SIAM Review, Volume 63, Issue 1, Page 165-166, January 2021.
This issue of SIAM Review presents three papers in the Education section. The first paper, “Improving the Accuracy of the Trapezoidal Rule,” is written by Bengt Fornberg. The trapezoidal rule is a basic technique for numerical calculation of definite one-dimensional integrals. The author provides a historical perspective of the developments around this rule. He starts with a reference tracing it back to Babylonian astronomers before 50 B.C., discusses the work of James Gregory (1638--1675) connecting to discoveries of Leonhard Euler (1707--1783) and Colin Maclaurin (1698--1746), and concludes with recent work published in the period 2012--2018. The author starts by introducing the basic ideas and observations that motivate the improvements of the trapezoidal rule. The leading numerical errors in that approximation method come from the integration bounds. Hence, one should adjust the quadrature weights to increase the accuracy in this type of procedure. Pioneering work in this direction is contained in the letters of James Gregory. His work is explained in section 2, where it is shown that Gregory's weights lead to Euler--Maclaurin coefficients while those coefficients became available many years later. This is a remarkable achievement because Gregory's work also precedes the first publications on calculus by Leibniz and Newton, as well as Brook Taylor's publication on what we call Taylor expansions. The main portion of the paper surveys recent developments in which Gregory's method has been revisited. A finite difference approach resulting from radial basis functions was proposed in 2018 by J. Reeger and B. Fornberg. While that work deals with more general numerical quadratures that use nodes over bounded curved surfaces in a three-dimensional space, a special case within the proposed approach produced Gregory-like methods with high orders of accuracy. The author goes further to explain the way of calculating the weights and provides some test results. An appendix contains MATLAB code for calculating quadrature weights. The presentation is very intuitive and accessible to a wide audience of students. The second paper is “Fluid-Structure Interaction for the Classroom: Interpolation, Hearts, and Swimming!” It is presented by Nicholas A. Battista. Here, interpolation of a function by splines is discussed in the context of mathematical modeling of biological fluid dynamics. The interpolation provides the description of the motion of objects in a fluid. Three examples of increasing complexity illustrate this approach. The first example discusses three circles and their movement within a fluid. In this context, various relevant questions are introduced. For example, do we have enough degrees of freedom to enforce continuous derivatives? How do linear interpolating functions compare to higher-order polynomials? The second example uses the same prescribed motion as in the first one but introduces different polynomials to interpolate between successive states. In this way, the motion of a beating heart can be simulated. The final example provides an idealized model of swimming. It is assumed that the swimmer's body bends so that it switches between two states of distinct curvature. The swimmer's body is approximated by connecting a line segment and a polynomial section in the interpolation procedure. Locomotion emerges due to the swimmer's interactions with the surrounding fluid. The article is addressed to teaching faculty. The author suggests how and when to bring up various relevant questions in this module. He provides guidance for the discussions and a code for the examples. References to recent scientific studies that have used this approach to successfully prescribe motion in many fields are included. The article also contains a link to a website with software that allows one to build and test spline interpolation for the purpose of modeling motion between one or more feature states. Two implementations, in MATLAB and in Python, are included, as well as links to tutorials helping with the software. The third contribution to the section is the paper “DeepXDE: A Deep Learning Library for Solving Differential Equations,” co-authored by Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. The paper elucidates recent developments in the numerical solutions of partial differential equations (PDEs) based on deep learning. The paper starts with an overview of physics-informed neural networks (PINNs) and the use of automatic differentiation. Most of the presentation is based on the simplest type of neural networks, called feed-forward (FNN), as they are considered sufficient for most PDE problems. However, the type of network is not crucial for the approach. The authors explain briefly how the algorithmic calculation of derivatives of the network outputs with respect to the network inputs is performed. Then the PINN algorithm is presented. Consider a PDE is given, which is parameterized by $\lambda$ for the solution $u(x)$ with $x$ on a given domain with suitable boundary conditions. A neural network $\hat{u}(x;\theta )$ is constructed as a surrogate of $u(x)$. The net takes the input $x$ and outputs a vector with the same dimension as $u$ depending on the net's parameters gathered in the vector $\theta$; the weight matrices and the bias vectors of all network layers. The derivatives of $\hat{u}(\cdot;\theta )$ are computed algorithmically as discussed before by applying the chain rule for differentiating compositions of functions. Two sets of randomly selected points in the domain of the PDE and its boundary are taken into account in order to reflect the physics imposed by the PDE and the boundary conditions. The loss function is the weighted $L^2$ norm of the residuals stemming from the PDE equation and its boundary condition on the selected points. The neural network is trained to find the best parameter $\theta^\ast$ by minimizing the loss function. The authors also propose a new adaptive refinement method to improve the training efficiency of PINNs. Further discussion in the paper includes the approximation properties and error analysis for the PINNs. The authors provide an implementation of PINNs in the form of a Python library, called DeepXDE, and discuss its features and customization abilities in section 3. They advocate the PINN algorithm as simple and widely applicable: it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, stochastic PDEs, as well as inverse problems. A representative example of each type is provided in section 4 of the paper. The concluding remarks contain a discussion on the advantages and limitation of the approach and on potential future extensions.

