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.

翻译：我们基于项重写引入了动力系统的代数模拟。通过将递归函数应用于迭代重写系统的输出，我们定义了一个形式化的模型类，动态机器学习的主要架构（包括循环神经网络、图神经网络和扩散模型）均可嵌入其中。从范畴论角度考量，我们同时证明了这些代数模型是描述动态模型组合性的自然语言。此外，我们提出这些模型为上述动态模型向结构化或非数值数据（包括"混合符号-数值"模型）学习问题的泛化提供了模板。