用几何工具统一解释自动微分，实现高效精确求导。
Jet Functors and Weil Algebras in Automatic Differentiation: A Geometric Analysis
- 基于喷泉函子与韦尔代数构建微分几何框架
- 单次前向传播即可计算所有混合导数，复杂度线性于代数维数
- 代码开源，适合研究高效自动微分系统者参考
我们提出一种基于喷泉函子和韦尔代数的自动微分（AD）微分几何表述。在此框架中，前向模式与反向模式微分自然对应于切映射与余切拉回，高阶微分则对应于在韦尔代数中的求值。该构造提供了一种统一、坐标无关的导数传播视角，并揭示了AD背后的代数结构。所有结果均在现代JAX代码中实现，其中韦尔模式在单次前向传播中计算所有混合导数，代价与代数维度呈线性关系。所得实现具有代数精确性与数值稳定性，且扩展性可预测，证明几何抽象可带来更高效透明的计算微分系统。代码已公开于 https://git.nilu.no/geometric-ad/jet-weil-ad。
原文摘要 · Abstract (English)
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
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