Nonlinear Problem Research Articles

Optimised Congestion Management Using Curative Measures in Combined AC/DC Systems with Flexible AC Transmission Systems

Due to the increasing demand for transport of electrical energy, measures for power flow control, congestion management, and higher utilisation of the existing grid play a decisive role in the transformation of the power system. Hence, enormous efforts must be undertaken using measures of congestion management. Modelling and integration of corresponding measures in optimisation tools to support grid and system operation and therewith reduce the resulting efforts become more important. This is especially true because of the high intermittency and decentralisation of renewable generation leading to increased complexity of the power system, higher loading of assets, and a growing need for control over flexible alternating current transmission systems (FACTS) and high-voltage direct current (HVDC) converters. This work therefore describes the implementation of optimised congestion management in an A Mathematical Programming Language (AMPL)-based nonlinear optimisation problem. AMPL is an effective tool to deal with highly complex problems of optimisation and scheduling. Therefore, the modelling of assets and flexibilities for power flow control in AC/DC systems in combination with an innovative grid operation strategy using predefined curative measures for the optimised use of the existing grid is introduced. The nonlinear mathematical optimisation aims at the optimal cost selection of flexibility measures. The application of the optimisation technique in a combined AC/DC system shows the optimal preventive and curative use of measures in operational congestion management. Simulation results prove that, by using predefined curative measures, the volume of cost-intensive preventive measures can significantly be reduced, especially in association with power flow control.

Small-disturbance input-to-state stability of perturbed gradient flows: Applications to LQR problem

This paper studies the effect of perturbations on the gradient flow of a general nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectories of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. We show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow.

Exploring Physics-Informed Neural Networks for the Generalized Nonlinear Sine-Gordon Equation

The nonlinear sine-Gordon equation is a prevalent feature in numerous scientific and engineering problems. In this paper, we propose a machine learning-based approach, physics-informed neural networks (PINNs), to investigate and explore the solution of the generalized non-linear sine-Gordon equation, encompassing Dirichlet and Neumann boundary conditions. To incorporate physical information for the sine-Gordon equation, a multiobjective loss function has been defined consisting of the residual of governing partial differential equation (PDE), initial conditions, and various boundary conditions. Using multiple densely connected independent artificial neural networks (ANNs), called feedforward deep neural networks designed to handle partial differential equations, PINNs have been trained through automatic differentiation to minimize a loss function that incorporates the given PDE that governs the physical laws of phenomena. To illustrate the effectiveness, validity, and practical implications of our proposed approach, two computational examples from the nonlinear sine-Gordon are presented. We have developed a PINN algorithm and implemented it using Python software. Various experiments were conducted to determine an optimal neural architecture. The network training was employed by using the current state-of-the-art optimization methods in machine learning known as Adam and L-BFGS-B minimization techniques. Additionally, the solutions from the proposed method are compared with the established analytical solutions found in the literature. The findings show that the proposed method is a computational machine learning approach that is accurate and efficient for solving nonlinear sine-Gordon equations with a variety of boundary conditions as well as any complex nonlinear physical problems across multiple disciplines.

Nonlinear Eigenvalue Problems for (p, 2)-Equations and Laplace Equations with Perturbations

We consider two types of nonlinear eigenvalue problems involving Laplace and p-Laplace operators (p>2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p>2)$$\\end{document}. The main result establishes the existence of at least two nontrivial weak solutions in the case of the perturbed equation and the existence of a continuous spectrum in the case of the (p, 2)-equation. In both cases, variational methods play a central role in our arguments.

