Automatic Differentiation With Higher Infinitesimals, or Computational Smooth Infinitesimal Analysis in Weil Algebra

We propose an algorithm to compute the $C^{\mathrm{\infty }}$-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 $C^{\mathrm{\infty }}$-rings. The notion of a $C^{\mathrm{\infty }}$-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 $C^{\mathrm{\infty }}$-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.