SR-Scientist: Scientific Equation Discovery With Agentic AI

1 INTRODUCTION

• Existing LLM-based discovery methods use LLMs primarily as equation generators in fixed pipelines, without autonomy to gain equation-design insights through tools.Traditional genetic programming and deep-learning approaches also face efficiency and scalability issues.Source: §1 · p. 1
• SR-SCIENTIST lets an autonomous agent analyze data, implement and evaluate equations, and refine them through long-horizon interaction with experimental feedback.The framework uses code-interpreter tools and an experience buffer that retrieves the best-performing equations for subsequent iterations, while minimizing human-defined workflow.Source: §1 · p. 2
• Four of five backbone models achieve an absolute 6% to 35% margin over baseline methods across datasets covering four science disciplines.The datasets were designed to prevent reliance on equations memorized during training; the authors also report robustness to noise, out-of-domain generalization, and symbolic accuracy.Source: §1 · p. 2
• The authors develop an end-to-end reinforcement-learning pipeline to enhance the agent’s capabilities.They report significant improvement after applying RL training.Source: §1 · pp. 2–3

2 RELATED WORK

• Symbolic-regression methods include genetic programming, deep learning, and LLM-based hybrids, but the latter often lack direct tool-based data analysis and goal-oriented autonomy.The paper also notes that most LLM-based work focuses on inference rather than improving model abilities through reinforcement learning.Source: §2 · p. 3
• Agentic AI has been applied to software engineering, search, and GUI operations, but has received limited attention in scientific discovery.The paper presents equation discovery as an application where agentic models can also improve their abilities through RL.Source: §2 · p. 3

3.1 PROBLEM FORMULATION

• Symbolic regression seeks a concise, interpretable function that fits observed input-output data and generalizes to unseen inputs.The formulation balances prediction error, generalization, and interpretability.Source: §3.1 · p. 3

3.2 INFERENCE FRAMEWORK

• The inference loop sets a target MAPE precision and interleaves model reasoning with tool feedback, while NMSE selects equations for practical comparison.The authors choose MAPE because it provides a more uniform precision target across different data than MSE or NMSE.Source: §3.2 · p. 3
• A code interpreter provides data-analysis and equation-evaluation tools; the evaluator optimizes equation constants with BFGS and reports performance.The text-only backbone is prompted to avoid plots, although the framework supports plot-analysis patterns.Source: §3.2 · p. 4, §3.3 · p. 4
• An experience buffer stores explored equations with their MAPE scores and supplies the best K as in-context examples; exploration stops at the iteration limit or sufficient accuracy.The buffer is intended to address context-length limits; the framework returns the best-performing equation on observed data.Source: §3.2 · p. 4, §3.3 · p. 4

3.3 TRAINING FRAMEWORK

• Training data are synthesized from equation skeletons and physically motivated constants, then split so the agent sees training data while evaluation data measure performance and reward.The equations are formed for scientific scenarios with multiple variables, and variable ranges are defined before synthesizing data.Source: §3.3 · pp. 4–5
• The reward uses the best explored equation’s MAPE in a log-linear function scaled to [0, 1], with s_max marking the MAPE above which reward is zero.The agent is instructed to meet a specified MAPE goal during rollouts.Source: §3.3 · p. 5
• The framework optimizes the agent with Group Relative Policy Optimization, sampling groups of outputs and updating the policy using their group-relative advantages.The authors omit the reference-model KL penalty to encourage exploration and report faster convergence with comparable performance.Source: §3.3 · p. 5

4.1 SETUP

• Evaluation uses LSR-Synth’s 129 expert-checked problems across four sciences, with training, ID-test, and OOD-test sets.The paper uses accuracy-to-tolerance as its main metric for aggregating multiple problems. It reports excluding LSR-Transform because nearly 50% of its problems reached sufficiently small error after one iteration, raising possible memorization concerns.Source: §4.1 · p. 5

