channel without small-scale fading was considered, we adopt a more general air-to-ground (A2G) Rician fading channel, where the LoS probability as well as the Rician factor is dependent on the UAV altitude. To tackle this highly nonconvex and nonlinear design problem, we first examine the scenario of a single EHD, drawing insights by deriving an optimal energy beamforming solution in closed form. We then devise efficient methods for jointly designing altitude and energy beamforming in scenarios with multiple EHDs. Our numerical results demonstrate that the proposed joint design considerably reduces the overall charging time while significantly lowering the computational complexity compared to conventional methods.