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A Review of Neural Network Solvers for Second-order Boundary Value Problems

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arXiv 2024年

作者： Sau, Ramesh Chandra Yin, Luowei Department of Mathematics Indian Institute of Technology Bombay India Department of Mathematics The Chinese University of Hong Kong New Territories Shatin Hong Kong

Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work reviews several deep learning solvers proposed a few years ago, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons amongst them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is also presented. Finally, we discuss the current limitations and bottlenecks of these methods. Copyright © 2024, The Authors. All rights reserved.

关键词： nonlinear programming

Vision and Contact based Optimal Control for Autonomous Trocar Docking

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arXiv 2024年

作者： Mower, Christopher E. Huber, Martin Tian, Huanyu Davoodi, Ayoob Poorten, Emmanuel Vander Vercauteren, Tom Bergeles, Christos School of Biomedical Engineering & Imaging Sciences King’s College London United Kingdom Department of Mechanical Engineering KU Leuven Leuven Belgium Beijing Institute of Technology Beijing China

Future operating theatres will be equipped with robots to perform varioussurgical tasks including, for example, endoscope control. Human-in-the-loopsupervisory control architectures where the surgeon selects from severalautonomous sequences is already being successfully applied in preclinicaltests. Inserting an endoscope into a trocar or introducer is a key step forevery keyhole surgical procedure - hereafter we will only refer to this deviceas a "trocar". Our goal is to develop a controller for autonomous trocardocking. Autonomous trocar docking is a version of the peg-in-hole problem. Extensivework in the robotics literature addresses this problem. The peg-in-hole problemhas been widely studied in the context of assembly where, typically, the holeis considered static and rigid to interaction. In our case, however, the trocaris not fixed and responds to interaction. We consider a variety of surgicalprocedures where surgeons will utilize contact between the endoscope and trocarin order to complete the insertion successfully. To the best of our knowledge,we have not found literature that explores this particular generalization ofthe problem directly. Our primary contribution in this work is an optimal control formulation forautomated trocar docking. We use a nonlinear optimization program to model thetask, minimizing a cost function subject to constraints to find optimal jointconfigurations. The controller incorporates a geometric model for insertion anda force-feedback (FF) term to ensure patient safety by preventing excessiveinteraction forces with the trocar. Experiments, demonstrated on a realhardware lab setup, validate the approach. Our method successfully achievestrocar insertion on our real robot lab setup, and simulation trials demonstrateits ability to reduce interaction forces. © 2024, CC BY.

关键词： nonlinear programming

Scalable Multi-Period AC Optimal Power Flow Utilizing GPUs with High Memory Capacities

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arXiv 2024年

作者： Shin, Sungho Rao, Vishwas Schanen, Michel Adrian Maldonado, D. Anitescu, Mihai Mathematics and Computer Science Division Argonne National Laboratory LemontIL United States

This paper demonstrates the scalability of open-source GPU-accelerated nonlinear programming (NLP) frameworks—*** and ***—for solving multiperiod alternating current (AC) optimal power flow (OPF) problems on GPUs with high memory capacities (e.g., NVIDIA GH200 with 480 GB of unified memory). There has been a growing interest in solving multi-period AC OPF problems, as the increasingly fluctuating electricity market requires operation planning over multiple periods. These problems, formerly deemed intractable, are now becoming technologically feasible to solve thanks to the advent of high-memory GPU hardware and accelerated NLP tools. This study evaluates the capability of these tools to tackle previously unsolvable multi-period AC OPF instances. Our numerical experiments, run on an NVIDIA GH200, demonstrate that we can solve a multi-period OPF instance with more than 10 million variables up to 10−4 precision in less than 10 minutes. These results demonstrate the efficacy of the GPU-accelerated NLP frameworks for the solution of extreme-scale multi-period OPF. We provide ***, an open-source modeling tool for multi-period AC OPF models for GPUs. Copyright © 2024, The Authors. All rights reserved.

关键词： nonlinear programming

Research on Optimal Allocation of Command Decision-making Authority Based on Operations Research Theory

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Research on Optimal Allocation of Command Decision-making Au...

第43届中国控制会议

作者： Zexuan Jin Jia Zhang Beijing Institute of Technology

ISBN: (数字)9789887581581

ISBN: (纸本)9798350366907

In order to improve the efficiency of command structure, it is necessary to study the optimization allocation of command and decision-making authority. Graph theory methods in operations research are used to describe the command relationship between command nodes and resource nodes and establish the optimization indexes such as decision transmission time, destructive resistance capability, and constraints such as resource constraints. Non-dominated sorting genetic algorithm Ⅱ(NSGA-Ⅱ) is applied to solve the multi-indicator nonlinear planning problem and obtain the optimal solution of the command decision authority allocation problem. Simulation experiments are carried out in three kinds of scenarios. The solution obtained by the algorithm proposed in this paper is compared with that based on the proximity principle and that obtained using the MOEA/D algorithm. The results show that the solution obtained by the algorithm proposed in this paper is more effective and can achieve better command effectiveness.