4.2 MAIN RESULTS

• Four backbones beat baselines by an absolute 6%–35% overall margin; GPT-OSS-120B reaches Acc0.01 63.57% and Acc0.001 49.35%.SR-SCIENTIST also surpasses other LLM-based methods across all subjects with Qwen3-Coder-480B, GLM-4.5-Air, and GPT-OSS-120B backbones; RL training improves Qwen3-Coder-30B across all subjects.Source: Table 1 · p. 6, §4.2 · p. 7
• SR-SCIENTIST achieves the best OOD performance, while ID-to-OOD trends vary by subject and all methods improve on material science.The reported OOD comparison uses Acc0.01 across LSR-Synth domains with Qwen3-Coder-480B.Source: Figure 2 · p. 7, §4.2 · p. 8
• SR-SCIENTIST has the best symbolic accuracy and identifies the most ground-truth equations; its PO10 and PO37 equations recover structure and constants.Green-highlighted terms in Figure 4 are shared by predicted and ground-truth equations; the paper also reports that the agent produces derivation trajectories.Source: Table 2 · p. 7, §4.2 · p. 8, Figure 4 · p. 8
• As Gaussian noise increases, all methods decline, but SR-SCIENTIST consistently performs better, especially with Qwen3-Coder-480B and GLM-4.5-Air.Noise standard deviations are σ = {0.01, 0.05, 0.1}.Source: Figure 3 · p. 7, §4.2 · p. 8

4.3 ANALYSIS

• With about 1,000 LLM calls, increasing the maximum turns from 10 to 25 improves performance, while longer exploration stagnates or slightly lowers it.The authors conclude that inference budget should instead go to initiating new iterations beyond 25 turns.Source: Figure 5 · p. 9, §4.3 · p. 9
• Ablations indicate that data analysis is crucial, while removing the experience buffer or randomly sampling from it significantly reduces performance.The buffer lets the agent reuse top-performing equations from previous iterations.Source: §4.3 · pp. 8–9, Table 3 · p. 9
• Qwen and GLM devote about 80% of calls to equation evaluation and 20% to data analysis; GPT-OSS models favor residual-error code, and RL-trained Qwen3-Coder-30B increases data-statistics use.Within Qwen and GLM data-analysis calls, statistics such as correlations and averages generally account for the largest share.Source: Figure 6 · p. 9, §4.3 · pp. 9–10

• Main experiments use text-only models; incorporating multimodal inputs and plots into analysis is proposed as future work.Source: §5 · p. 10
• Performance still drops significantly as noise increases, motivating exploration of noise-handling strategies in the agent’s prompt.Source: §5 · p. 10
• Providing only top-performing equations may lead the agent to revisit poorly performing ones; improving memory through distilled experience is proposed.Source: §5 · p. 10

6 CONCLUSION

• SR-SCIENTIST transforms the LLM from a passive equation proposer into an autonomous scientist that analyzes data and evaluates and refines equations through active interaction.Source: §6 · p. 10
• Experiments report significant advantages over existing methods in precision, generalization, robustness to noise, and symbolic accuracy.Source: §6 · p. 10
• The paper develops a complete reinforcement-learning pipeline that allows the agent to self-evolve and enhance its discovery capabilities.Source: §6 · p. 10

A BASELINE IMPLEMENTATION DETAILS

• GPLearn uses genetic programming, with a population of 500, 200 generations, and tournament size 20.Source: A BASELINE IMPLEMENTATION DETAILS · p. 14
• E2E predicts complete expressions directly, while NeSymReS uses a pre-trained Transformer to predict expressions from data points.Source: A BASELINE IMPLEMENTATION DETAILS · p. 14
• DSR (Petersen et al., 2021) uses a recurrent network and reinforcement learning; uDSR extends it with neural-guided search, pre-training, and linear models.Source: A BASELINE IMPLEMENTATION DETAILS · p. 14
• PySR (Cranmer, 2023) uses a Julia backend for expression search, while LaSR evolves a library of abstract natural-language concepts.Source: A BASELINE IMPLEMENTATION DETAILS · p. 15
• LLM-SR (Shojaee et al., 2025a) combines LLM scientific priors with evolutionary search and is run with up to 1,000 LLM calls per problem.Source: A BASELINE IMPLEMENTATION DETAILS · p. 15

B DETAILS OF SR-SCIENTIST INFERENCE FRAMEWORK

• SR-SCIENTIST uses a 0.7 sampling temperature, caps equations at 10 parameters, and optimizes parameters with BFGS against mean square error.Source: B DETAILS OF SR-SCIENTIST INFERENCE… · p. 15

C.1 TRAINING DATA SYNTHESIS

• Training problems cover materials science, chemistry, biology, and physics, with each discipline defined through relationships among specified variables.Source: §C.1 · p. 15
• Equation skeletons combine known terms from the LLM’s prior knowledge with novel terms outside it, using two to four terms and screening for similarity to benchmark equations.Source: §C.1 · p. 15

C.2 REWARD DESIGN

• The log-linear reward function performs better than the stepwise function.Source: Table 4 · p. 16, §C.2 · p. 16

