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

• URL: http://arxiv.org/abs/2510.14342v1
• Date: Thu, 16 Oct 2025 06:25:24 GMT
• Title: Jet Functors and Weil Algebras in Automatic Differentiation: A Geometric Analysis
• Abstract summary: We present a geometric formulation of automatic differentiation (AD) using jet bundles and Weil algebras.<n>We derive concise statements on correctness, stability, and complexity.<n>This framework interprets AD theory through the lens of differential geometry.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We present a geometric formulation of automatic differentiation (AD) using jet bundles and Weil algebras. Reverse-mode AD emerges as cotangent-pullback, while Taylor-mode corresponds to evaluation in a Weil algebra. From these principles, we derive concise statements on correctness, stability, and complexity: a functorial identity for reverse-mode, algebraic exactness of higher-order derivatives, and explicit bounds on truncation error. We further show that tensorized Weil algebras permit one-pass computation of all mixed derivatives with cost linear in the algebra dimension, avoiding the combinatorial blow-up of nested JVP/VJP schedules. This framework interprets AD theory through the lens of differential geometry and offers a foundation for developing structure-preserving differentiation methods in deep learning and scientific computing. Code and examples are available at https://git.nilu.no/geometric-ad/jet-weil-ad.

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