SR-Scientist: Scientific Equation Discovery With Agentic AI
Authors: Shijie Xia
Yuhan Sun
Pengfei Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Symbolic regression
Equation Discovery
Large Language Models
Agentic AI
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We present SR-Scientist, a framework with a corresponding RL training strategy, in which an autonomous agent discovers scientific equations through long-horizon, tool-driven data analysis and equation evaluation.

Abstract: Recently, Large Language Models (LLMs) have been applied to scientific equation discovery, leveraging their embedded scientific knowledge for hypothesis generation. However, current methods typically confine LLMs to the role of an equation proposer within search algorithms like genetic programming. In this paper, we present SR-Scientist, a framework that elevates the LLM from a simple equation proposer to an autonomous AI scientist that writes code to analyze data, implements the equation as code, submits it for evaluation, and optimizes the equation based on experimental feedback. Specifically, we wrap the code interpreter into a set of tools for data analysis and equation evaluation. The agent is instructed to optimize the equation by utilizing these tools over a long horizon with minimal human-defined pipelines. Empirical results show that SR-Scientist outperforms baseline methods by an absolute margin of 6\% to 35\% on datasets covering four science disciplines. Additionally, we demonstrate our method's robustness to noise, the generalization of the discovered equations to out-of-domain data, and their symbolic accuracy. Furthermore, we develop an end-to-end reinforcement learning framework to enhance the agent's capabilities.

Published as a conference paper at ICLR 2026
SR-SCIENTIST: SCIENTIFIC EQUATION DISCOVERY WITH AGENTIC AI
Shijie Xia Yuhan Sun 1.3- Pengfei Liu 1,233
Shanghai Jiao Tong University 2Shanghai Innovation Institute (shi mixiia, sunmouren, pengfeilesjtu.
3GAIR
e edu..
ABSTRACT
Recently, Large Language Models (LLMs) have been applied to scientific equa- tion discovery, leveraging their embedded scientific knowledge for hypothesis generation. However, current methods typically confine L.LSs to the role of an equation proposer within search algorithms like genetic programming. In this pa- per, we present RR-SCIENT,ST, a framework that clevates the LLM from a sim- ple equation proposer
autonomous AI scientist that writes code to analyze data, implements the equation as code, submits for evaluation, and optimizes the wrap the code inter-
equation based on experimental feedback. Specifically, preter into a set of tools for data analysis and equation evaluation. The agent is in- structed optimize the equation by utilizing these tools over long horizon with minimal human-defined pipelines. Empirical results show that SR-SCIENTIST outperforms baseline methods by an absolute margin of 6% to 35% on datasets covering four science disciplines. Additionally, we demonstrate our method's 's ro- bustness to noise, the generalization of the discovered equations out-of-domain data, and their symbolic accuracy. Furthermore, we develop an end-to-end rein- forcement learning framework enhance the agent's s capabilities'.
INTRODUCTION
In the era of Agentic AI, Large Language Models (LLMs) have evolved from simple knowledge retrievers to agentic models capable of completing complex tasks by interacting with their envi- ronments (Schneider, 2025), as exemplified by products like Claude Code (Anthropic, 2025) and Gemini CLII (Google, 2025). These agents exhibit many characteristics of human scientists, such as engaging in long-horizon interaction with environmental feedback, operating with autonomy, and relying less on predefined pipelines (Newell et al., 1972). However, most current work using LLMs for scientific discovery (Shojaee al., 2025a; Novikov et al., 2025; Romera-Pareda et al., 2024), LLMS serve as static components within human carnan- pipelines, lacking the autonomy to generate and refine hypotheses through active environmental interaction. Building scientific dis- covery framework around agentic models could therefore shift the paradigm from using LLMs as passive tools to empowering them as autonomous agents that drive the entire discovery lifecycle. In this paper, we focus on equation discovery, a fundamental task in science.
Mathematical equations central to scientific progress, serving concise and interpretable mod- els of physical phenomena (Lemos et al., 2022; Batra et al., 2021: Hernandez et al., 2019). The task of data-driven equation discovery, also known as symbolic regression (SR), is NP-hard problem due to its vast search space (Virgolin & Pissis, 2022). Traditional methods have relied on Genetic Programming (GP) for combinatorial search (Cranmer, 2023) or on deep neural networks trained on large-scale synthetic data for direct prediction (Kamienny et al., 2022; Biggio et al., 2021), which often suffer from efficiency and scalability issues. Recently, the research community has begun to embed L.LSs into GP algorithms equation proposers, leveraging their scientific prior knowledge to generate more effective hypotheses (Shojace et al., 2025a; Grayeli et al., 2024). While this can enhance scarch efficiency. LLMs primarily serve equation generators in fixed pipeline, which lacks the autonomy to gain insight for equation design through tools.
"Co-first authors. Corresponding author. Code and data available: https: //g/ www.bccoooocte.eee cwm/ MI-NLP/SR--- Saiet---SSSniii

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Published as a conference paper at ICLR 2026
Long-horizon Optimization
Your goal is to optimize the equation until the mean absolute percentage error less than 0.1%. Previously Explored Equations
Fetch Tap-K Equations
First explore data.
Import ison dista Fist json loedss impul, dista
print/first 5 rows, data Iss(:5))
first [[0.010, 16.543, 95. 159).
Experience Buffer
S₁
outputs can negative small. We can start simple? output = params[(] p (1-P/ params{1])
Update
numpy equation perams) nnderayy
Eqx
8x
[" params(0); 00: params
returnr P
Mean absolute percentage error: 12234%
Lat's explors correlation between output and
Figure 1: The inference framework of SR-SCIENTIST. At each iteration, the LLM agent lonomously conducts long-horizon optimization using code interpreters for data analysis and equa- tion evaluation. To overcome the context length limitations of current LLMs, we implement an experience buffer to fetch the best-performing equations for subsequent iterations. 'Eq' denotes the equation and 'S' denotes the equation score.
To address these limitations, we introduce ER-SCIENT,ST, framework designed to enable LLMs to find scientific equations through long-horizon optimization driven by experimental feedback and analysis. To achieve this, we wrap the code interpreter as tools to analyze data and evaluate equa- tions. As shown in Figure 1, the LLM agent is instructed to solve problems by utilizing these tools, including writing code to analyze data, implementing the equation as code, submitting it for evalu- ation, and optimizing the equation based on experimental feedback. In the process, long-horizon optimization, allowing the agent to interact with data through tools over multiple turns (e.g., more than twenty) gather sufficient information for equation design. To overcome the con- text length limitations current LLMs, we implement an experience buffer to store the explored equations and fetch the berf-preno equations for subsequent iterations. Moreover, we adhere to the principle of minimal human-defined pipelines (Sutton, 2019), in which the agent is free to determine its own workflow for given problem. Within the flexible framework, we develop a rein- forcement learning (RL) pipeline from training data construction to reward design to help the LLM agent evolve its capabilities.
emphasize
To validate the effectiveness of our framework, we evaluate SR-SCIENTIST using 5 different models as backbone L.LMs on datasets covering science disciplines carefully designed to prevent the LLMs from relying memorized equations from their training data (Shojaee al., 2025b). Results demonstrate that SR-SCIENTIST consistently outperforms state-of-the-art SR methods, with four of the models achieving an absolute margin of 6% to 35% over baselines. We also observe significant improvement after applying RL training. Additionally, we demonstrate our method's robustness to noise, the generalization of the discovered equations to out-of-domain data, and their symbolic accuracy. Furthermore, our ablation studies highlight the importance of data analysis, the experience buffer, and long-horizon optimization.
Overall, our contributions are as follows:
We develop SR-SCIENTIST, a framework where autonomous agent discovers scientific equa- tions through long-horizon, tool-driven data analysis and equation evaluation.