Planar Reconstruction of Indoor Scenes from Sparse Views and Relative Camera Poses

Planar reconstruction detects planar segments and deduces their 3D planar parameters (normals and offsets) from the input image; this has significant potential in the fields of digital preservation of cultural heritage, architectural design, robot navigation, intelligent transportation, and security monitoring. Existing methods mainly employ multiple-view images with limited overlap for reconstruction but lack the utilization of the relative position and rotation information between the images. To fill this gap, this paper uses two views and their relative camera pose to reconstruct indoor scene planar surfaces. Firstly, we detect plane segments with their 3D planar parameters and appearance embedding features using PlaneRCNN. Then, we transform the plane segments into a global coordinate frame using the relative camera transformation and find matched planes using the assignment algorithm. Finally, matched planes are merged by tackling a nonlinear optimization problem with a trust-region reflective minimizer. An experiment on the Matterport3D dataset demonstrates that the proposed method achieves 40.67% average precision of plane reconstruction, which is an improvement of roughly 3% over Sparse Planes, and it improves the IPAA-80 metric by 10% to 65.7%. This study can provide methodological support for 3D sensing and scene reconstruction in sparse view contexts.

Ensemble variational Bayesian approximation for the inversion and uncertainty quantification of Darcy flows in heterogeneous porous media with random parameters

The inversion and uncertainty quantification of physical parameters are important for many science and engineering problems. For nonlinear parameter inversion problems of flows in hydrocarbon reservoirs, estimating the true posterior probability distribution function for the random parameter is difficult when the dimension of the parameter is relatively high. Inversion methods such as gradient-based or heuristic optimization methods searching for the maximum a posterior are generally point-estimate, while uncertainty quantification using MCMC is computationally expensive. In the current study, a new ensemble variational Bayesian approximation method is developed to estimate the posterior probability density using an ensemble of realizations regarded as a trial sample distribution. The objective is to optimize the trial sample distribution such that the evidence lower bound is maximized indicating that the distance between the trial and true posterior distribution is minimized. A subspace is built using principle component analysis based on finite samples to obtain a diagonal positive-definite covariance matrix, and ensemble-based data assimilation is used to optimize the trial distribution efficiently. Heterogeneous 2D and 3D test cases of transient single- and two-phase Darcy flows are presented for validation.

A Method for State of Charge and State of Health Estimation of LithiumBatteries Based on an Adaptive Weighting Unscented Kalman Filter

This paper considers the estimation of SOC and SOH for lithium batteries using multi-innovation Levenberg–Marquardt and adaptive weighting unscented Kalman filter algorithms. For parameter identification, the second-order derivative of the objective function to optimize the traditional gradient descent algorithm is used. For SOC estimation, an adaptive weighting unscented Kalman filter algorithm is proposed to deal with the nonlinear update problem of the mean and covariance, which can substantially improve the estimation accuracy of the internal state of the lithium battery. Compared with fixed weights in the traditional unscented Kalman filtering algorithm, this algorithm adaptively adjusts the weights according to the state and measured values to improve the state estimation update accuracy. Finally, according to simulations, the errors of this algorithm are all lower than 1.63 %, which confirms the effectiveness of this algorithm.

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

Determining the parameters of the functioning for a nonlinear ballistic system in a real external environment

GeoTaichi: A Taichi-powered high-performance numerical simulator for multiscale geophysical problems

This study introduces GeoTaichi, an open-source high-performance numerical simulator designed for addressing multiscale geophysical problems. By leveraging the power of the Taichi parallel language, GeoTaichi maximizes the utilization of modern computer resources on multicore CPU and GPU architectures. It offers robust and reliable modules for the discrete element method (DEM), material point method (MPM), and coupled material point-discrete element method (MPDEM). These modules enable efficient solving of large-scale problems while being implemented in pure Python. The design philosophy of GeoTaichi focuses on creating a framework that is readable, extensible, and user-friendly. This paper highlights the coupling procedure of MPDEM, the code structures, and the most important features of GeoTaichi. Rigorous benchmark tests have been conducted to verify the validity and robustness of GeoTaichi. Additionally, the performance of GeoTaichi is compared with similar software tools in the field, underscoring a notable improvement in both computational efficiency and memory savings when compared to existing alternatives. Program summaryProgram title: GeoTaichiCPC Library link to program files:https://doi.org/10.17632/858bmcf7j6.1Developer's repository link:https://github.com/Yihao-Shi/GeoTaichiLicensing provisions: GNU General Public License v3.0Programming language: PythonNature of problem: The simulations of large-deformation geophysical flows and their interaction with structures play a crucial role in the field of geophysics. To address the complexities of these nonlinear problems, the discrete element method (DEM), material point method (MPM), and their coupling (MPDEM) have proven to be highly suitable numerical schemes. However, these schemes impose substantial computational demands, necessitating the development of an efficient framework that can harness modern computer resources on multicore CPU and GPU architectures.Solution method: The open-source code GeoTaichi implements the DEM, MPM, and coupled MPDEM, encompassing a range of constitutive models and contact laws for different geologic materials. The clump particle model is also introduced in DEM to solve granular mechanics involving complex-shaped particles. One significant advantage of GeoTaichi is its utilization of the Taichi parallel language, which is designed to be user-friendly and easily extensible for customized applications.

Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems

Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.

GBDT Method Integrating Feature-Enhancement and Active-Learning Strategies-Sea Ice Thickness Inversion in Beaufort Sea.

Sea ice, as an important component of the Earth's ecosystem, has a profound impact on global climate and human activities due to its thickness. Therefore, the inversion of sea ice thickness has important research significance. Due to environmental and equipment-related limitations, the number of samples available for remote sensing inversion is currently insufficient. At high spatial resolutions, remote sensing data contain limited information and noise interference, which seriously affect the accuracy of sea ice thickness inversion. In response to the above issues, we conducted experiments using ice draft data from the Beaufort Sea and designed an improved GBDT method that integrates feature-enhancement and active-learning strategies (IFEAL-GBDT). In this method, the incident angle and time series are used to perform spatiotemporal correction of the data, reducing both temporal and spatial impacts. Meanwhile, based on the original polarization information, effective multi-attribute features are generated to expand the information content and improve the separability of sea ice with different thicknesses. Taking into account the growth cycle and age of sea ice, attributes were added for month and seawater temperature. In addition, we studied an active learning strategy based on the maximum standard deviation to select more informative and representative samples and improve the model's generalization ability. The improved GBDT model was used for training and prediction, offering advantages in dealing with nonlinear, high-dimensional data, and data noise problems, further expanding the effectiveness of feature-enhancement and active-learning strategies. Compared with other methods, the method proposed in this paper achieves the best inversion accuracy, with an average absolute error of 8 cm and a root mean square error of 13.7 cm for IFEAL-GBDT and a correlation coefficient of 0.912. This research proves the effectiveness of our method, which is suitable for the high-precision inversion of sea ice thickness determined using Sentinel-1 data.

Multiplicity of solutions for a nonlinear nonlocal problem with variable exponent

This work deals with a class of value problem involving the $p(x)$-biharmonic elliptic equation \begin{equation*} \begin{gathered} M_1(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}dx)\Delta^2_{p(x)}u-M_2(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\Delta_{p(x)}u=\lambda f(x,u)+\mu g(x,u) \quad\text{in }\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega, \end{gathered} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N,~N\geq1.$ with smooth boundary $\partial \Omega.$ Our technical method is based on a theorem obtained by B. Ricceri.

“Research on music emotion analysis and dance creation based on neural network”

The one-of-a-kind avenues for self-expression that dancing provides as a form of physical art. Movements of the body, both internal and external, and other forms of expression make up the bulk of a dancer's lexicon. Through passive observation, the viewer is able to catch up on the dancer's intended meaning through their body language. It's the one-of-a-kind way that each dancer finds to convey feeling and story through their body. In this piece, we intelligently oversee dance training and take on challenging nonlinear control problems by combining AI technology and the BP neural networks (BPNN) approach.In this study, the training dance teachers receive in dance language, talking music for dancing, and stage art was evaluated using the BPNN technique and the PCA-BPNN strategy. Median accuracy for the BPNN research model is 85.35 percent as age 80 approaches, whereas for the PCA-BPNN research model it is just 65.64 percent. This demonstrates that the BPNN grading model provides more precise results than the PCA-BPNN model. Using BPNN algorithm-based dancing performed within the context of artificial intelligence technology is one way to attain two goals: the spread and pleasure of beauty, and the harmonious combination of athletics and the arts. These two aims are not mutually exclusive.

