# DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience

arXiv: https://arxiv.org/abs/2506.04282 · HTML: https://mlsanity.com/paper/2506.04282/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Placed in Knowledge- and tool-augmented reasoning, one of 17 directions in LLM reasoning.
- Builds on 32 of its 49 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Placed in

- LLM reasoning / [Knowledge- and tool-augmented reasoning](https://mlsanity.com/lr/knowledge-and-tool-augmented-reasoning/)

## Builds on (most cited first, 30 of 32)

- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Language Model Crossover: Variation through Few-Shot Prompting](https://mlsanity.com/paper/2302.12170/) (2023)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Symbolic Regression with a Learned Concept Library](https://mlsanity.com/paper/2409.09359/) (2024)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2504.10415/) (2025)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Generating Synergistic Formulaic Alpha Collections via Reinforcement Learning](https://mlsanity.com/paper/2306.12964/) (2023)
- [In-Context Symbolic Regression: Leveraging Large Language Models for Function Discovery](https://mlsanity.com/paper/2404.19094/) (2024)
- [Llama-3.1-FoundationAI-SecurityLLM-Base-8B Technical Report](https://mlsanity.com/paper/2504.21039/) (2025)
- [Fast, accurate, and transferable many-body interatomic potentials by symbolic regression](https://mlsanity.com/paper/1904.01095/) (2019)
- [Taylor Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2205.09751/) (2022)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data](https://mlsanity.com/paper/2309.13705/) (2023)
- [Improving exploration in policy gradient search: Application to symbolic optimization](https://mlsanity.com/paper/2107.09158/) (2021)
- [CoEvo: Continual Evolution of Symbolic Solutions Using Large Language Models](https://mlsanity.com/paper/2412.18890/) (2024)
- [PruneSymNet: A Symbolic Neural Network and Pruning Algorithm for Symbolic Regression](https://mlsanity.com/paper/2401.15103/) (2024)
- [NIERT: Accurate Numerical Interpolation through Unifying Scattered Data Representations using Transformer Encoder](https://mlsanity.com/paper/2209.09078/) (2022)
- [DisCo-DSO: Coupling Discrete and Continuous Optimization for Efficient Generative Design in Hybrid Spaces](https://mlsanity.com/paper/2412.11051/) (2024)

## Built on by (most cited first, 22 of 22)

- [A Survey of Scientific Large Language Models: From Data Foundations to Agent Frontiers](https://mlsanity.com/paper/2508.21148/) (2025)
- [SR-Scientist: Scientific Equation Discovery With Agentic AI](https://mlsanity.com/paper/2510.11661/) (2025)
- [A Survey of Inductive Reasoning for Large Language Models](https://mlsanity.com/paper/2510.10182/) (2025)
- [Can Large Language Models Adequately Perform Symbolic Reasoning Over Time Series?](https://mlsanity.com/paper/2508.03963/) (2025)
- [MATRIX: A Multimodal Benchmark and Post-Training Framework for Materials Science](https://mlsanity.com/paper/2602.00376/) (2026)
- [STRIDE: A Self-Reflective Agent Framework for Reliable Automatic Equation Discovery](https://mlsanity.com/paper/2605.17790/) (2026)
- [NSR-Boost: A Neuro-Symbolic Residual Boosting Framework for Industrial Legacy Models](https://mlsanity.com/paper/2601.10457/) (2026)
- [Prior-Guided Symbolic Regression: Towards Scientific Consistency in Equation Discovery](https://mlsanity.com/paper/2602.13021/) (2026)
- [Back to the Beginning of Heuristic Design: Bridging Code and Knowledge with LLMs](https://mlsanity.com/paper/2605.06123/) (2026)
- [Discovery of Interpretable Surrogates via Agentic AI: Application to Gravitational Waves](https://mlsanity.com/paper/2605.11280/) (2026)
- [Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether](https://mlsanity.com/paper/2509.23004/) (2025)
- [PASM: Population Adaptive Symbolic Mixture-of-Experts Model for Cross-location Hurricane Evacuation Decision Prediction](https://mlsanity.com/paper/2604.00074/) (2026)
- [LLM-driven design of physics-constrained constitutive models: two agents are better than one](https://mlsanity.com/paper/2605.23754/) (2026)
- [SciText2Eq: Assessing LLMs for Explainable Equation Generation for Scientific Creativity](https://mlsanity.com/paper/2606.16003/) (2026)
- [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026)
- [What Should a Large Language Model See? Physical Invariants as a Data Representation for PDE Discovery](https://mlsanity.com/paper/2608.25189/) (2026)
- [LLMs as Post-hoc Auditors of Physiological Plausibility in Symbolic Regression: A Clinician-Evaluated Case Study](https://mlsanity.com/paper/2609.11431/) (2026)
- [TNLearn: An Open Source Python Package for Task-based Neurons](https://mlsanity.com/paper/2609.27564/) (2026)
- [SRHarness: A Harness for Agentic Symbolic Regression](https://mlsanity.com/paper/2609.35501/) (2026)
- [EvoMO-SR: Multiobjective LLM-based Evolution of Symbolic Expressions with substructure guidance](https://mlsanity.com/paper/2609.36187/) (2026)
- [How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes](https://mlsanity.com/paper/2610.00435/) (2026)
- [CueRator: Agentic Search for Symbolic Rules to Adapt Frozen Multimodal Encoders](https://mlsanity.com/paper/2610.07868/) (2026)

## Abstract (arXiv)

Symbolic regression is a fundamental tool for discovering interpretable mathematical expressions from data, with broad applications across scientific and engineering domains. Recently, large language models (LLMs) have demonstrated strong performance in this task, leveraging embedded scientific priors and reasoning capabilities to surpass traditional methods. However, existing LLM-based approaches, such as LLM-SR, often over-rely on internal priors, lacking explicit data understanding and systematic reflection during equation generation. To address these limitations, we propose DrSR (Dual Reasoning Symbolic Regression), a framework that combines data-driven insight with reflective learning to enhance both robustness and discovery capability. Specifically, DrSR guides LLMs to analyze structural relationships (e.g., monotonicity, nonlinearity, and correlation) within the data to generate structured descriptions. Simultaneously, it monitors equation performance and establishes a feedback loop to refine subsequent generations. By integrating data understanding and generation reflection in a closed loop, DrSR enables more efficient exploration of the symbolic expression space. Experiments across interdisciplinary datasets in physics, chemistry, biology, and materials science demonstrate that DrSR substantially improves the valid equation rate and consistently outperforms both classical and recent LLM-based methods in terms of accuracy, generalization, and search efficiency. These results underscore its potential for scientific equation discovery.