We propose an algorithm to compute the $C^\infty$-ring structure of arbitrary Weil algebra. It allows us to do some analysis with higher infinitesimals numerically and symbolically. To that end, we first give a brief description of the (Forward-mode) automatic differentiation (AD) in terms of $C^\infty$-rings. The notion of a $C^\infty$-ring was introduced by Lawvere and used as the fundamental building block of smooth infinitesimal analysis and synthetic differential geometry. We argue that interpreting AD in terms of $C^\infty$-rings gives us a unifying theoretical framework and modular ways to express multivariate partial derivatives. In particular, we can "package" higher-order Forward-mode AD as a Weil algebra, and take tensor products to compose them to achieve multivariate higher-order AD. The algorithms in the present paper can also be used for a pedagogical purpose in learning and studying smooth infinitesimal analysis as well.

中文翻译：

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更高无穷小的自动微分，或 Weil 代数中的计算平滑无穷小分析

我们提出了一种算法来计算任意 Weil 代数的 $C^\infty$-ring 结构。它允许我们在数值和符号上对更高的无穷小进行一些分析。为此，我们首先根据 $C^\infty$-rings 简要描述（前向模式）自动微分（AD）。$C^\infty$-ring 的概念由 Lawvere 引入，用作平滑无穷小分析和合成微分几何的基本构建块。我们认为，用 $C^\infty$-rings 来解释 AD 为我们提供了一个统一的理论框架和模块化方法来表达多元偏导数。特别是，我们可以将高阶 Forward-mode AD“封装”为 Weil 代数，并用张量积将它们组合起来，实现多元高阶 AD。