发表机构
Institute of Automation, Chinese Academy of Sciences; University of the Chinese Academy of Sciences; School of Advanced Interdisciplinary Sciences; School of Astronomy and Space Science, University of the Chinese Academy of Sciences（中国科学院自动化研究所; 中国科学院大学; 先进交叉科学学院; 中国科学院大学天文与空间科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
MOT-SR是整合外部分析工具、采用双协同LLM模块的多目标工具增强型符号回归框架，在40项标准任务及EMRI轨道建模中均展现出更优性能，可实现长程科学动力学的可靠建模。
AI 中文摘要
符号回归（Symbolic Regression，SR）旨在从观测数据中解析出解析方程，是科学建模的核心环节。尽管近期基于大语言模型（Large Language Model，LLM）的方法展现出潜力，但存在两大局限：其一，这类方法缺乏用于揭示变量依赖关系的数据分析机制，降低了方程发现的效率；其二，多数方法依赖仅聚焦拟合误差的单目标评估，对结构复杂度与泛化性的忽视常导致模型过早收敛至局部最优，限制了其探索更广阔方程空间的能力。本文提出多目标工具增强型符号回归（Multi-Objective Tool-augmented Symbolic Regression，MOT-SR），这一统一框架整合外部分析工具以提取结构先验并指导方程生成，同时通过维护动态帕累托最优前沿的多目标评估模块，联合优化准确性、复杂度与泛化性。MOT-SR采用两个协同的LLM模块：元策略生成器（Meta Strategy Generator）基于帕累托最优方程选择工具并合成结构优化策略，方程生成器（Equation Generator）据此生成新的候选方程。该系统以闭环方式运行，持续优化策略与方程结构。在40项标准任务中，MOT-SR在准确性、泛化性与效率上均优于现有SR方法。我们进一步在极端质量比旋进（extreme mass-ratio inspiral，EMRI）轨道建模任务中验证MOT-SR的性能，该任务是空间引力波天文学的重要问题，微小的局部误差会在长期演化中大幅累积。所发现的可解释修正项在保留配置上实现了最低的轨迹级积分误差，这些结果证明MOT-SR具备实现长程科学动力学可靠建模的潜力。
英文摘要
Symbolic Regression (SR) aims to discover analytical equations from observational data and plays a central role in scientific modeling. While recent Large Language Model (LLM) based approaches show promise, they face two limitations. First, they lack data analysis mechanisms for uncovering variable dependencies, which reduces the efficiency of equation discovery. Second, most methods rely on single-objective evaluation focused solely on fitting error. This neglect of structural complexity and generalization often causes models to converge prematurely to local optima, limiting their ability to explore the broader equation space. We propose Multi-Objective Tool-augmented Symbolic Regression (MOT-SR), a unified framework that integrates external analytical tools to extract structural priors and guide equation generation, while jointly optimizing for accuracy, complexity, and generalization via a multi-objective evaluation module that maintains a dynamic Pareto front. MOT-SR employs two collaborative LLM modules: a Meta Strategy Generator, which selects tools and synthesizes structural optimization strategies based on Pareto-optimal equations, and an Equation Generator, which produces new candidate equations accordingly. The system operates in a closed-loop manner, continuously refining both strategies and equation structures. Across 40 standard tasks, MOT-SR outperforms existing SR methods in accuracy, generalization, and efficiency. We further validate MOT-SR on extreme mass-ratio inspiral (EMRI) orbital modeling, an important problem in space-based gravitational-wave astronomy where small local errors can accumulate substantially over long-term evolution. The discovered interpretable correction achieves the lowest trajectory-level integration error on held-out configurations. These results demonstrate the potential of MOT-SR to enable reliable modeling of long-horizon scientific dynamics.
CommentsCode is available at https://github.com/wswbx/MOT-SR
