Rapid and Robust High Order Spectral Solvers for Learning Photonic Structures

在过去的二十年里，工程师们建造了具有惊人的探测、传感和成像能力的光学设备。虽然直觉一直是推动可能性极限的宝贵指南，但日益复杂的设计和种类繁多的可用组件使这种方法变得更加困难。正因为如此，数值模拟已成为不可估量的工具。随着这种方法变得越来越普遍，对快速、健壮和高精度算法的搜索变得非常迫切。PI将通过探索数值逼近的新途径、实施控制方程的新途径以及利用机器学习(尤其是深度学习)的强大方法来发现设备设计的最佳参数值，从而增强他的模拟工具类别。PI还将对这些新的数值格式进行严格的分析，以评估和验证它们在现实世界中的表现。该项目每年将为一名研究生提供支持。在过去的二十年里，工程师们建造了具有惊人的探测、传感和成像能力的光学设备。虽然直觉和线性化模型在推动可能性的极限方面起到了无价的指导作用，但设计的日益复杂和可用组件的广泛种类使这种方法变得更加困难。正因为如此，数值模拟已成为不可估量的工具。随着这种方法变得越来越普遍，对快速、健壮和高精度算法的搜索变得非常迫切。PI将增强他的高阶谱求解器的类别，目标如下：(1)将二维材料加入到他的快速且高精度的三维“场展开”向量Maxwell方程求解器中；(2)将他的几何/频率联合摄动的“曲面的高阶摄动/渐近波形估计”算法扩展到三维向量Maxwell方程；(3)将他的结构摄动的“高阶包络摄动”算法推广到这种三维情况；(4)对后两种情况提供严格的分析证明；以及(5)利用深度学习的强大新工具来发现设备设计的最佳参数值。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Over the past two decades engineers have built optical devices with stunning capabilities of detection, sensing, and imaging. While intuition has been an invaluable guide in pushing the envelope of what is possible, the increasing complexity of design and the wide array of available components is rendering this approach ever more difficult. For this reason, numerical simulation has become an invaluable tool. As this approach becomes ever more pervasive, the search for rapid, robust, and highly accurate algorithms has become quite acute. The PI will enhance his class of simulation tools by exploring new avenues of numerical approximation, novel paths to enforcing the governing equations, and appealing to the powerful methods of Machine Learning (in particular, Deep Learning) to discover optimal parameter values for device design. The PI will also develop a rigorous analysis of these new numerical schemes in order to evaluate and validate their real-world performance. This project will provide support for one graduate student each year of the three year award.Over the past two decades engineers have built optical devices with stunning capabilities of detection, sensing, and imaging. While intuition and linearized models have been invaluable guides in pushing the envelope of what is possible, the increasing complexity of design and the wide array of available components is rendering this approach ever more difficult. For this reason, numerical simulation has become an invaluable tool. As this approach becomes ever more pervasive, the search for rapid, robust, and highly accurate algorithms has become quite acute. The PI will enhance his class of High-Order Spectral solvers with the following objectives: (1) Incorporating two-dimensional materials into his rapid and highly accurate three-dimensional "Field Expansions" vector Maxwell equation solver; (2) Expand his "High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation" algorithm for joint geometry/frequency perturbation to the three-dimensional vector Maxwell equations; (3) Extend his "High-Order Perturbation of Envelopes" algorithm for structural perturbations to this three--dimensional case; (4) Provide rigorous analytical justification for each of the latter two; and (5) Bring to bear the powerful new tools of Deep Learning to discover optimal parameter values for device design.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.