We show that ERISCIENTIST significantly outperforms baseline methods in precision, general- ization, robustness to noise, and symbolic accuracy.
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Published as a conference paper at ICLR 2026
We develop corresponding end-to-end reinforcement learning pipeline to enhance the agent's capabilities.
2 RELATED WORK
Symbolic Regression SR aims to identify interpretable mathematical expression that accu- rately describes the relationship between variables observed data (Makke & Chawla, 2024; Cava et al., 2021). Current popular methods can be categorized as follows: 1) GP-based Methods: This line of methods uses constrained representations, such as expression trees, to define a search space composed of symbolic operators, constants, and variables (Crammer, 2023). Evolutionary or genetic programming is then applied to search for solutions. 2) Deep learning Methods: These methods leverage deep neural networks learn direct mapping from numerical observations to their un- derlying mathematical expressions through online (Petersen et al., 2021; Landajuela et al., 2022) or offline learning (Kamienny et al., 2022; Biggio al., 2021). 3) LLM-based methods: Due to the efficiency and scalability issues of the above methods, the research community has integrated LLMs with traditional techniques like GP, leveraging the models' extensive knowledge
generate more informed hypotheses (Shojaee et al., 2025a; Grayeli et al., 2024; Ma et al., 2024; Wang et al., 2025b). While promising, current hybrid methods still lack the direct analysis of observed data through tools to gain insights. Furthermore, the LLM's behavior is often fixed, lacking the autonomy to decide its actions in a goal-oriented manner. Moreover, most work focuses on using LLMs for inference, without exploring how these models evolve their abilities through methods like RL.
Agentic AI Agentic AI can autonomously execute complex tasks in dynamic environments re- quiring adaptation, interaction, and reasoning in a goal-oriented manner (Schneider, 2025). The open-source community has built powerful agentic models for tasks like software engineering (Team et al., 2025; Zeng et al., 2025a), search (Gao et al., 2025), and GUI operations (Qin et al., 2025; Wang al., 2025a), enabling them solve tasks in end-to-end manner with minimal human interven- tion. However, there has been limited focus on scientific discovery. In this paper, we demonstrate that with careful design, these agentic models can be powerful for equation discovery and can also improve their abilities through RL.
3 SR-SCIENTIST
3.1 PROBLEM FORMULATION
Symbolic regression aims to find a symbolic expression that describes the underlying relationship in a dataset. Formally, given a dataset D {(x; y/) yr)))²,, where x; E Rd an input vector and y: E TR. is scalar output, an SK method seeks to find concise symbolic function f such that f(x;) R yi- The goal SR is not only to discover a function that minimizes the error between the predicted values and the true values but also one that generalizes well to unseen inputs while maintaining interpretability.
3.2 INFERENCE FRAMEWORK
We outline the overall inference framework in Algorithm 1. At each iteration, we set a goal G for desired optimization precision and the agent is instructed to find an equation that satisfies the precision. We choose the mean absolute percentage error (MAPE) as the goal: MAPE n Σ -=f)(x)] Then, the LLM agent solves the problem by interleaving internal reason- ing with external environment feedback. Formally, the agent's trajectory is structured follows, typically within a ReAct (Yao et al., 2023) framework:
100%
(r1. o1) (r2, T2, o2,, (rk. Ts. oR)
(I)
"This is slightly different from other SR methods that usually use the mean squared error (MSE) or nor- malized mean squared error (NMSE) as the score, as we find MAPE provides more uniform precision target across different data than the aforementioned metrics. To ensure fair comparison with other SR methods, we select the equation with the lowest NMSE in practical evaluation.
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where r; denotes the model's natural language reasoning at step i. Ti S T is the set of tools invoked at step i, and o, is the observation received after executing the tools in T. We also set a maximum number of interaction turns M for the trajectory to avoid excessively long inference times without improved performance.
Tool Design We provide a code interpreter as the primary tool, enabling the agent to interact with the data and validate hypotheses. Specifically, we wrap the code interpreter into two common tools for the tasks, data ana yzer and equat inn eva luator, denoted as Ti and T2, respec- tively. For Ti, we link it to the observed data and include a code example in the prompt demon- strating how to access the data. Through this tool, the agent can write and execute diverse code to analyze the data, such inspecting data samples, conducting statistical analysis, residuals between the predicted value and the true value". Notably, we do not constrain the code to these examples, allowing for diverse and emergent analysis patterns. For T2, following the approach of Shojaee et al. (2025a), we design the tool to accept an equation skeleton with placeholders for constants in code format (See Figure
analyzing the
for the equation example). During the execution phase, the BFGS algorithm is implemented to optimize these constants and then report the performance of the equation. It also accepts equations with constants decided through other methods like data analysis. This tool prevents the agent from writing repetitive code for equation evaluation during the long-horizon exploration process.
Experience Buffer While the agent can con- tinue the optimization process over multi- ple turns, the model's limited context length presents a challenge. Moreover, the equations can still perform poorly after long-horizon op- timization. To overcome this, we maintain an experience buffer E the equations the agent has explored, where e; denotes the equation and 8; is its correspond- ing MAPE. At the beginning of each iteration, we fetch the best K equations from the buffer and provide them to the agent as in-context examples. The optimization goal is also up- dated if it was reached in the previous iter- ation. This mechanism effectively bypasses the context length limitation. Additionally, we explored using GP algorithms for experience management but observed no significant im- provement.
Algorithm SR-SCIENTIST inference frame- work.
Input:
Iterations N; maximum turns M; number of fetched equations K; the initial goal G1
I: # Store ranked equations by heap
st)) J.Y to record
2: H +- heap()
3: for i = iori=1,.... L..... do
4:
Generate candidate equations from. IIM
5:
if = 1 then
6:
F + L.M G,)
7:
else
E. (H. (K) M, G.)
end if
10:
H. append
11:
# Check stopping condition and
update goal
12:
if stopping.conditioon (I) then
13:
break
14:
else
15:
G - UpdateGaallG
Stopping Condition and Submission We stop the exploration process when the maxi- mum number of iterations is reached or the equation's 's error is sufficiently small. Then, we select the equation with the best performance on the observed data and submit it for final eval- uation.
16: end if
17:
end for
Output: H. t.topk (1) # Return the best
3.3 TRAINING FRAMEWORK
Training Data Construction Following the approach of Shojaee et al. (2025b), we employ a mixed strategy of rule-based and model-based data synthesis. For each scientific scenario covering multiple variables, we instruct an LLM to synthesize potential relationships between the variables, typically in the form of an equation skeleton. For each skeleton, we use the model to determine the values for the constants based on their physical significance and thereby construct the full equation.
'Since we primarily use a text-only model that cannot accept figures as input, we explicitly instruct the model via prompts avoid generating plots. However, the model hackbone is orthogonal to our framework, which intrinsically supports such plot analysis patterns.
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Once the complete equation is formulated, define appropriate range of values for its variables and synthesize the observed data accordingly. This dataset is then split into training and evaluation sets. The training data is accessible to the agent during rollouts, while the evaluation data is used to measure performance and calculate the reward.
Reward Design In the rollout process, the LLM agent instructed to find an equation that satisfies the precision specified by a MAPE goal Sgoal- Then, the LLM agent interacts with the environment and generates multiple equations. To simplify the RL infrastructure, the agent conducts the opti- mization process for one iteration. Unlike math or code tasks that typically assign binary rewards based on the outcome (DeepSeck-AA AI et al., 2025; Zeng al., 2025b), the performance of equations can be measured by continuous metrics such as MAPE. This makes it possible to assign continuous rewards to avoid reward sparsity. Since we only submit the best equation at inference (Petersen et al., 2021), we select the best equation from those explored and use its score for the calculation. We employ log-linear reward function that maps & to the range [0, 1] as follows:
ig 5max ig
= clip(
0, 1)
lg 8max
lg 8ynal
where smax represents the maximum MAPE for which a non-zero reward can be gained. Addition- ally, explore other reward functions and present the discussion in Appendix C.2.