关键词： command decision authority optimal assignment graph theory nonlinear programming NSGA-II algorithm

Designing gradient coils with the shape derivative and the closed B-spline curves

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arXiv 2024年

作者： Takahashi, Toru Department of Mechanical Systems Engineering Nagoya University Furo-cho Chikusa-ku Nagoya city Aichi464-8603 Japan

This study proposes a versatile and efficient optimisation method for discrete coils that induce a magnetic field by their steady currents. The prime target is gradient coils for MRI (Magnetic Resonance Imaging). The derivative (gradient) of the z-component the magnetic field, which is calculated by the Biot-Savart's law, with respect to the z-coordinate in the Cartesian xyz coordinate system is considered as the objective function. Then, the derivative of the objective function with respect to a change of coils in shape is formulated according to the concept of shape optimisation. The resulting shape derivative (as well as the Biot-Savart's law) is smoothly discretised with the closed B-spline curves. In this case, the control points (CPs) of the curves are naturally selected as the design variables. As a consequence, the shape derivative is discretised to the sensitivities of the objective function with respect to the CPs. Those sensitivities are available to solve the present shape-optimisation problem with a certain gradientbased nonlinear-programming solver. The numerical examples exhibit the mathematical reliability, computational efficiency, and engineering applicability of the proposedmethodology based on the shape derivative/sensitivities and the closed B-spline curves. © 2024, CC BY-NC-ND.

关键词： nonlinear programming

OPTIBENCH MEETS RESOCRATIC: MEASURE AND IMPROVE LLMS FOR OPTIMIZATION MODELING

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arXiv 2024年

作者： Yang, Zhicheng Wang, Yiwei Huang, Yinya Guo, Zhijiang Shi, Wei Han, Xiongwei Feng, Liang Song, Linqi Liang, Xiaodan Tang, Jing The Hong Kong University of Science and Technology Guangzhou China The Hong Kong University of Science and Technology Hong Kong University of California Merced United States City University of Hong Kong Hong Kong Huawei Noah’s Ark Lab Hong Kong Sun Yat-sen University China MBZUAI United Arab Emirates Chongqing University China

Large language models (LLMs) have exhibited their problem-solving abilities in mathematical reasoning. Solving realistic optimization (OPT) problems in application scenarios requires advanced and applied mathematics ability. However, current OPT benchmarks that merely solve linear programming are far from complex realistic situations. In this work, we propose OPTIBENCH, a benchmark for end-to-end optimization problem-solving with human-readable inputs and outputs. OPTIBENCH contains rich optimization problems, including linear and nonlinear programming with or without tabular data, which can comprehensively evaluate LLMs’ solving ability. In our benchmark, LLMs are required to call a code solver to provide precise numerical answers. Furthermore, to alleviate the data scarcity for optimization problems, and to bridge the gap between open-source LLMs on a small scale (e.g., Llama-3-8b) and closed-source LLMs (e.g., GPT-4), we further propose a data synthesis method namely ReSocratic. Unlike general data synthesis methods that proceed from questions to answers, ReSocratic first incrementally synthesizes formatted optimization demonstrations with mathematical formulations step by step and then back-translates the generated demonstrations into questions. Based on this, we synthesize the RESOCRATIC-29K dataset. We further conduct supervised fine-tuning with RESOCRATIC-29K on multiple open-source models. Experimental results show that RESOCRATIC-29K significantly improves the performance of open-source models. The code and data can be found at https://***/yangzhch6/ReSocratic. © 2024, CC BY-NC-SA.

关键词： nonlinear programming

A novel approach for solving neutrosophic fractional transportation problem with non- linear discounting cost

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Neutrosophic Sets and Systems 2023年 61卷 149-164页

作者： El- Wahed Khalifa, Hamiden Abd Broumi, Said Edalatpanah, S.A. Alburaikan, Alhanouf Department of Mathematics College of Science and Arts Qassim University Al- Badaya51951 Saudi Arabia Department of Operations and Management Research Faculty of Graduate Studies for Statistical Research Cairo University Giza12613 Egypt Laboratory of Information Processing Faculty of Science Ben M’Sik University Hassan II Casablanca Morocco Department of Applied Mathematics Ayandegan Institute of Higher Education Tonekabon Iran

