主成分分析在异向介质带隙优化问题中的应用
Principal Component Analysis Applied to Gradient Fields in Band Gap Optimization Problems for Metamaterials
摘要：
一种很有前途的声学超材料的频谱设计技术是基于制定适当的约束非线性优化问题。不幸的是，由于底层物理模型的复杂性，将经典的基于梯度的迭代优化算法直接应用于此类问题的数值解通常要求很高。然而，监督机器学习技术可以减少这样的计算工作量，例如，通过用更容易计算的近似值替换这种优化问题的原始目标函数。在此框架下，本文介绍了相关的无监督机器学习技术，即主成分分析，近似的声学超材料的带隙优化问题的目标函数的梯度的应用，其目的是使连续应用的梯度为基础的迭代优化算法更快。数值结果表明了该方法的有效性。
英文摘要：
A promising technique for the spectral design of acoustic metamaterials is based on the formulation of suitable constrained nonlinear optimization problems. Unfortunately, the straightforward application of classical gradient-based iterative optimization algorithms to the numerical solution of such problems is typically highly demanding, due to the complexity of the underlying physical models. Nevertheless, supervised machine learning techniques can reduce such a computational effort, e.g., by replacing the original objective functions of such optimization problems with more-easily computable approximations. In this framework, the present article describes the application of a related unsupervised machine learning technique, namely, principal component analysis, to approximate the gradient of the objective function of a band gap optimization problem for an acoustic metamaterial, with the aim of making the successive application of a gradient-based iterative optimization algorithm faster. Numerical results show the effectiveness of the proposed method.
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