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov–Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

Set inversion and box contraction on Lie groups using interval analysis

Set inversion and box contraction consist in finding guaranteed solutions to an equation where variables belong to bounded sets. These problems are classic in Euclidean spaces and can be solved using interval analysis. However, they quickly become complex when dealing with non-Euclidean manifolds. This paper investigates set inversion and box contraction problems when variables belong to generic Lie groups (e.g. the set of rotation matrices SO(3)). The concept of Lie group boxes is introduced and discussed. The core idea is to treat Lie group subsets via interval analysis in the Euclidean space of the Lie algebra. The concept of inclusion function on a Lie group is defined as an extension of classic inclusion functions in Euclidean interval analysis. It is shown that set inversion problems on Lie groups can be solved via the Lie algebra using Euclidean interval analysis and that the solution set has a finite volume in the group. On the same principle, the box contraction problem is extended to Lie groups via a Euclidean contractor in the Lie algebra. Numerical simulations demonstrate the high interest of using Lie group boxes instead of classic Euclidean parametrization for nonlinear problems (such as the Euler angles for rotations) both in terms of computational efficiency and accuracy. The proposed approach is of practical interest for state estimation and navigation in robotics since it makes it possible to solve highly nonlinear problems with a limited complexity and a high accuracy, in particular for attitude and pose estimation.

Development of a linear decoupling cable-driven manipulator with independent driving joints,

A cable-driven manipulator demonstrates significant application in cramped environments, such as space maintenance and equipment monitoring, owing to its slender body and excellent flexibility. However, in traditional designs, the mapping between the operational space and the joint space is nonlinear and non-consistent, and the driving cables are also coupled. Consequently, the kinematics and dynamics become highly complex, posing challenges in enhancing efficiency and precision in trajectory planning and control. This paper introduces a novel linear decoupling cable-driven manipulator with independent driving joints. Two sets of nonlinear transmission mechanisms are designed and serially connected to form an equivalent linear transmission mechanism. This arrangement establishes a proportional relationship between the motor angle and joint angle, with the proportionality coefficient representing the equivalent transmission ratio. Moreover, a two-way wire-pulling mechanism is designed to achieve one-to-one driving between the motor and the joint. The nonlinear coupling problem between driving cables is solved by connecting the driving cable to the target joint through a constant-length cable sleeve. Based on the aforementioned design, the linear and consistent mapping between the operational space and the joint space is realized, significantly simplifying the kinematic model. Prototype experiments validate the manipulator's extensive range of motion and high motion accuracy.

Machine learning for determination of activity of water and activity coefficients of electrolytes in binary solutions

Activity of water and electrolytes in aqueous solutions is of utmost importance for multiple industrial applications. However, experimental determination of such values is time-consuming, while calculation of activity coefficients using numerical methods is challenging. By training neural networks models on literature data, one could predict activity of water and electrolytes easily, without requiring any experiment. In this paper, multiple descriptors (or features) are compared to predict activity coefficients of electrolytes and activity of water in electrolyte solutions. A neural network based on the Levenberg-Marquardt algorithm (LM-NN) showed high accuracy to calculate values, despite the small size of the training datasets. Both activity coefficients of electrolytes and activity of water in electrolyte solutions can be predicted accurately even on unseen data, using simple descriptors such as electrolyte concentration, ion sizes and charges. However, some discrepancies were observed due to the lack of representativeness of the training dataset. This could be solved by selecting training data sets that are similar (e.g. same group of the periodic table) to the unknown values, or by including available experimental data for the salt considered. The ability of the LM-NN to solve non-linear least square curve fitting problems makes it a good candidate to fit experimental activity coefficient data, with the advantage of simplicity as compared to e-NRTL or UNIQUAC methods. This method paves the way for accurate and quick determination of thermodynamic data for electrolyte solutions (and beyond) using machine learning, without necessitating large training datasets.

EResNet-SVM: an overfitting-relieved deep learning model for recognition of plant diseases and pests.

