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

• Main Author: Ishii, Hiromi
• Format: Journal Article
• Language: English
• Published: 05-07-2021
• Subjects: Computer Science - Mathematical Software; Computer Science - Numerical Analysis; Computer Science - Symbolic Computation; Mathematics - Category Theory; Mathematics - Differential Geometry; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.2106.14153

Computer Algebra in Scientific Computing, pp. 174-191. CASC 2021.
Lecture Notes in Computer Science, vol 12865. Springer, Cham 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.