Training Algorithm We apply the Group Relative Policy Optimization (GRPO) (Shao et al., 2024) algorithm for optimization. To encourage exploration, we omit the KL penalty term against a reference model and observe that this leads to faster convergence and comparable performance. Specifically, for each question q, we sample a group of outputs (01, .2. .oo) from the old policy Toai and then optimize the policy by maximizing the following objective:
JGRPO (0) 18gg ((Q),oo.)E
Tota (O]9)
G
(T(0:1g) Ad, clip
To(0i19) - e, (e)+) A1 4))
(3)
Σ
T
min
Toara (o:lg)
Ta (0:1g)
where E is the hyper-parameter, and A₁ is the advantage computed using a group of rewards corre- sponding the outputs within each group.
4 EXPERIMENTS
4.1 SETUP
Dataset Due to the vast corpus that L.LMs have been pretrained on, the evaluation dataset must be carefully curated to prevent LLMs from having memorized the equations. We evaluate methods on the synthetic part of the LLM-SRBench (Shojaee et al., 2025b), denoted as LSR-Synth', combines known terms in the underlying equation with synthetic, novel terms to create problems that go beyond memorization. Furthermore, all equations and visualizations of their generated data points were verified by two subject matter experts to ensure scientific rigor. It contains 129 problems, spanning four scientific domains: chemistry (36), biology (24), physics (44), and material science (25). For each problem, it has three types of datasets: training set accessible the SR method, an in-domain (ID) test set, and an out-of-domain (OOD) test set.
Evaluation Metrics We use accuracy-to-tolerance as our main metric. We find it serves as more robust metric when aggregating the overall results covering multiple problems than others, like Normalized Mean Squared Error. is also more challenging than other metrics like R² Please refer to Appendix D. for detailed analysis. Given a desired tolerance threshold the metric calculates whether the predicted values and the ground truth values satisfy it as follows:
Acc, =1
max ISSS Nise
ji
Vi
< T
(4)
*We also considered LSR- Transform (Shojaee et al., 2025b) as candidate for the evaluation dataset. How- ever, we found that might still have memorization issues, as nearly 50% of the problems achieved sufficiently small after the first iteration.
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Overall
Material Science
Chemistry
Biology
Physics
Method
Acce.0
Accumm
Accim
AcCaui
AcCa,01
Acceu01
Acca.1
Accuei
Without LLMs
GPLcam
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
E2E
0.26
0.00
1.33
0.00
0.00
0.00
0.00
0.00
0.00
0.00
NeSymReS
3.10
0.78
8.00
4.00
2.78
0.00
4.17
0.00
0.00
0.00
DSR
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
29,46
12.40
25.00
29.17
6.82
PySR
29.46
14.47
53.33
22.67
25.93
11.11
16.67
6.95
25.76
16.67
Qwen3 Coder-480BB 4---4-8888----- BAMB--A3BB--sseee
LaSR
11.89
7.49
13.33
16.67
9.26
6.95
6.95
9.85
6.82
LLM-SR
41.08
18.09
80.00
52.00
36.11
9.26
30.56
18.06
28.79
6.06
SR RR-SCIENTIST
49.09
24.55
86.67
69.33
40.74
5,56
50.00
26.39
34.09
13.64
GL.M4.55-Ar
LaSR
14.21
8.53
18.67
12.00
16.67
10.18
11.11
5.56
11,36
6.82
LLM-SR
35.92
14.47
61.33
38.67
30.56
3.71
18.06
4.17
35.61
15.15
SR RR-SCIENTIST
48.32
25.06
81.33
70.67
45.37
11.11
40.28
16.66
36.37
15.15
GPT-OSS- //00
LaSR
16.02
10.08
20.00
12.00
18.52
12.96
9.72
5.56
15.15
9.09
LLM-SR
28.16
11.37
64.00
38.67
22.22
3.71
11.11
21.97
SR-SCIENTIST
63.57
49.35
74.67
60.00
81.48
64.81
66.67
43.05
40.91
34.09
GPT-OSS-200
LaSR
12.66
8.53
14.67
8.00
15.74
12.96
11.11
6.95
9.85
6.06
LLM-SR
33.33
12.92
70.67
46.67
28.71
5.56
22.22
2.78
21.97
5.30
SR- RE-SCIENTIST
42.64
23.00
62.67
40.00
49.07
23.15
34.72
19.44
30.30
15.15
LaSR
12.66
8.27
20.00
10.67
17.59
12.04
8.33
6.95
6.82
4.55
LLM-SR
24.55
7.24
29.33
6.67
30.55
10.18
15.28
5.55
21.97
6.06
SR-SCIENTIST
32.30
16.02
81.33
52.00
22.22
5.56
22.22
8.33
18.18
8.33
RL
40.92
20.69
85.33
65.33
37.38
7.46
29.17
11.11
25.00
11.37
Table 1: Comparison of SR-SCIENTIST and SR bascline models on different scientific benchmark
problems measured by Acco.01 and Acco.001-
We discard the 5% worst predictions prevent outliers due to the presence of the max operator, following Biggio et al. (2021); Kamienny et al. (2022). Additionally, we present symbolic accuracy, which checks if the discovered equations identical to the ground truth equations. We present the details for the symbolic accuracy calculation in Appendix D.6.
Baselines We comprehensively compare our methods against several baseline methods. For meth- ods without LLMs, we include GPLearn, E2E (Kamienny et al., 2022), NeSymReS (Biggio et al., 2021), DSR (Petersen al., 2021), uDSR (Landajuela et al., 2022), and PySR (Cranmer, 2023). For the LLM-based methods, we include LLM-SR and LaSR for comparison. For methods without LLMs, we constrain the number equation candidates for each problem to 100,000. For methods with LLMs, constrain the LLM calls to 1,000, with at most 1,000 equation candidates from each problem. For SR-SCIENTIST, we set the maximum number of turns per iteration to 25, with a total of 40 iterations. We present the configuration details of the baseline methods in Appendix and the configuration of SR-SCIENTIST in Appendix B. We evaluate our method with different mod- els backbone LLMs, including Swen3-Coder-480B-2 et al., 2025a), GPT-OSS- (OpenAI, 2025), GPT-OSS-20D and Swen3-Coder-30B-AA computation costs and training infrastructure limitations, we use only Qwen3-Coder: Codeerrrrr--BBB for RL training, while the other models are used for inference only. However, we also verify the effectiveness RI. training on other models and present the results in Appendix C.4. For each experiment, we repeat it three times and report the average value to reduce noise.
(Qwen, 2025), GLM-4.5 --ir (Zeng
Due
Training Details For training data construction, we synthesize training data for the scientific sce- marios in LSR-Sy To prevent data contamination, authors of this paper independently and carefully checked the equation skeletons to prevent duplicates and confirmed their agreement. Fi- nally, obtained 1024 problems for RL training. Please refer Appendix C. for details. During

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Method
Dataset
PySR
00D
uDSR
ID
80-
LLM-SR
SR SR-Seientist
ACCuu
20-
Material Science
Chemistry
Blology
Physics
Overall
Figure 2: Detailed results of in-domain (ID) and out-of-domain (OOD) performance using Acco.ot across various LSR-Synth scientific domains. (with Qwen3-Coder-480B as LLM backbone)
PySR
UDSR
Method
SA
SR-Scientist (Qwen)
LLM-SR (Qwen)
SR-Scientist (GLM)
LLM-SR (GLM)
uDSR
0.77
SR-Scientist (GPT)
LLM-SR (GPT)
PySR
4.65
LLM-SR (GLM)
5.43
80
LLM-SR (GPT)
4.65
30
LLM-SR (Qwen)
5.43
AcCouss
SR RR-SCIENTIST (GLM)
7.75
PR-SCIENTIST (GPT)
7.00
20
SR-SCIENTIST (Qwen)
7.00
10
Table 2:
Symbolic accuracy (SA)
0.01
0.05 Noise Level
0.1
of different methods LSR-Synth. GLM, GPT, and Qwen represent GLM- 4.5-Air, GPT-OSS- 120B, and Qwen3- Coder-480B, respectively.
