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SR-Scientist: Scientific Equation Discovery With Agentic AI

Conference: ICLR 2026 arXiv: 2510.11661 Code: GitHub Area: LLM Agent Keywords: symbolic regression, agentic AI, equation discovery, reinforcement-learning, scientific discovery

TL;DR

This paper proposes the SR-Scientist framework, which elevates LLMs from simple equation proposers to autonomous AI scientists. By leveraging a code interpreter tool for data analysis and equation evaluation, the framework autonomously discovers scientific equations through long-horizon interactions, with reinforcement learning further enhancing its capabilities.

Background & Motivation

Symbolic Regression (SR) aims to discover interpretable mathematical expressions from observational data and is a foundational task in scientific discovery. Existing approaches fall into three main categories: - Genetic Programming (GP) methods: e.g., PySR and GPLearn, which perform combinatorial search over expression trees - Deep learning methods: e.g., E2E, NeSymReS, and DSR, which train neural networks to map numerical data to expressions - LLM-augmented methods: e.g., LLM-SR and LaSR, which embed LLMs within GP pipelines as equation proposers

Key limitations of existing LLM-based approaches include: 1. LLMs serve only as equation generators within fixed pipelines, lacking autonomy 2. They cannot directly analyze observational data via tools to extract insights 3. Most works focus solely on inference and do not explore self-improvement through methods such as RL

The core motivation of this paper is to build a scientific discovery framework centered on Agentic AI, transforming the LLM from a passive tool into an autonomous agent capable of driving the entire discovery lifecycle.

Method

Overall Architecture

The inference framework of SR-Scientist adopts an iterative design (Algorithm 1):

  1. Each iteration sets a precision target \(G_i\) (based on MAPE)
  2. The LLM agent operates under the ReAct framework, alternating between reasoning and tool invocation: \((r_1, \mathcal{T}_1, o_1), (r_2, \mathcal{T}_2, o_2), \ldots\)
  3. An Experience Buffer propagates the best equations across iterations
  4. Upon meeting the stopping criterion, the best equation is submitted

The objective function uses MAPE (Mean Absolute Percentage Error):

\[\text{MAPE} = \frac{100\%}{n} \sum_{i=1}^{n} \left| \frac{y_i - f(\mathbf{x}_i)}{y_i} \right|\]

Key Designs

Tool Design: The code interpreter is encapsulated into two core tools: - Data Analyzer \(T_1\): Linked to observational data, allowing the agent to write code for statistical analysis, residual analysis, and various forms of data exploration - Equation Evaluator \(T_2\): Accepts equation skeletons with constant placeholders, internally optimizes constants via the BFGS algorithm, and reports performance

Experience Buffer: Maintains \(E = \{(e_i, s_i)\}_{i=1}^{N}\) to record explored equations and their MAPE scores. At the start of each iteration, the top-\(K\) equations are retrieved as in-context examples, elegantly circumventing LLM context length constraints.

Long-Horizon Optimization: Each iteration allows the agent up to \(M=25\) rounds of interaction (exceeding 20 turns), providing sufficient time for data analysis and equation refinement.

Loss & Training

The training framework adopts the GRPO algorithm with a log-linear reward function:

\[\mathcal{R} = \text{clip}\left(\frac{\lg s_{\max} - \lg s}{\lg s_{\max} - \lg s_{\text{goal}}}, 0, 1\right)\]

where \(s\) is the MAPE of the best equation, \(s_{\max}=100\%\), and \(s_{\text{goal}}=0.1\%\). This continuous reward design avoids the sparsity of binary rewards. Training data is constructed via a hybrid strategy combining rule-based and model-based synthesis, covering four domains: materials science, chemistry, biology, and physics.

