A Categorical Compositional Framework for Generative Systems and Modular Equation Decomposition

This article introduces a categorical framework for modeling generative systems, denoted as $G_Θ$, which interprets system dynamics through a modular decomposition into states, inputs, generators, and optimizations. We propose treating equations such as those in physics, machine learning, or neural networks not as atomic entities but as composable structures built from functions, transitions, symmetries, constraints, and stochastic kernels. This approach extends the Kolmogorov Arnold representation theorem from functions to entire equations, enabling systematic generation of new mathematical laws via module recombination. Unlike multilayer perceptrons (MLPs), Kolmogorov-Arnold Networks (KANs), or Physics-Informed Neural Networks (PINNs), our framework provides a unified, typed calculus for compositional reasoning, drawing from applied category theory and probabilistic programming. We demonstrate how this modular abstraction facilitates principled exploration of “what-if” scenarios in equation design, bridging optimization, compositionality, and hybrid probabilistic-deterministic modeling.