# Jet Functors and Weil Algebras in Automatic Differentiation: A Geometric Analysis

arXiv: https://arxiv.org/abs/2510.14342 · HTML: https://mlsanity.com/paper/2510.14342/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 10 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Builds on (most cited first, 3 of 3)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [The simple essence of automatic differentiation](https://mlsanity.com/paper/1804.00746/) (2018)

## Built on by (most cited first, 1 of 1)

- [Coordinate-Free Automatic Differentiation](https://mlsanity.com/paper/2610.12776/) (2026)

## Abstract (arXiv)

We present a differential-geometric formulation of automatic differentiation (AD) based on jet functors and Weil algebras. In this framework, forward- and reverse-mode differentiation arise naturally as pushforward and cotangent pullback, while higher-order differentiation corresponds to evaluation in a Weil algebra. This construction provides a unified, coordinate-free view of derivative propagation and clarifies the algebraic structure underlying AD. All results are realized in modern JAX code, where the Weil-mode formulation computes all mixed derivatives in a single forward pass with cost linear in the algebra dimension. The resulting implementation achieves algebraically exact and numerically stable differentiation with predictable scaling, demonstrating that geometric abstraction can yield more efficient and transparent computational differentiation systems. Code is available at https://git.nilu.no/geometric-ad/jet-weil-ad