A Stochastic Differential Equation Framework for Multi-Objective LLM Interactions: Dynamical Systems Analysis with Code Generation Applications

TL;DR

The paper tackles the problem of unpredictable and competing objectives in iterative LLM interactions by introducing a stochastic differential equation framework that captures drift and diffusion in objective vectors: $d\mathbf{x} = \boldsymbol{\mu}(\mathbf{x}, \pi)dt + \boldsymbol{\sigma}(\mathbf{x}, \pi)d\mathbf{W}$. It provides a rigorous theoretical foundation including Euler-Maruyama discretization, an interference matrix to quantify cross-objective coupling, and eigenvalue-based criteria to characterize convergence regimes. A code-generation proof-of-concept validates the framework across security, efficiency, and functionality, showing strategy-dependent dynamics with convergence rates $\rho$ in $[0.15,1.29]$ and predictive power up to $R^2=0.74$. The work demonstrates the value of dynamical-systems analysis for guiding multi-objective LLM algorithm design and suggests broad applicability beyond code generation to domains like content optimization, reasoning, and human–AI collaboration.

Abstract

We introduce a general stochastic differential equation framework for modelling multiobjective optimization dynamics in iterative Large Language Model (LLM) interactions. Our framework captures the inherent stochasticity of LLM responses through explicit diffusion terms and reveals systematic interference patterns between competing objectives via an interference matrix formulation. We validate our theoretical framework using iterative code generation as a proof-of-concept application, analyzing 400 sessions across security, efficiency, and functionality objectives. Our results demonstrate strategy-dependent convergence behaviors with rates ranging from 0.33 to 1.29, and predictive accuracy achieving R2 = 0.74 for balanced approaches. This work proposes the feasibility of dynamical systems analysis for multi-objective LLM interactions, with code generation serving as an initial validation domain.