中文翻译：

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教育

SIAM 评论，第 63 卷，第 1 期，第 165-166 页，2021 年 1 月。
本期 SIAM 评论在教育部分介绍了三篇论文。第一篇论文“提高梯形规则的准确性”由 Bengt Fornberg 撰写。梯形法则是一维定积分数值计算的基本技术。作者提供了围绕这一规则发展的历史视角。他从追溯到公元前 50 年之前巴比伦天文学家的参考文献开始，讨论了詹姆斯·格雷戈里 (1638--1675) 与莱昂哈德·欧拉 (1707--1783) 和科林·麦克劳林 (1698--1746) 的发现相关的工作，以及以 2012--2018 年期间发表的近期工作结束。作者首先介绍了推动梯形规则改进的基本思想和观察。该近似方法中的主要数值误差来自积分边界。因此，应该调整正交权重以提高此类程序的准确性。詹姆斯·格雷戈里 (James Gregory) 的信中包含了朝这个方向开展的开创性工作。他的工作在第 2 节中进行了解释，其中表明 Gregory 的权重导致欧拉-麦克劳林系数，而这些系数在多年后才可用。这是一项了不起的成就，因为格雷戈里的工作还早于莱布尼茨和牛顿关于微积分的第一次出版物，以及布鲁克泰勒关于我们所谓的泰勒展开式的出版物。论文的主要部分调查了最近重新审视格雷戈里方法的发展。J. Reeger 和 B. Fornberg 于 2018 年提出了一种由径向基函数产生的有限差分方法。虽然这项工作涉及在三维空间中在有界曲面上使用节点的更一般的数值求积，但所提出的方法中的一个特殊情况产生了具有高准确度的类似 Gregory 的方法。作者进一步解释了计算权重的方法并提供了一些测试结果。附录包含用于计算正交权重的 MATLAB 代码。该演示文稿非常直观，可供广大学生使用。第二篇论文是“课堂流固交互：插值、心和游泳！” 它由 Nicholas A. Battista 提出。这里，在生物流体动力学数学建模的背景下讨论了通过样条对函数进行插值。插值提供了流体中物体运动的描述。三个日益复杂的例子说明了这种方法。第一个例子讨论了三个圆及其在流体中的运动。在此背景下，介绍了各种相关问题。例如，我们是否有足够的自由度来执行连续导数？线性插值函数与高阶多项式相比如何？第二个示例使用与第一个相同的指定运动，但引入了不同的多项式以在连续状态之间进行插值。通过这种方式，可以模拟跳动的心脏的运动。最后一个例子提供了一个理想化的游泳模型。假设游泳者的身体弯曲，以便在两种不同曲率的状态之间切换。通过在插值过程中连接线段和多项式段来近似游泳者的身体。由于游泳者与周围流体的相互作用，运动出现。这篇文章是针对教师的。作者建议如何以及何时在本模块中提出各种相关问题。他为讨论提供了指导，并为示例提供了代码。引用了最近的科学研究，这些研究使用这种方法在许多领域成功地规定了运动。这篇文章还包含一个网站的链接，该网站的软件允许人们构建和测试样条插值，以便对一个或多个特征状态之间的运动进行建模。包括在 MATLAB 和 Python 中的两个实现，以及指向帮助该软件的教程的链接。该部分的第三个贡献是由 Lu Lu、Xuhui Meng、Zhiping Mao 和 George Em Karniadakis 合着的论文“DeepXDE：用于求解微分方程的深度学习库”。本文阐述了基于深度学习的偏微分方程 (PDE) 数值解的最新进展。本文首先概述了物理信息神经网络 (PINN) 和自动微分的使用。大多数演示文稿基于最简单的神经网络类型，称为前馈 (FNN)，因为它们被认为足以解决大多数 PDE 问题。但是，网络类型对于该方法并不重要。作者简要解释了网络输出相对于网络输入的导数的算法计算是如何执行的。然后介绍PINN算法。考虑给定一个偏微分方程，对于具有合适边界条件的给定域上的解 $u(x)$ 和 $x$，由 $\lambda$ 参数化。神经网络 $\hat{u}(x;\theta )$ 被构造为 $u(x)$ 的代理。网络接受输入 $x$ 并根据在向量 $\theta$ 中收集的网络参数输出与 $u$ 具有相同维度的向量；所有网络层的权重矩阵和偏置向量。$\hat{u}(\cdot;\theta )$ 的导数是通过算法计算的，如前所述，通过应用链式法则来区分函数的组合。考虑了 PDE 及其边界域中随机选择的两组点，以反映 PDE 和边界条件施加的物理特性。损失函数是来自 PDE 方程的残差的加权 $L^2$ 范数及其在选定点上的边界条件。训练神经网络通过最小化损失函数来找到最佳参数 $\theta^\ast$。作者还提出了一种新的自适应细化方法来提高 PINN 的训练效率。论文中的进一步讨论包括 PINN 的近似属性和误差分析。作者以 Python 库的形式提供了 PINN 的实现，称为 DeepXDE，并在第 3 节中讨论了它的特性和定制能力。他们主张 PINN 算法简单且应用广泛：它可以应用于不同类型的偏微分方程，包括积分微分方程、分数阶偏微分方程、随机偏微分方程以及逆问题。本文的第 4 节提供了每种类型的代表性示例。结束语包含对该方法的优点和局限性以及未来可能扩展的讨论。