用多目标优化和外部工具辅助，提升科学方程发现的准确性与泛化能力。
MOT-SR: Multi-Objective Tool-Augmented Scientific Equation Discovery with Large Language Models
- 引入外部分析工具提取变量关系，指导方程生成。
- 同时优化精度、复杂度与泛化性，避免陷入局部最优。
- 在40个标准任务及引力波轨道建模中表现优异，适合长期动态建模。
符号回归（SR）旨在从观测数据中发现解析方程，在科学建模中具有核心作用。现有基于大语言模型（LLM）的方法存在两大局限：一是缺乏数据解析机制以揭示变量依赖关系，降低方程发现效率；二是普遍依赖单一目标评估，仅关注拟合误差，忽视结构复杂度与泛化能力，常导致模型过早收敛至局部最优，限制对更广泛方程空间的探索。本文提出多目标工具增强符号回归（MOT-SR），一个统一框架，通过集成外部分析工具提取结构先验并引导方程生成，同时通过多目标评估模块联合优化精度、复杂度与泛化性，并维护动态帕累托前沿。MOT-SR包含两个协同工作的LLM模块：元策略生成器根据帕累托最优方程选择工具并合成结构优化策略；方程生成器据此生成新候选方程。系统以闭环方式持续优化策略与方程结构。在40个标准任务上，MOT-SR在准确率、泛化性和效率方面均优于现有方法。进一步在极端质量比旋进（EMRI）轨道建模中验证，该问题为星基引力波天文学中的关键挑战，微小局部误差在长期演化中会显著累积。所发现的可解释修正项在未见配置下实现了最低的轨迹级积分误差。结果表明，MOT-SR具备实现长时序科学动态可靠建模的潜力。
原文摘要 · Abstract (English)
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.
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