Figure 3: Noise robustness analysis on LSR-Synth. Qwen, GLM. and GPT represent Wwen3-Coder-480B, GLM-4.5-Air, and GPT-OSS- 120B, respectively.
rollout, we set the maximum number of turns to 20 and train for steps. We set Smax to 100% and Sgoal to 0.1 % in the reward design. The detailed training configure can be found in the Appendix C.3.
4.2
MAIN RESULTS
Precision Table presents the accuracy results. In terms of overall accuracy, SR- SCIENTIST consistently outperforms the baseline methods, with four of the models achieving an absolute per- formance margin of 6% to 35% over the baselines. Notably, when using GPT-OSS- as back- bone, SR-SCIENTIST achieves the highest overall performance, with an Acco.01 of 63.57% and an Acco.001 49.35%. In terms of performance across different subjects, with backbones such as Qwen3-Coder-4800. GLM-4.5-Air. and GPT-OSS-120B. ER-SCIENTIST surpasses other LLM- based methods in all subjects. For Qwen3-Coder-300 its performance compared to the original model in all subjects, highlighting that the agent can evolve its abilities through its own experience. We provide further analysis the LLM backbone for RL training in Appendix C.4 and the computation cost discussion in Appendix D.2.
end-to-end RL training significantly improves

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Published as a conference paper at ICLR 2026
PO10
Ground Truth
PySR
3.%(() 0uS3:(j 0.8sin)))
(0.04944191
0.7894445mmm( sini
EOSSEO SONOSS
SR-Scientist (Qwen3-Coder-480B)
LLM-SR (8wen3-Codar-4003)
def Rouatiion parame]:
der sevation (s. parsee)
oulpu: parass [D) puranu x
outple pazaral [0]"x pacass Tam
parezs [2]
pazana (2) top. (paciana (3)
paccaro [3) 2p. [4]
pazond [4] "ap-expp exp (-paicamo 21
(arane(51)
return outgus
outgaa
60))
3399 2:(f) 0.05000 + 0.8innf)
-4966t 1185 6.020oo (500) 0.79%6ithh--mm
PO37
Ground Truth
PySR
11xx(()) - 1.92e)) 0.72xin(() 0.I D.08Ge
1.09900 0.C528)
110-a:000----120000088-------ppp 1,136
SR-Scientist (GPT-OSS-120B)
LLM-SR LLM-SR (GPT-OSS-120E)
def eglation parses):
V, parans):
xecuzn (-1.73
phi, d, kappa, qazza pacsas Leff
+0.22 np-ein(t)
rostinear -beta xe*3 gamms ye45 =k nanlinwer
-0.14
-0.086 v*13)
173xlt) 072aintt) 0.14x())
(0
- (1.725 0.13%-8--00 0.047-2093 0.25440117 1.918-100
0.716+508( 6.28) C.0000117
Figure 4: Equations discovered for the PO10 and PO37 physics problems. The variables i(t), t, x, and represent acceleration in non-linear harmonic oscillator, time, position, and velocity, respec- tively. Terms highlighted in green are common to both the predicted and ground truth equations.
OOD Performance Figure 2 shows the performance the discovered equations on OOD data. For material science, the performance of all methods improves when shifting from ID OOD data, while for other subjects, the methods exhibit varying trends. Among them, SR-SCIENTIST still achieves the best performance on OOD data, demonstrating its strong generalization capabilities. The results of other models are presented in Appendix D.3 and show similar trends.
Robustness to Noise To test the robustness to noise in the observed data, we add Gaussian noise with different standard deviations (a = (0.01, 0.05, 0.1)) to each training data point the model has access to and report the overall performance. As shown in Figure 3, while the performance of all methods drops as the noise level increases, SR-SCIENTIST consistently performs better than other methods, especially with Dwen3-Coder-480B and 4LM-45-Air LLM backbones.
Symbolic Accuracy Table 2 shows the symbolic accuracy of representative methods, evaluated on all problems the dataset. Overall, it challenging to identify equations identical to the ground truth from observed data. Among these methods, ECIENIINTT achieves the best performance, correctly identifying the most equations. For a more qualitative assessment, Figure 4 presents the discovered equations for the PO10 and PO37 physics problems, which are related nonlinear oscil- lators. For both problems, SR-SCIENTIST identifies the structure of the equation and its constants, while the equations from other methods are usually more complex and less accurate. Additionally, SR-SCIENTIS produces trajectories that illustrate its derivation process, providing information that can inspire human scientists design better equations.
4.3 ANALYSIS
Ablation studies Table presents the results of our ablation studies on the data analysis and experience buffer components. The findings indicate that allowing the agent to perform data analysis crucial for performance, particularly for the GPT-OSS- 120B and Qwen3-Coder-4805 models. Moreover, the experience buffer plays important role in the long-term memory of the agent,
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Method
Accp.01
AcCp.001
SR- SCIENTIST (GPT)
63.57
49.35
w/o
35.66
16.28
w/o experience
57.36
41.86
Twen3-Coder--ee
w/o top-k
58.14
41.86
SPT-0SS-1208
SR-SCIENTIST (Qwen)
49.09
24.55
w/o Vi
41.08
14.73
w/o experience
35.66
16.28
ACCo.01
33
w/o top-k
26.36
13.70
SCIENIINTIT (GLM)
48.32
25.06
w/o Ti
46.51
22.48
w/o experience
37.21
18.61
45
w/o top-k
41.96
21.71
40
Table 3: Ablation studies. In 'w/o Ti the agent can not utilize the data anal yzer tool. In 'w/o experience", the agent can not utilize the experience buffer and optimize from scratch for each iteration. In 'w/o top-k', we randomly sample equations from the experience buffer in- stead using top-k sampling. GPT, Qwen, and GLM represent GPT-OSS- PEF-OSS--10B,, Qwen3- Coder-48OB, and GLMM-.5-Air, respectively.
20
Maximum Tums
Figure 5: Overall performance under different maximum turns. We keep the total number of LLM calls at around 1,000 and trade off be- tween maximum turns and iterations.
Qwenn3-Code-44808
80
GLM-4.5-Air
Qwen33Coder-300
Qwen3-Coder (RL)
GPT-055-20D
Percentage
GPT-OSS-120B
40
Equation Evaluation
Data Statistics
Residual Error Analysis
Data Print
Figure 6: The tool call behaviors of different models.
allowing it to utilize the experience of top hop-performing equations from previous iterations. Either directly removing the experience buffer or randomly sampling equations from it leads to significant performance drop.
The Effect of the Turn Number We fixed the LLM calls at around 1,000 and traded off between maximum turns and iterations investigate the effect of different exploration lengths at each iter- ation. Increasing the maximum turns from 10 to 25, as shown in Figure 5, significantly improves performance, highlighting the value of long-horizon optimization. However, performance stagnates or slightly drops beyond 25 turns. indicating that excessively long exploration does not bring addi- tional performance and that the inference budget should be allocated initiating new iterations.
Tool Call Analysis We analyzed the distribution of tool call types and instructed LLMs classify the data analyzer behaviors, presenting the results Figure 6. Overall, the Qwen and GLM families exhibit similar patterns, with around 80% of their calls dedicated to equation evaluation and 20% to data analysis. Within the data analysis calls, data statistics generally account for a higher percentage, showing the importance of calculating relevant values such as correlations and averages.

