MOT-SR: Multi-Objective Tool-Augmented Scientific Equation Discovery with Large Language Models

2026-07-31

AI Summary

This work addresses critical limitations in existing large language model (LLM)-driven symbolic regression methods, which typically neglect variable dependency analysis and optimize solely for fitting error while disregarding structural complexity and generalization capability, often leading to suboptimal local solutions. To overcome these issues, the authors propose MOT-SR, a novel framework that introduces multi-objective optimization and tool-augmented mechanisms into symbolic regression for the first time. MOT-SR leverages external analytical tools to extract structural priors and employs dual LLM modules in a closed-loop iterative process to generate candidate equations. It jointly optimizes accuracy, complexity, and generalization, dynamically maintaining a Pareto front to enable co-evolution of search strategies and equation structures. Evaluated on 40 standard benchmarks, MOT-SR significantly outperforms current approaches and achieves the lowest trajectory integration error in modeling extreme mass-ratio inspiral orbits, demonstrating its effectiveness and reliability in scientific dynamical modeling.

Abstract

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.

Problem

• Symbolic Regression
• Multi-Objective Optimization
• Large Language Models
• Equation Discovery
• Generalization

Innovation

• Symbolic Regression
• Large Language Models
• Multi-Objective Optimization
• Tool Augmentation
• Pareto Front