SIAM Review, Volume 65, Issue 3, Page 733-733, August 2023.
The three articles in this issue's Research Spotlight section highlight the breadth of problems and approaches that have differential equations as a central component. In the first article, “Neural ODE Control for Classification, Approximation, and Transport,” authors Domènec Ruiz-Balet and Enrique Zuazua seek to expand understanding of some of the main properties of deep neural networks. To this end, the authors develop a dynamical control theoretical analysis of neural ordinary differential equations, a discretization of which is commonly known as a ResNet in machine learning. In this approach, time-dependent parameters are defined by piecewise-constant controls used to achieve targets associated with classification and regression tasks. A key aspect of the article's treatment is the reliance on an activation function characterization that only deforms one half space, leaving the other half space invariant; the rectified linear unit (ReLU) is a popular example of such an activation function. The authors derive constructive universal approximation results that can be used to understand how the complexity of the control depends on the target function's properties. Among other applications, these results are used to control a neural transport equation with the Wasserstein distance, common in optimal transport problems, measuring the approximation quality. Ruiz-Balet and Zuazua conclude with a number of open problems. Differential equation--based compartment models date back at least a century, when they were used to model the dynamics of malaria in a mixed population of humans and mosquitoes. Since then, compartment models have been used in areas far beyond epidemiology, typically with the simplifying assumption that each compartment is internally well mixed. As a consequence, all members in a compartment are treated the same, independent of how long they have resided in the compartment. In “Compartment Models with Memory,” authors Timothy Ginn and Lynn Schreyer expand the fields for which compartment models can provide insight by incorporating age in compartment in the underlying rate coefficients. This has the benefit of being able to account for a wide array of residence time distributions and comes at a cost of having to numerically solve a system of Volterra integral equations instead of a system of ordinary differential equations. The authors demonstrate and validate this approach on a number of examples and conclude by incorporating a delay in contagiousness of infected persons in a nonlinear SARS-CoV-2 transmission model. The authors also summarize several open questions based on this approach of allowing model parameters to be written as functions of age in compartment. “Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?” This is the question posed by (and the title of) the final Research Spotlights article in this issue. Authors Jeffrey Galkowski and Euan A. Spence consider Helmholtz problems that arise when a plane wave is scattered by a smooth obstacle. Of particular interest are very high frequency waves, which necessarily require a large number of discretized degrees of freedom to accurately resolve solutions. A so-called pollution effect arises if the number of degrees of freedom required grows faster than a particular polynomial of the wave number as the wave number tends to infinity. The authors examine finite element and boundary element methods with a meshwidth that varies like the inverse of the asymptotically increasing wave number. While such finite element methods suffer from the pollution effect, Galkowski and Spence establish that the corresponding boundary element methods do not.

中文翻译：

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研究热点

《SIAM 评论》，第 65 卷，第 3 期，第 733-733 页，2023 年 8 月。
本期研究聚焦部分的三篇文章强调了以微分方程为核心组成部分的问题和方法的广度。在第一篇文章“分类、近似和传输的神经常微分方程控制”中，作者 Domènec Ruiz-Balet 和 Enrique Zuazua 试图扩展对深度神经网络的一些主要属性的理解。为此，作者开发了神经常微分方程的动态控制理论分析，其离散化在机器学习中通常称为 ResNet。在这种方法中，时间相关参数由分段常数控制定义，用于实现与分类和回归任务相关的目标。本文处理的一个关键方面是依赖激活函数表征，该函数仅使一半空间变形，而使另一半空间保持不变；修正线性单元 (ReLU) 是此类激活函数的一个常见示例。作者得出了建设性的通用逼近结果，可用于理解控制的复杂性如何取决于目标函数的属性。在其他应用中，这些结果用于控制具有 Wasserstein 距离的神经传输方程，这在最优传输问题中很常见，用于测量近似质量。Ruiz-Balet 和 Zuazua 最后提出了一些悬而未决的问题。基于微分方程的隔室模型可以追溯到至少一个世纪之前，当它们被用来模拟人类和蚊子混合种群中的疟疾动态时。从那时起，隔室模型已被用于远远超出流行病学范围的领域，通常采用简化的假设，即每个隔室内部充分混合。因此，隔间中的所有成员都会受到相同的对待，无论他们在隔间中居住了多长时间。在“具有记忆的房室模型”中，作者 Timothy Ginn 和 Lynn Schreyer 通过将房室年龄纳入基础速率系数，扩展了房室模型可以提供洞察力的领域。这样做的好处是能够考虑各种停留时间分布，但代价是必须对 Volterra 积分方程组而不是常微分方程组进行数值求解。作者通过多个例子证明并验证了这种方法，并通过将感染者的传染性延迟纳入非线性 SARS-CoV-2 传播模型得出结论。作者还总结了基于这种方法的几个悬而未决的问题，该方法允许将模型参数写为隔室中年龄的函数。“亥姆霍兹边界元法会受到污染影响吗？” 这是本期最后一篇研究热点文章提出的问题（也是其标题）。作者 Jeffrey Galkowski 和 Euan A. Spence 考虑了当平面波被光滑障碍物散射时出现的亥姆霍兹问题。特别令人感兴趣的是非常高频的波，它必然需要大量的离散自由度来准确地求解解。当波数趋于无穷大时，如果所需的自由度数增长快于波数的特定多项式，就会出现所谓的污染效应。作者研究了有限元和边界元方法，其网格宽度的变化类似于渐近增加波数的倒数。虽然此类有限元方法会受到污染效应的影响，但 Galkowski 和 Spence 证明相应的边界元方法不会受到污染效应的影响。