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

2025-10-12

AI Summary

This paper addresses the challenge of dynamically modeling safety-efficiency-functionality trade-offs in multi-objective large language model (LLM) interactions. Methodologically, it introduces the first stochastic differential equation (SDE)-based dynamic analysis framework: the drift term captures cooperative target evolution, the diffusion term explicitly models the intrinsic stochasticity of LLM responses, and a learnable interference matrix quantifies systemic competition among objectives. Its key contribution lies in formulating multi-objective LLM optimization as a noisy continuous-time dynamical system, enabling interpretable modeling and prediction of interference mechanisms. Empirical evaluation on a 400-iteration code generation task demonstrates a convergence rate range of 0.33–1.29 and achieves an R² of 0.74 for balanced strategy prediction—substantially outperforming conventional discrete optimization paradigms.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Search and Optimization: Learning to Search
• Natural Language Processing: (Large) Language Models

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

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.

Problem

• Modeling stochastic dynamics in multi-objective LLM interactions
• Analyzing systematic interference patterns between competing objectives
• Validating framework through iterative code generation applications

Innovation

• Stochastic differential equation framework for multi-objective LLM modeling
• Explicit diffusion terms capture LLM response stochasticity
• Interference matrix formulation reveals objective competition patterns