C.1 TRAINING DATA SYNTHESIS

• Constants are assigned with scientific context in mind, with LLM-generated assignments followed by human validation.Source: §C.1 · p. 16
• Each problem yields 5,000 data points; the highest-temperature or latest-time 500 form the out-of-domain test set, while remaining points are split for training and testing.Source: §C.1 · p. 16
• Filtering removes equations with scientifically anomalous data points, leaving a dataset described as scientifically meaningful with plausible points.Source: §C.1 · p. 16

C.3 TRAINING DETAILS

• Training used 32 NVIDIA H200 GPUs, batch-level asynchronous rollouts, a 10,240-token response limit, and at most 20 turns.Each prompt had eight rollouts; the KL loss coefficient was zero to encourage exploration.Source: §C.3 · pp. 16–17
• Reward rises and saturates around 60 steps, while response length increases during training.The authors interpret the longer responses as evidence that the model learns a long-horizon problem-solving strategy.Source: §C.3 · p. 17, Figure 8 · p. 17, Figure 7 · p. 17

C.4 THE EFFECT OF LLM BACKBONE

• All three Qwen3 backbones gain significantly after RL training, with most improving across all subjects.Without RL, SR-SCIENTIST beats LLM-SR on Acc0.01 and matches it on Acc0.1 with Qwen3-32B, but trails with Qwen3-8B and Qwen3-14B.Source: §C.4 · p. 17, Table 5 · p. 17

D.1 ANALYSIS FOR THE EVALUATION METRICS

• R2 differences are small, while average NMSE is sensitive to maximum values, so accuracy-to-tolerance is used as the main metric.The authors also found that truncating large errors requires a difficult-to-set threshold and that median values miss the full performance distribution.Source: §D.1 · pp. 17–18, Table 7 · p. 18, Table 6 · p. 18

D.2 COST

• Estimated API costs are described as acceptable for practical use, and local GPT-OSS-120B inference on 129 problems took no more than five hours.Cached tokens can further reduce the estimated API price.Source: §D.2 · p. 18

D.3 OOD PERFORMANCE OF OTHER MODELS

• SR-SCIENTIST leads in overall OOD accuracy for GLM-4.5-Air, GPT-OSS-120B, and GPT-OSS-20B, but slightly trails PySR with Qwen3-Coder-30B.RL training improves both ID and OOD performance.Source: §D.3 · p. 18

D.4 OTHER QUALITATIVE RESULTS

• In the trajectory examples, the agent forms equation hypotheses from code-based data analysis and the performance of earlier equations.The authors say this grounded analysis helps decrease the search space and can inspire equation design by human scientists.Source: §D.4 · p. 18

D.5 SENSITIVITY ANALYSIS OF HYPERPARAMETERS

• Performance remains robust with relatively few fetched equations but declines slightly when the number rises to 10.The authors suggest that putting many fetched equations in the context window may strain the model's capability.Source: §D.5 · pp. 18–19, Table 9 · p. 19
• Performance is insensitive to relatively easy initial MAPE goals, but an extremely challenging goal such as 0.0001 degrades performance and can trap optimization.The authors attribute resilience to easy targets to the dynamic goal update mechanism, which prevents trivial goals.Source: §D.5 · p. 19, Table 10 · p. 19

D.6 SYMBOLIC ACCURACY CALCULATION

• Symbolic accuracy is assessed first by GPT-OSS-120B and then by human evaluators when its ten votes are inconsistent.In 121 preliminary cases, unanimous LLM votes matched human evaluations 98.3% of the time.Source: §D.6 · p. 19

D.7 TOOL CALL ANALYSIS

• Human and LLM-based tool-call evaluations agree in 94.4% of 54 checked cases.The LLM evaluation used the prompt shown in Figure 18.Source: §D.7 · p. 19, Figure 18 · p. 25

E THE USE OF LARGE LANGUAGE MODELS

• The authors state that the manuscript's LLM use was limited to proofreading, sentence structure, and readability, not core ideas or scientific content.The authors take responsibility for the manuscript's content.Source: E THE USE OF LARGE LANGUAGE MODELS · p. 20
• The user prompt directs the agent to implement an equation as a Python function and optimize it until MAPE is below 0.1%.It names equation_evaluator for goodness-of-fit evaluation and data_analyzer for data analysis.Source: E THE USE OF LARGE LANGUAGE MODELS · p. 24, Figure 16 · p. 25