The accurate recognition and early warning for plant diseases and pests are a prerequisite of intelligent prevention and control for plant diseases and pests. As a result of the phenotype similarity of the hazarded plant after plant diseases and pests occur, as well as the interference of the external environment, traditional deep learning models often face the overfitting problem in phenotype recognition of plant diseases and pests, which leads to not only the slow convergence speed of the network, but also low recognition accuracy. Motivated by the above problems, the present study proposes a deep learning model EResNet-support vector machine (SVM) to alleviate the overfitting for the recognition and classification of plant diseases and pests. First, the feature extraction capability of the model is improved by increasing feature extraction layers in the convolutional neural network. Second, the order-reduced modules are embedded and a sparsely activated function is introduced to reduce model complexity and alleviate overfitting. Finally, a classifier fused by SVM and fully connected layers are introduced to transforms the original non-linear classification problem into a linear classification problem in high-dimensional space to further alleviate the overfitting and improve the recognition accuracy of plant diseases and pests. The ablation experiments further demonstrate that the fused structure can effectively alleviate the overfitting and improve the recognition accuracy. The experimental recognition results for typical plant diseases and pests show that the proposed EResNet-SVM model has 99.30% test accuracy for eight conditions (seven plant diseases and one normal), which is 5.90% higher than the original ResNet18. Compared with the classic AlexNet, GoogLeNet, Xception, SqueezeNet and DenseNet201 models, the accuracy of the EResNet-SVM model has improved by 5.10%, 7%, 8.10%, 6.20% and 1.90%, respectively. The testing accuracy of the EResNet-SVM model for 6 insect pests is 100%, which is 3.90% higher than that of the original ResNet18 model. This research provides not only useful references for alleviating the overfitting problem in deep learning, but also a theoretical and technical support for the intelligent detection and control of plant diseases and pests. © 2024 Society of Chemical Industry.

State-dependent trust region for successive convex programming for autonomous spacecraft

AbstractSpacecraft trajectory optimization is essential for all the different phases of a space mission, from its launch to end-of-life disposal. Due to the increase in the number of satellites and future space missions beyond our planet, increasing the level of autonomy of spacecraft is a key technical challenge. In this context, traditional trajectory optimization methods, like direct and indirect methods are not suited for autonomous or on-board operations due to the lack of guaranteed convergence or the high demand for computational power. Heuristic control laws represent an alternative in terms of computational power and convergence but they usually result in sub-optimal solutions. Successive convex programming (SCVX) enables to extend the application of convex optimization to non-linear optimal control problems. The definition of a good value of the trust region size plays a key role in the convergence of SCVX algorithms, and there is no systematic procedure to define it. This work presents an improved trust region based on the information given by the nonlinearities of the constraints which is unique for each optimization variable. In addition, differential algebra is adopted to automatize the transcription process required for SCVX algorithms. This new technique is first tested on a simple 2D problem as a benchmark of its performance and then applied to solve complex astrodynamics problems while providing a comparison with indirect, direct, and standard SCVX solutions.

A Multi-Agent RL Algorithm for Dynamic Task Offloading in D2D-MEC Network with Energy Harvesting.

Delay-sensitive task offloading in a device-to-device assisted mobile edge computing (D2D-MEC) system with energy harvesting devices is a critical challenge due to the dynamic load level at edge nodes and the variability in harvested energy. In this paper, we propose a joint dynamic task offloading and CPU frequency control scheme for delay-sensitive tasks in a D2D-MEC system, taking into account the intricacies of multi-slot tasks, characterized by diverse processing speeds and data transmission rates. Our methodology involves meticulous modeling of task arrival and service processes using queuing systems, coupled with the strategic utilization of D2D communication to alleviate edge server load and prevent network congestion effectively. Central to our solution is the formulation of average task delay optimization as a challenging nonlinear integer programming problem, requiring intelligent decision making regarding task offloading for each generated task at active mobile devices and CPU frequency adjustments at discrete time slots. To navigate the intricate landscape of the extensive discrete action space, we design an efficient multi-agent DRL learning algorithm named MAOC, which is based on MAPPO, to minimize the average task delay by dynamically determining task-offloading decisions and CPU frequencies. MAOC operates within a centralized training with decentralized execution (CTDE) framework, empowering individual mobile devices to make decisions autonomously based on their unique system states. Experimental results demonstrate its swift convergence and operational efficiency, and it outperforms other baseline algorithms.