Research Spotlights

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.