Fractional transportation problem that includes source and destination may have fractional objective functions in real- world applications to maximize the profitability ratio like profit/cost or profit/time. We refere to such transportation problems as fractional transportation *** paper considers the interval- valued neutrosophic numbers and its aritemematic operations. This paper deals with fractional transportation problem having discounting cost in neutrosophic environment, where the supply, demand and transportation costs are uncertain. The problem is considered by introducing all the parameters as neutrosophic numbers. Using the benefits of the score function definition, the problem is transformed into the corresponding deterministic form which can be illustrated by any method. and hence by applying of least cost method with the help of Kuhn- Tucker’ optimality conditions, the optimal solution is resulted. Our strategy is to assess the issue and can rank different sort of neutrosophic numbers. To claify the proposed technique, a numerical example is given to show the adequacy of the new model. © (2023), (Neutrosophic Sets and Systems). All rights reserved.

关键词： nonlinear programming

Multi-Agent Disk Inspection

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arXiv 2024年

作者： Conley, James Georgiou, Konstantinos Department of Mathematics Toronto Metropolitan University TorontoON Canada

We consider n unit-speed mobile agents initially positioned at the center of a unit disk, tasked with inspecting all points on the disk’s perimeter. A perimeter point is considered covered if an agent positioned outside the disk’s interior has unobstructed visibility of it, treating the disk itself as an obstacle. For n ＝ 1, this problem is referred to as the shoreline problem with a known distance and was originally proposed as a more tractable variant of Bellman’s famous lost-in-the-forest problem [14]. Isbell [46] derived an optimal trajectory that minimizes the worst-case inspection time for that problem. For n ≥ 2 agents, worst-case optimal trajectories were shown in [1,30]. Our contributions are threefold. First, and as a warm-up, we extend Isbell’s findings by deriving worst-case optimal trajectories addressing the partial inspection of a section of the disk, hence deriving an alternative proof of optimality for inspecting the disk with n ≥ 2 agents. Second, we analyze the average-case inspection time, assuming a uniform distribution of perimeter points (equivalent to randomized inspection algorithms). Using spatial discretization and nonlinear programming (NLP), we propose feasible solutions to the continuous problem and evaluate their effectiveness compared to NLP solutions. Third, we establish Pareto-optimal bounds for the multi-objective problem of jointly minimizing the worst-case and average-case inspection times. Copyright © 2024, The Authors. All rights reserved.

关键词： nonlinear programming

Insights on the Cenozoic Tectonic Evolution and Heat Sources in the Eastern Tibet Plateau Revealed by Magnetotelluric Imaging

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SSRN

SSRN 2024年

作者： Ji, Leizhe Jin, Sheng Ye, Gaofeng Sun, Xiaolei Wei, Wenbo Qi, Jialin School of Geophysics and Information Technology China University of Geosciences Beijing100083 China Key Laboratory of Intraplate Volcanoes and Earthquakes China University of Geosciences Beijing Ministry of Education Beijing100083 China Land Resources Exploration Center of Hebei Bureau of Geology and Mineral Exploration and Development Shijiazhuang050081 China

The continued collision of the Indian and Asian continents has formed the Himalayas and Tibetan Plateau through various deformation processes. The uplift and deformation are also accompanied by extensive geothermal activity. However, the geodynamic mechanism is still unclear and debated on the rigid-block extrusion model and the mid-lower crustal flow model. To address this challenge, we obtained a 3D resistivity model in the eastern Tibetan Plateau using a nonlinear conjugate gradient method based on the Magnetotelluric sounding data. Two low-resistivity anomalies (C1 and C2) are identified in the Ganzi-Litang suture and near the Xianshuihe fault, isolated by the crustal high-resistivity body and an upper mantle medium-resistivity body. We propose that the low-resistivity C1 is caused by the re-activation of the Ganzi-Litang suture, and the low-resistivity C2 can be interpreted as partial melting due to the upwelling of mantle material. Our resistivity model, combined with other geological and geophysical evidences, supports the stepwise uplift propagation style in the eastern Tibetan Plateau, suggesting the tectonic evolution is dominated by the pre-existing lithospheric structural weak zone in the early stage, the late-stage crust deformation is controlled by the mid-lower crustal flow and upwelling of lithospheric mantle material. Furthermore, the Eastern Tibetan Plateau Geothermal Belt (ETGB) can be divided into two types of geothermal belts, each with distinct deep heat source mechanisms, and the thermal channels of the ETGB are unified and controlled by the northwest deep faults in the eastern Tibetan Plateau. © 2024, The Authors. All rights reserved.

关键词： nonlinear programming