加权Helmholtz方程特征值反问题的多人工神经网络

IF 3.4 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2024-12-19 DOI:10.1016/j.cnsns.2024.108544

Zhengfang Zhang, Shizhong Zou, Xihao Zhou, Xinping Shao, Mingyan He, Weifeng Chen

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引用次数: 0

摘要

研究了加权亥姆霍兹方程的特征值反问题。密度函数由有限光谱数据的观测恢复。将其重新表述为优化问题，并定义了多目标损失函数。提出了一种多人工神经网络（multi-ANN）算法。证明了该优化问题解的存在性和稳定性，以及多神经网络解对原优化问题解的收敛性。给出了加权亥姆霍兹方程一维和二维特征值反问题的数值结果。通过与传统有限元方法的比较，验证了该方法的鲁棒性和有效性。

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Multi-artificial neural network for inverse eigenvalue problem with the weighted Helmholtz equation

The inverse eigenvalue problem of weighted Helmholtz equations is investigated. The density function is recovered from the observation of the limited spectral data. It is reformulated as an optimization problem and a multi-objective loss function is defined accordingly. A multi-artificial neural network (multi-ANN) algorithm is proposed. The existence and stability of the solution of the optimization problem, and the convergence of the multi-ANN solution towards that of the original optimization problem are proved. The numerical results of one-dimensional and two-dimensional inverse eigenvalue problems of the weighted Helmholtz equation are given. Compared with the traditional finite element method, the robustness and effectiveness of the proposed multi-ANN method are illustrated.

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.