We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Finally, we sketch how the analysis extends to other AD methods by considering a continuation-based method.

中文翻译：

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通过 Diffeologies 和 Categorical Gluing 自动区分的正确性

我们提出了自动微分（AD）的语义正确性证明。我们考虑在具有代数数据类型的高阶语言上的前向模式 AD 方法，并且我们将其表征为独特的结构保留宏，给定基本运算的导数选择。我们描述了基于差异空间的可微编程的丰富语义。我们证明它可以解释我们的语言，并且我们描述了 AD 方法在此语义方面正确的含义。我们表明，我们对 AD 的表征基于差异空间上的粘合构造，产生了对其正确性的优雅语义证明。我们解释了这在本质上如何是一个逻辑关系论证。最后，我们通过考虑基于延续的方法概述了分析如何扩展到其他 AD 方法。