Key Experimental Results

Main Results

Accuracy results on the LSR-Synth benchmark (129 problems across 4 disciplines):

Method Overall Acc₀.₀₁ Overall Acc₀.₀₀₁ Materials Acc₀.₀₁ Chemistry Acc₀.₀₁ Biology Acc₀.₀₁ Physics Acc₀.₀₁
PySR 29.46 14.47 53.33 25.93 16.67 25.76
LLM-SR (Qwen-480B) 41.08 18.09 80.00 36.11 30.56 28.79
SR-Scientist (GPT-120B) 63.57 49.35 74.67 81.48 66.67 40.91
SR-Scientist (GLM) 48.32 25.06 81.33 45.37 40.28 36.37
SR-Scientist (Qwen-480B) 49.09 24.55 86.67 40.74 50.00 34.09
SR-Scientist (30B) 32.30 16.02 81.33 22.22 22.22 18.18
SR-Scientist (30B+RL) 40.92 20.69 85.33 37.38 29.17 25.00

Core Finding: SR-Scientist outperforms all baselines by 6%–35% across all four backbone models, with GPT-OSS-120B achieving the highest overall performance. RL training yields consistent and significant improvements across all disciplines.

Ablation Study

Method Acc₀.₀₁ Acc₀.₀₀₁
SR-Scientist (GPT) 63.57 49.35
w/o Data Analyzer \(T_1\) 35.66 16.28
w/o Experience Buffer 57.36 41.86
w/o top-k (random sampling) 58.14 41.86

Key findings from the ablation: - The data analysis tool has the largest impact on the GPT model (a drop of ~28 percentage points) - The experience buffer has the largest impact on the Qwen model (a drop of 13.4 percentage points) - Top-k sampling outperforms random sampling

Key Findings

  1. Symbolic Accuracy: SR-Scientist achieves the best performance in fully recovering ground-truth equations (SA=7.75–8.00), surpassing PySR (4.65) and LLM-SR (5.43)
  2. Noise Robustness: SR-Scientist consistently outperforms other methods under Gaussian noise with varying standard deviations
  3. OOD Generalization: Discovered equations maintain the best performance on out-of-distribution test data
  4. Optimal Interaction Length: 25 rounds is the optimal value; too few (10 rounds) is insufficient for thorough exploration, while too many yields diminishing returns
  5. Tool Usage Behavioral Differences: GPT-series models tend to directly write residual analysis code, whereas Qwen/GLM models more frequently use descriptive statistics

Highlights & Insights

  1. Paradigm Shift: Transforming the LLM from a passive equation proposer to an autonomous AI scientist represents a significant conceptual advance in scientific discovery
  2. Elegant Experience Buffer Design: A simple heap structure resolves the LLM context length limitation while enabling cross-iteration knowledge transfer
  3. Continuous Reward Design: Leveraging the continuous measurability of equation performance, the log-linear reward avoids sparsity and is better suited to this task than the binary rewards typical of math or code tasks
  4. Minimal Human Pipeline Principle: The agent freely determines its own workflow; different backbone models exhibit distinct analytical strategies (e.g., GPT favors residual analysis, Qwen favors statistical analysis)
  5. Effective RL Self-Improvement: The 30B model trained with RL approaches the performance of larger non-RL models, validating the feasibility of agent self-improvement

Limitations & Future Work

  1. Only text-based models are used; multimodal inputs (e.g., chart analysis) are not exploited
  2. Performance degrades significantly under noisy conditions
  3. The agent may repeatedly explore previously identified poor-performing equations across iterations; the memory system has room for improvement
  4. Although the evaluation set is designed to mitigate memorization, LSR-Synth remains synthetic data and may not fully reflect the complexity of real-world scientific discovery
  • Compared to works such as FunSearch (Romera-Paredes et al., 2024) and AlphaEvolve, SR-Scientist places greater emphasis on agent autonomy and long-horizon interaction
  • The combination of experience buffer and GRPO provides a blueprint for RL training of scientific discovery agents
  • The modular framework design (pluggable tools, swappable backbone models) is highly extensible and can be generalized to other scientific discovery tasks

Rating

  • Novelty: ⭐⭐⭐⭐ — Introducing the agentic AI paradigm into symbolic regression, combined with RL-based self-improvement, constitutes a significant contribution
  • Experimental Thoroughness: ⭐⭐⭐⭐⭐ — Comprehensive evaluation across 4 disciplines, 5 backbone models, and multiple metrics including accuracy, generalization, noise robustness, and symbolic accuracy
  • Practicality: ⭐⭐⭐⭐ — Open-source code and modular framework, though the reliance on extensive LLM calls entails non-trivial computational cost
  • Writing Quality: ⭐⭐⭐⭐ — Clear structure and well-specified algorithms, though some sections could be more concise
  • Overall Rating: ⭐⭐⭐⭐ (4/5)