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Published as a conference paper at ICLR 2026
In contrast, GPT-OSS-20B and GPT-OSS-I20E tend directly write their code to perform residual error analysis for more fine-graine aee-grained information. Further case studies show that GPT-OSS- 120B also tends to directly define equation constants through data analysis, demonstrating greater flexibility. Moreover, after RI training, the Qwen3-Coder-300 also increases its use of data statistics for more advanced data analysis.
LIMITATIONS AND FUTURE DIRECTIONS
While SR-SCIENTIST demonstrates significant advantages in equation discovery, identify sev- eral limitations that open avenues for future research. First, in our main experiments, we only used text-only models. Expanding the model to incorporate multi-modal inputs and include plots in the analysis phase is promising direction. Second, in noisy scenarios, although SR-SCIENTIST performs best, still suffers from significant performance drop. Future work could explore incor- porating common noise-hanae criie-handling strategies into the agent's prompt to guide its data analysis. Third, since we only provide the top-performing equations to the agent in each iteration, the agent may still re-explore equations that previously performed poorly. Optimizing the agent's s memory system, such as by distilling high-level experience from the equations, is an interesting future direction.
CONCLUSION
In this paper, we introduce SR-SCIENTIST, a framework that transforms the LLM from a passive equation proposer into autonomous scientist for symbolic regression. By analyzing data, evaluat- ing and refining equations, the agent generates and refines hypotheses through active environmental interaction. Our experiments show that SR-SCIENTIST significantly outperforms existing methods in precision, generalization, robustness to noise, and symbolic accuracy. Furthermore, we develop complete reinforcement learning pipeline that allows the agent to self-evolve and enhance its dis- covery capabilities.
ACKNOWLEDGMENTS
We would like to express our great gratitude to Yan Ma for assisting with the writing and to Xuefeng Li for providing advice RL Infra. This work partially funded by the National Natural Sci- ence Foundation of China (62476168) the National High Technology Research and Development Program of China (2015AA01 5408), the Shanghai Science and Technology Development Funds (14ZR 1403200), and the AI for Science Program of the Shanghai Municipal Commission of Econ- omy and Informatization
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APPENDIX
A BASELINE IMPLEMENTATION DETAILS
GPLearn
The open-source splearn? library is a Python package built on scikit-learn that uses genetic programming to perform SR. We set the population size 500, the number of generations to 200, and the tournament size to 20. The function set includes addition, subtraction, multiplication, division, sine, cosine, square root, logarithm, absolute value, negation, and inverse.
End-to-End Symbolic Regression (E2E) E2E (Kamienny et al., 2022) is a method that uses single deep learning model, often a Transformer, to directly predict a complete mathematical expression, including both its structure and numerical constants. We implement the method using the symbol i wwwgressiioooooooooo Facebook repository For this method, set the maximum input points to 200 and the number of trees for refinement to 10.
NeSymReS NeSymReS (Biggio et al., 2021) is deep learning approach that uses pre-trained Transformer model to directly predict a mathematical expression from data points. We use the Neura 15mbol cRegress i on Thatttcalll es* repository to implement this method. We set the number of data points passed to the Transformer 500 and the beam size (expression sampling size) to 32.
Deep Symbolic Regression (DSR) DSR (Petersen al., 2021) is deep learning approach that uses a recurrent neural network and reinforcement learning to search for the math- ematical expression that best fits given dataset. We implement the method using the deep- synbolic-ooggoooo imi mniiatioon ion repository. For this method, we set the number of samples 100,000, the batch size to 512, and the learning rate 0.0005. The function set includes addition, subtraction, multiplication, division, sine, cosine, exponentiation, and logarithm, with expression lengths ranging from 1 to 9.
Unified Deep Symbolic Regression (uDSR) uDSR (Landajuela et al., 2022) is a deep learning framework that unifies multiple symbolic regression strategies into single modular system. It ex- tends DSR by combining approaches like neural-guided search, pre-training, and linear models. We implement it using the deep-symbol immooiccopt... imi zat i repository. For the genetic program- ming configuration, we set the population size to 100, generations to 20, crossover probability to 0.5, mutation probability 0.5, tournament size and maximum mutation tree depth 3. Other parameters are the same as DSR's. s. The function set is: add, sub, mul, div, sin, cos, exp, log, and poly.
http::///gtthin ctm http://hhw.....ooooooceeeeeeeete Symbol cieerres Thats
cale las httpp:://gi whtgggtthhb..gogg wttpttttcooooooorggg///ooo/oo detps/ybooppeeeoloeeeeeoo 1 c- imi z
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Python Symbolic Regression (PySR) The PySR Python package" (Cranmer, 2023) uses pow- erful Julia backend to find simple, interpretable mathematical expressions that best fit the observed data. We set the number of iterations to 125, cycles per iteration to 550, populations to 15 with population size of 33, maximum size to 30, and the randomization weight to 0.1. Its binary operators +, - / pow, and its unary operators are exp, log, sqrt,sin, cos.
Library-Augmente Symbolic Regression (LaSR) LaSR (Grayeli et al., 2024) uses LLMs to discover and evolve library of abstract, natural-language concepts. We implement the method using the set the number of iterations 25, cycles per iteration to 550, populations to 10 with a population size of 33, maximum size to 30, and the maximum number of concepts to 20. The LLM operation weights for crossover, mutation, and randomization are each set 0.02. The binary operators, unary operators, nested constraints, and other constraints are the same those in PySR.
cRegression on ji repository. For this method, we
LLM-SR LLM-SR (Shojace et al., 2025a) leverages the extensive scientifie knowledge and code- generation capabilities of LLMs to discover mathematical equations. It treats equations as programs and combines an LLM's scientific priors with an evolutionary search to find accurate and inter- pretable formulas. We implement it using the 11m- immmsrrbnch" repository with its default configu- ration and set the maximum number of LLM calls to 1,000 for each problem.
B DETAILS OF SR- TR-SCIENTIST INFERENCE FRAMEWORK
For each iteration, we set the initial MAPE goal 0.1% and the termination MAPE threshold to 0.0001% %. For each LLM, we set the sampling temperature to 0.7 and the maximum completion length per call to 8,192. For the equation evaluator tool, to control the length and the com- plexity of the generated equations, we set the maximum number of parameters to 10 in all experi- ments, following Shojaee al. (2025a). To ensure fair comparison with methods like LLM-SR, the optimization goal for the BFGS algorithm is the mean square error. For parameter optimization, we use the sci scipy py library for nonlinear optimization with the BFGS algorithm. The prompt for the LLM agent is shown in Figure 15 and Figure 16.
C DETAILS OF -RSSCIENTIS TRAINING FRAMEWORK
C.I TRAINING DATA SYNTHESIS
We synthesize our training data following the approach of Shojace et al. (2025a), using a hybrid model-based and rule- busebbseed method. The problems cover four scientific disciplines: materials sci- ence (Stress with respect to Strain and Temperature), chemistry (Reaction rate with respect to Time and Concentration). biology (Growth rate with respect to Time and Population size), and physics (Acceleration with respect to Time, Displacement, and Velocity). The detailed procedure is as fol- lows:
(1) Equation Skeleton Generation: Each equation skeleton contains at least one known and one novel term. The known terms refer to common concepts from an LLM's pre-training knowledge, and the novel terms refer to terms outside the LLM's prior knowledge. We synthesize both types of terms by prompting Claude 4 Sonnet. These terms are then combined into equation skeletons using additinn. To ensure a moderate level of complexity, each equation is limited to 2-4 total terms. To prevent potential data leakage between our training set and the benchmark data, two authors of the paper independently identify equations that are identical or too similar and discuss them for further reconciliation. Any equations deemed too similar are discarded.
