We introduce an algebraic analogue of dynamical systems, based on term
rewriting. We show that a recursive function applied to the output of an
iterated rewriting system defines a formal class of models into which all the
main architectures for dynamic machine learning models (including recurrent
neural networks, graph neural networks, and diffusion models) can be embedded.
Considered in category theory, we also show that these algebraic models are a
natural language for describing the compositionality of dynamic models.
Furthermore, we propose that these models provide a template for the
generalisation of the above dynamic models to learning problems on structured
or non-numerical data, including 'hybrid symbolic-numeric' models.

引入了一种基于术语重写的动力系统的代数模拟，证明了递归函数应用于迭代重写系统的输出定义了一类模型，其中包括循环神经网络、图神经网络和扩散模型等所有主要的动态机器学习模型架构。从范畴论的角度来看，这些代数模型也是描述动态模型组成性的一种自然语言。此外，我们提出这些模型为将上述动态模型推广到结构化或非数值数据（包括 “混合符号 - 数值” 模型）的学习问题提供了一个模板。