Many problems in Physics and Chemistry are formulated as the minimization of a functional. Therefore, methods for solving these problems typically require differentiating maps whose input and/or output are functions -- commonly referred to as variational differentiation. Such maps are not addressed at the mathematical level by the chain rule, which underlies modern symbolic and algorithmic differentiation (AD) systems. Although there are algorithmic solutions such as tracing and reverse accumulation, they do not provide human readability and introduce strict programming constraints that bottleneck performance, especially in high-performance computing (HPC) environments. In this manuscript, we propose a new computer theoretic model of differentiation by combining the pullback of the $\mathbf{B}$ and $\mathbf{C}$ combinators from the combinatory logic. Unlike frameworks based on the chain rule, this model differentiates a minimal complete basis for the space of computable functions. Consequently, the model is capable of analytic backpropagation and variational differentiation while supporting complex numbers. To demonstrate the generality of this approach we build a system named CombDiff, which can differentiate nontrivial variational problems such as Hartree-Fock (HF) theory and multilayer perceptrons.

翻译：物理学和化学中的许多问题被表述为泛函的极小化问题。因此，求解这些问题的方法通常需要对输入和/或输出为函数的映射进行微分——这通常被称为变分微分。现代符号微分和算法微分（AD）系统所基于的链式法则，在数学层面上并未处理此类映射。尽管存在诸如追踪法和反向累积法等算法解决方案，但它们不具备人类可读性，并且引入了严格的编程约束，从而成为性能瓶颈，尤其是在高性能计算（HPC）环境中。在本文中，我们通过结合组合逻辑中 $\mathbf{B}$ 和 $\mathbf{C}$ 组合子的回拉，提出了一种新的计算机理论微分模型。与基于链式法则的框架不同，该模型对可计算函数空间的一个极小完备基进行微分。因此，该模型能够进行解析反向传播和变分微分，同时支持复数。为了证明该方法的通用性，我们构建了一个名为 CombDiff 的系统，该系统能够对非平凡的变分问题进行微分，例如 Hartree-Fock（HF）理论和多层感知机。