(2) Parameter Instantiation: For each equation skeleton, transform it into a complete equation by assigning values to each constant. This process is not a random assignment; rather, it is
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Published as a conference paper at ICLR 2026
Overall
Material Science
Chemistry
Biology
Physies
Method
Acce.01
Acco,00
Acca.u
Acc
Accau1
Ace9.001
Accom
Accwum
Accu.a1
Actumm
RE-ECIENTIST
32.30
16.02
81.33
$2.00
22.22
5.56
22.22
8.33
18.18
8.33
RI. (ll((lo-iinear)
40.92
20.69
85.33
65.33
37.38
7.46
29.17
11.11
25.00
11.37
RL (slepwise)
37.21
17.83
84.00
56.00
27.78
2.78
33.33
16.67
20.45
9.09
Table 4: The performance of different reward functions. 'log-linear' refers the reward function corresponding to Equation 2 and 'stepwise" refers to the reward function corresponding to Equa- tion 5.
guided by the scientific context and significance of each term. For example, in the context of material science, the reference temperature To is set to 273.15 K (the triple point of water). We instruct the LLMs to perform value assignment, which is then followed by human validation.
(3) Data Poins Generation: For the material science equations, which represent static systems, we generate data points by uniformly sampling across defined range of parameters. The temper- ature sampled from the range of 273 K, 573 KJ. and strain e is sampled from [O, 0.6]. A total of 5000 points are generated using evenly spaced grid. The OOD test set is created by taking the 500 data points with the highest temperature values. The remaining 4500 points are then randomly shuffled, with 500 points allocated to the test and the rest forming the training set. For the chemistry, biology, and physics equations, which represent dynamic sys- tems, we use the solve_ivp function from the SciPy Python package with the RK45 method to solve the differential equations. Given initial conditions at t = O, we solve the equations to obtain the relationship between variables over time (equations that cannot be solved are simply discarded). We generate 5000 data points by uniformly sampling the time variable from the range of [o, 60). The OOD test set is created by selecting the 500 data points with the highest time values. The remaining 4500 points are randomly partitioned into 500-point test set and the remaining training set.
(4) Filtering and Final Dataset Assembly: the filtering stage, we compute the statistical properties of the resulting data. Equations with data points that are scientifically anomalous are filtered out. This process ensures that our final dataset contains only scientifically meaningful equations with plausible data points. Finally, the instantiated equations, their corresponding numerical data points, and other relevant information (variable symbols, equation names, etc.) are packed for the subsequent training.
C.2 REWARD DESIGN
Besides the log-lineat reward design, we also explore the stepwise reward function as follows:
1.0
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0.5
0.001 I < 0.01
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0.25
if < 0.1
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where denotes the MAPE value. better than the stepwise function.
As shown in Table 4, the log-linear reward function performs
C.3 TRAINING DETAILS
We conduct RL training on 32 NVIDIA H200 GPUs. For the infrastructure, we use the verl 13 framework, employing SGLang as the rollout engine and FSDP the training engine. For rollouts, we implement batch-level asynchronous rollouts to reduce the rollout time. We set the temperature to 1.0, the maximum response length to 10,240, and the maximum number turns to 20. We use
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Published as a conference paper at ICLR 2026
M h Mmv
mmp
0.35
9500
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90.00
0.75
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8500
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0.15
8030
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
Stap
Stap
The change in response length during training.
Figure 7:
Figure 8: The change in reward score during training.
Overall
Material Science
Chemistry
Biology
Physics
Method
Acce.1
Accanet
Accau
Accuan
AcCaui
AcCuan
Accu.i
AcCa.00
Acca.1
Accure
Qwent-332
LLM-SR
30.49
9.04
5.56
15.28
29.55
9.09
SR-SCIENTIST
29.46
15.50
72.00
48.00
22.22
8.33
20.83
4.17
15.91
9.09
+RL
39.53
20.16
80.00
64.00
38.89
8.33
25.00
4.17
25.00
13.64
Qwen3-14B
LLM-SR
27.91
9.30
$6.00
32.00
16.67
2.78
8.33
0.00
31.82
6.82
SR 3R-SCIENTiST
18.60
6.98
48.00
"8.00
13385
0..00
12.50
4.17
9,09
2227
RI. 23.26
10.08
52.00
40.00
16.67
2.78
20.83
0.00
13.64
4.55
Qwen3-8B
LLM-SR
22.48
8.53
48.00
28.00
16.67
5,56
8.33
0.00
20.45
4.55
SR BR-SCIENTIST
13.30
4.05
40.00
16.00
8.35
5.36
16.67
0.00
6.82
00.00
RI. 20.93
9.30
68.00
44.00
16.67
0.00
4.17
0.00
6.82
2.27
Table 5: Performance comparison of different LLM backbones for RL training, including Qwen3- 8B, Qwen3- 14B, and Qwen3-32B.
a prompt batch size of 32 and generate rollouts per prompt. The KL loss coefficient set to 0 to encourage exploration. The remaining parameters the default settings.
Figures 7 and 8 present the training dynamics. As shown, the reward progressively increases and saturates around 60 steps. The response length also increases, demonstrating that the model leams a long-horizon problem-solving strategy.
C.4 THE EFFECT OF LLM BACKBONE
To further investigate the effect RL training on different models, we conduct RL training on Qwen3-8B, Qwen3-14B, and Qwen3-32B models using the same training configuration detailed in Appendix C.3 and present the results Table 5. All models achieve significant performance gains after RL training, with most improving their performance across all subjects. When comparing LLM-SR and SR-SCIENTIST without additional RI. training, DR-SCIENTIST outperforms LLM- SR on Acco.01 and matches it Acca,1 when using Qwen3-32B as the LLM backbone. However, it lags behind when using the Qwen3-8B and Qwen3-14B models. This highlights the importance of model's intrinsic ability in a framework that provides the model with greater autonomy.
D SUPPLEMENTARY ANALYSIS AND EVALUATION DETAILS
D. ANALYSIS FOR THE EVALUATION METRICS
Table 7 presents the results measured by the R² metric. As shown, the differences between the vari- ous methods are small, making it difficult to distinguish their performance. Additionally, we present the NMSE results in Table 6. The average NMSE is sensitive maximum values, which makes it
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Published as a conference paper at ICLR 2026
Method
NMSE (Average)
NMSE (Max)
NMSE (Min)
NMSE (Medium)
LaSR
1.53e-02
1.67c-01
2.66e-13
1.84c-03
LLM-SR
3. 3.199-03
7.83e-02
1.72e-14
8.87e-06
SR-SCIENTIST
1.29e-03
4.06e-02
1.57e-14
2.64e-06
Table 6: Comparison of SR- RR-SCIENTIST and LLM-basee SR baseline methods measured by NMSE.
We report the result of Ppwe3-CCoder-405 on the physics subject.
Method
Overall (R*)
Material Science (R*)
Chemistry (R")
Biology (R²)
Physics (R²)
LaSR
0.991
0.998
0.996
0.988
0.985
LLM-SR
0.998
0.996
1.000
1.000
0.997
SR- SCIENNITSS
0.998
0,996
1.000
0.995
0.999
Table 7: Comparison of SR- PR-SCIENTIST and LLM-based SR baseline methods measured by R We report the result of Qwen3-Coder-480B.
an unreliable indicator of overall performance. Although quantitative methods could be employed, such as truncating errors that exceed a certain constant, this threshold is difficult to define. Using the median value is also insufficient, as cannot account for the entire performance distribution. Therefore, we abandoned these metrics and instead used socuacy-to-tolerame as the main metric.
