Education

Abstract

SIAM Review, Volume 66, Issue 1, Page 147-147, February 2024.
In this issue the Education section presents two contributions. The first paper, “Resonantly Forced ODEs and Repeated Roots,” is written by Allan R. Willms. The resonant forcing problem is as follows: find $y(\cdot)$ such that $L[y(x)]=u(x)$, where $L[u(x)]=0$ and $L=a_0(x) + \sum_{j=1}^n a_j(x) \frac{d^j}{dx^j}$. The repeated roots problem consists in finding $mn$ linearly independent solutions to $L^m[y(x)]=0$ under the assumption that $n$ linearly independent solutions to $L[y(x)]= 0$ are known. A recent article by B. Gouveia and H. A. Stone, “Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods” [SIAM Rev., 64 (2022), pp. 485--499], discusses a method for finding solutions to these two problems. This new contribution observes that by applying the same mathematical justifications, one may get similar results in a simpler way. The starting point consists in defining operators $L_\lambda := \hat L -g(\lambda)$ with $L_{\lambda_0}=L$ for some $\lambda_0$ and of a parameter-dependent family of solutions to the homogeneous equations $L_\lambda[y(x;\lambda)]=0$. Under appropriate assumptions on $g$, differentiating this equality allows one to get solutions to problems of interest. This approach is illustrated on nine examples, seven of which are the same as in the publication of B. Gouveia and H. A. Stone, where for each example $g$ and $\hat L$ are appropriately chosen. This approach may be included in a course of ordinary differential equations (ODEs) as a methodology for finding solutions to these two particular classes of ODEs. It can also be used by undergraduate students for individual training as an alternative to variation of parameters. The second paper, “NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators,” is presented by Zongren Zou, Xuhui Meng, Apostolos Psaros, and George E. Karniadakis. In machine learning uncertainty quantification (UQ) is a hot research topic, driven by various questions arising in computer vision and natural language processing, and by risk-sensitive applications. Numerous machine learning models, such as, for instance, physics-informed neural networks and deep operator networks, help in solving partial differential equations and learning operator mappings, respectively. However, some data may be noisy and/or sampled at random locations. This paper presents an open-source Python library (https://github.com/Crunch-UQ4MI) for employing a reliable toolbox of UQ methods for scientific machine learning. It is designed for both educational and research purposes and is illustrated on four examples, involving dynamical systems and high-dimensional parametric and time-dependent PDEs. NeuralUQ is planned to be constantly updated.