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.

中文翻译：

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教育

SIAM Review，第 66 卷，第 1 期，第 147-147 页，2024 年 2 月。
在本期中，教育部分提出了两项​​贡献。第一篇论文“共振强制常微分方程和重复根”由艾伦·R·威尔姆斯 (Allan R. Willms) 撰写。共振强迫问题如下：找到 $y(\cdot)$ 使得 $L[y(x)]=u(x)$，其中 $L[u(x)]=0$ 且 $L=a_0 (x) + \sum_{j=1}^n a_j(x) \frac{d^j}{dx^j}$。重根问题在于找到 $L^m[y(x)]=0$ 的 $mn$ 个线性独立解，假设 $L[y(x)]= 0$ 的 $n$ 个线性独立解为已知。 B. Gouveia 和 HA Stone 最近发表的文章“使用扰动方法生成普通微分方程的共振和重复根解”[SIAM Rev., 64 (2022), pp. 485--499] 讨论了一种寻找解的方法针对这两个问题。这项新贡献指出，通过应用相同的数学论证，人们可以以更简单的方式得到类似的结果。起点在于为某些 $\lambda_0$ 定义运算符 $L_\lambda := \hat L -g(\lambda)$ 和 $L_{\lambda_0}=L$ 以及依赖于参数的一系列解决方案齐次方程$L_\lambda[y(x;\lambda)]=0$。在对 $g$ 进行适当假设的情况下，对这个等式进行微分可以得到感兴趣问题的解决方案。这种方法通过九个例子进行说明，其中七个与 B. Gouveia 和 HA Stone 的出版物中的相同，其中每个例子 $g$ 和 $\hat L$ 被适当选择。这种方法可以包含在常微分方程 (ODE) 课程中，作为寻找这两类特定 ODE 解的方法。本科生也可以使用它进行个人训练，作为参数变化的替代方案。第二篇论文“NeuralUQ：神经微分方程和算子中不确定性量化的综合库”由邹宗仁、孟旭辉、Apostolos Psaros 和 George E. Karniadakis 发表。在机器学习中，不确定性量化（UQ）是一个热门研究课题，受到计算机视觉和自然语言处理中出现的各种问题以及风险敏感应用的推动。许多机器学习模型，例如基于物理的神经网络和深度算子网络，分别有助于求解偏微分方程和学习算子映射。然而，一些数据可能有噪声和/或在随机位置采样。本文提出了一个开源 Python 库 (https://github.com/Crunch-UQ4MI)，用于采用可靠的 UQ 方法工具箱进行科学机器学习。它专为教育和研究目的而设计，并通过四个示例进行说明，涉及动力系统以及高维参数和瞬态偏微分方程。 NeuralUQ 计划不断更新。