D.2 COST
For LLM-based methods, the cost mainly comes from API calls to LLMs. In Table 8, we present the estimated cost of SR-SCIENTIST with GPT-OSS- 120B and GPT-OSS-200 as the LLM backbones. We calculated the cost for each problem based on token consumption and common API pricing. As shown, the cost is acceptable for practical usage, and when considering cached tokens, the price can be further reduced. Additionally, we deployed GPT-OSS-1200 H100s for batch-level inference on 129 problems and recorded the wall-clock times. The maximum time more than 5 hours, duration that is acceptable for practical usage, typical scenarios involve only a few problems.
local server with 2 NVIDIA
D.3 OOD PERFORMANCE OF OTHER MODELS
We illustrate the OOD performance using other models as LLM backbones in Figures 9, 10, 11, and 12. For models including GLM-4.5-Air, GPT-OSS-120B, and GPT-OSS-20B, SR-SCIENTIST consistently outperforms the other methods in overall accuracy and most subjects. For Qwen3- Coder-3OB, slightly lags behind the method PySR on the OOD data. After RL training, not only enhances its performance on ID data but also on OOD data, showing the generalization of the discovered equations.
D.4 OTHER QUALITATIVE RESULTS
To demonstrate how the agent gains insight from data analysis and long-horizon exploration, we provide two snippets of the agent trajectories in Figure 13 and Figure 14. As shown by the reasoning process in the dashed box, the agent forms its hypothesis for the equation based on its analysis of the data through code, along with the performance of historical equations. This analysis grounded in the data helps decrease the search space compared to previous methods. The reasoning and analysis process can also inspire human scientists towards better equation design.
D.S SENSITIVITY ANALYSIS OF HYPERPARAMETERS
We present sensitivity analysis of the hyperparameters in Table 9 and Table 10, covering the num- ber of equations fetched from the experience buffer and the initial MAPE goal. Regarding the num- ber of fetched equations, method remains robust when this value is relatively small. However,
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Published as a conference paper at ICLR 2026
Method
API Price
API Cost Per Problem
SR-SCIENTIST (GPT-OSS-I22) 120B))
Input: M0:05/IM; Output: $0.25/IM
$0.25 $0.1
SR-SCIENTIST (GPT-OSS-2000 2002
Input: $0.03/1M: Output: $0.15/IM
Table 8: The estimated cost of SR- DR-SCIENTIST. This calculation does consider cached tokens; including them would reduce the cost further.
PySR
000
SR--ccientist
Acdim
Material Science
Chemistry
Blology
Physics
Overall
Figure 9: Detailed results of in-domain (ID) and oft-o--domain (OOD) performance using Accp.01 across various LSR-S LSR-Synth scientific domains. (with GLM-445- as LLM backbone)
K
Accp.01
Acco.00 Ac11.001
8gaal
AcCo.01
Acco.001
1
63.57
49.61
0.1
64.34
48.84
3
63.57
49.35
0.01
63.31
49.61
5
63.31
50.90
0.001
63.57
49.35
10
61.50
47.29
0.0001
51.74
37.70
Table 9: Sensitivity analysis of the number of equations K fetched from the experience buffer. (with GPT-OSS- 120B as LLM backbone)
Table 10:
Sensitivity analysis of the initial MAPE goal 8gual- (with GPT-OSS-120B as LLM backbone)
performance declines slightly when is increased to 10. This suggests that including a large number of fetched equations in the context window may strain the model's capability. Regarding the initial MAPE goal, performance is insensitive to relatively easy targets, as our dynamic goal update mech- anism effectivvly prevents trivial goals. However, an extremely challenging goal (e.g.. 0.0001) leads to degraded performance, causing the model to become trapped during the optimization process.
D.6 SYMBOLIC ACCURACY CALCULATION
We calculate symbolic accuracy using two-step evaluation strategy: an initial assessment by an LLM. followed by human verification. Specifically, we instruct GPT-OSS 120B to determine the equivalence between a predicted equation and ground truth counterpart using the prompt shown in Figure 17. To ensure reliability, the LLM evaluates each problem 10 times. Our preliminary studies on 121 cases reveal a 98.3% consistency rate between LLM and human evaluations for instances where the 10 voting results were unanimous. Therefore, practical workflow involves first using the LLM for evaluation. However, for any case where the LLM's 's votes are inconsistent, pass the problem to human evaluators for final decision.
D.7 TOOL CALL ANALYSIS
We instruct GPT-OSS- 120B to perform tool call analysis using the prompt in Figure 18. We check 54 cases and find 94.4 % consistency between human and L.LM-based evaluation.
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Published as a conference paper at ICLR 2026
E THE USE OF LARGE LANGUAGE MODELS
The large language model was utilized as a writing assistant during the preparation of this manuscript. Its role was strictly limited proofreading for grammatical errors, improving sen- lence structure, and enhancing readability. The large language model was not used for generating any core ideas, methodology, or scientific content. The authors take full responsibility for all content presented.
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Published as a conference paper at ICLR 2026
PySR
2222
OOD
SR-Scientist
Acti
Material Science
Chemistry
Biology
Physics
Overall
Figure 10: Detailed results of in-domain (ID) and out-of-domain (OOD) performance using Acco.01 across various LSR-Synf scientific domains. (with GPT-OSS-1200 LLM backbone)
140
Methed
Detaset
LLM SR
3R-Scientist
Atten
Biology
Physits
Figure 11: Detailed results of in-domain (ID) and out-of-domain (OOD) performance using Acco.01 across various LSR-Synth scientific domains. (with GPT-OSS-20B as LLM backbone)
PySR
COD
LOSSR
SR, Scientintt
Silk Scientiinnttn (RL)
Acti
Material Science
Chemstry
Biology
Physks
Overell
Figure 12: Detailed results of in-domain (ID) and out-of-domain (OOD) performance using Acco.o1 across various LSR-Synth scientific domains. (with Qwen3-Coder-30B as LLM backbone)
21

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Published as a conference paper at ICLR 2026
Agent: Now I1 enalyze the data further to understand relatlonehip better, Let look how the growth changes respect populat
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Figure 13: The snippet of the agent trajectory for the BPG18 biology problem. (with Qwen3-Coder- 480B LLM backbone)
22

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Published as a conference paper at ICLR 2026
Agenz:
No aeed to improve model. Let's inspect data to see relacionshipe. Use data ast_aancyyse to view
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[-0,0:20 0.01094399
Minear coeffs (onsst) (-1.019377996131 -1.01219659e -2.8811-11188880888 2.881435444 03 -6,03552468+ 01]
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Figure 14: The snippet of the agent trajectory for the PO29 physics problem. (with GPT-OSS-120B L.LM backbone)
23

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Published as a conference paper at ICLR 2026
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print (*Success: The absolute
lealler then 0.18.";
print print l"Feilure: The neon absolute percentage error Larger than 0,28.")
equation's goodness of optinizes tind optimal parameters your equation
SciPy's dataset. then provides an output with perfosmance marries (Rean Squared Error, Nornalized Mean Squared Beroc, and Rean Ancolute Forcentag: Error), the success status, and details of any bags. utilizing
entire cunation, including sunction
For the 'data_snalyzer" tool, st is code interpretor that can run your data explorstion snispet. You can sccess the data shown in the exanple.
Rowaver, >>gythor.
fosbidden
any libraries like Matplotiih plotcing fiqures anslysis.
import laon
impart
lood
iapur_data_sta sys.atdin.readf]
sata_list 1son. loads (sapur_data_stts)
print the tirst entries
data_iist, the first value
prediet, and inputs.
Figure 15: System prompt for the agent.
24

---

Published as a conference paper at ICLR 2026
Find the mathenacical function ckelezen that represente Acecleration in Hen--1inear Harmonic dacilla.or, given Fosition by Pine, and Velceilyy Line
Follow these ataps to solve the problen:
wl. Inplement the Equacion in Coders
Bazed on your kneviedge and analysis, idencidy the standard equatior. and imploment it in the coda. 'parans"
likely have (0.g-, "pazans [C]', "pazams [1]', "parana[2] 1) to represent these constanta, as the squacion_evalucca"" taol is designed to opt intinize then, Note that the "parans" 'ist has a fixed size of 10 ("HAX_NPRRRS 10"),
equat
10 pasanelecs in model.
A.?. Teat, Analyse, and Befiners
x Evaluate the eguat iun's goudness of fil using the "eguat oer Laul. You need to pasa the entire cunctior, incleding the fanetion header, to the tool. Bare is an example: hor
equalion|ss op.ndascay, Lt op.rdesray, vi op-ndacray, paranas ng.ndacrayl ng.mderray
Mechenscical function for Acceleration in Nonl-linear Harnonic Oscillator Args:
aunpy acray representing absesval of Pasition ti numpy acrey represerting observations of Time.
vs A manyy scray representing diservat inns of Velocity at time t.
parant: kcray of nuneris conatants paramelera to be sptimized Recurn:
à rumpy arrey represent ing kocelerat
Noni lineer Hamonii Oscil as the result of applying the mathenatica. tunction to the inputs.
putpurt parans
paraus
parens
parans(3)
return output
You modi fy the Junct fan. body, bet the Function beadle most remain unchanged. abeo_ute percentage error Meet ing this condit on indicates that your equation is gond fit for the date.
Your gonl
requce
than 0.10038,
* If this goal ls nut mely te your equat fon in Python and observe its perfusmatio,
You write
exploration. snippet and the 'cata aanaiyzee' tool to execute it, potent
alllowing you inspect
ps
*+3. Subrit Your Final Anewere.
Once you are contident your equation has net the concition, or 15 you conclude atter numercus
atteepts that you cannot it, provide the completed Python Funct fon your
Figure 16: User prompt for the agent.
Siven the ground truth nathemstical expresssen and the hypothesis a, determine it there exist any corszanz parameter values that would make the hypotheeis equi ent to the given ground truth expression on. Rxplain your ressoning and then provide the finel anaver
fat think ssep by H. "reasoning"; "Step-by-step analysie", "answer"; "Yes/So" 13
Ground Truth As (gtuegueet ion) Hyposhesis ipred_ equstionn
Figure 17: Prompt for symbolic assessment.
You are en expert code anslyst. Your task is to classi fy Python code en ppets written by an AT agent that is trying to discover scientitic equations from sata.
Analyze
purpose ot the code snippet assign category assed the detinitsens belex. category refinstions:
Bata Staristics: The coda colculates a staristionl property of the data, Examples inclede
calculacing the nean, variance, correlasion, minimum, or maximum values.
category, you For example: Data staristies: corroistion Data stacistios: Mean.
:MPORTANT:
tornat:
Statisties: (SINTISTIC_MNN))
Bata Print: The code prints zaN eata samples for the purpoce at initial inspection and understanding. common example printing the first feu ross.
Recidual Error Aralysie:
evalcates preposed nathensrical equation colculating ecror netrics deternine ite
recidcale (the diftesence between precicted and actual valuesi goodness of fit to the data.
It the code' a purpose does not fit any she categorias Listed above, create new, concise, and category all casegorics except <Data stacistics', respond wish ONLY the catagory name.
Classity the tollswing Python code onippet. coda snippaz:
[code]
Figure 18: Prompt for tool call analysis.
25


Reviews:
[{"review_id": "N0KEFRVqDh", "reviewer": "Reviewer_zbXv", "summary": "This paper introduces SR-SCIENTIST, a framework for applying large language models (LLMs) to the scientific equation discovery (SED) process. The framework integrates evolutionary search with an LLM-as-heuristic component but offers greater flexibility than prior methods by allowing LLMs to perform a wider range of actions in each iteration. The paper demonstrates that enabling LLMs to interact directly with observational data (via tool use) and to engage in multi-turn interaction improves performance on SED tasks. Additionally, it explores finetuning LLMs with GRPO for this domain.", "questions": "1. Figure 6 shows that Equation Evaluation dominates tool usage. Does this suggest that the agents primarily engage in a “guess-and-check” process rather than deriving insights from observation data through the Data Analyzer tool?\n2. The analysis indicates that increasing the maximum number of turns beyond 25 results in stagnating or slightly declining performance. Is this due to context length limitations, or does it reflect the agent’s inability to explore new strategies (e.g., requiring periodic resetting or reinitialization)?\n3. Could the authors include representative failure cases? A qualitative analysis of typical failure modes - such as difficulties with specific mathematical structures, high noise levels, or certain data domains - would provide valuable insights into the framework’s limitations.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The experiments present strong results across multiple metrics and settings, consistently favoring the proposed approach.\n2. Comprehensive ablation studies highlight the effects of two key components - tool calls and experience buffer - as well as the influence of interaction length and the distribution of tool usage.\n3. The paper provides a detailed discussion of the reinforcement learning (RL) process, including the methodology for constructing synthetic training data and a comparative analysis of different reward function designs.\n4. The writing is clear, well-structured, and easy to follow.", "weaknesses": "1. The contribution in terms of novelty is somewhat incremental. The framework closely resembles prior approaches such as LLM-SR, particularly in its use of an experience buffer, evaluation module, and parameter optimization function. The main components - tool use and multi-turn interaction - can be seen as adaptations of existing agentic LLM paradigms (e.g., GPT-OSS, Qwen3-Coder) to the symbolic regression task, rather than a fundamentally new agentic framework. The RL fine-tuning pipeline largely follows established practices, including synthetic data construction and the use of the GRPO loss.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JdTsqFFm6I", "reviewer": "Reviewer_FL8G", "summary": "This paper presents a novel framework, SR-Scientist, for LLM-based scientific equation discovery. Unlike prior approaches that primarily treat LLMs only as equation proposers within search algorithms, SR-Scientist extends their role to act as agentic AI scientists capable of performing complementary pre- and post-processing tasks such as data analysis, equation implementation, and submission for evaluation. The framework integrates a set of wrapped tools (e.g., data analysis and equation evaluation) that are used in an agentic and long-horizon manner to autonomously navigate the full equation discovery process.\nEmpirical results demonstrate that the proposed approach outperforms existing baselines on standard benchmarks, exhibits improved robustness to noise, and shows better generalization across diverse tasks. The paper also includes experiments with a smaller Qwen-30B model augmented with end-to-end RL fine-tuning which shows their results could even be strengthen with the fine-tuning and adaptation of LLM backbone.", "questions": "Included in the weaknesses section", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- The paper is well-written and clearly motivated, making it an engaging read.\n- The experiments are comprehensive on recent benchmarks and with thorough analysis. \n- The reported results demonstrate substantial performance improvements over state-of-the-art baselines.", "weaknesses": "- The proposed data analysis tool is well-motivated and conceptually sound. However, I am unclear about the necessity of using an LLM as an equation evaluator, given that evaluation in this task is typically data-driven and follows a consistent procedure. Introducing an agentic LLM into the evaluation loop could potentially introduce unnecessary variability or LLM generation errors. I would appreciate clarification on why this component is needed and what advantages it offers over standard predefined evaluation methods for generated hypotheses. Additionally, Table 3 seem to not include an ablation for this tool (T2), making its specific role and impact questioning. \n\n- I think additional qualitative examples would help clarify what the agentic steps contribute beyond the baseline LLM-SR approach. Showing intermediate output such as snippets of generated code or key decision points the agent follows during the discovery process would help to understand how these steps influence the final discovered equations.\n\n- For the results reported in Table 2 and Figure 3, it is unclear whether they are evaluated on all LLM-SRBench tasks or only a specific subset or category. It would be helpful to include this information explicitly in the captions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iQo3nIbksA", "reviewer": "Reviewer_E5Ep", "summary": "The paper proposes SR-Scientist, an agentic large-language-model (LLM) framework for symbolic regression. It combines two tools (a data analyzer and an equation evaluator) in a long-horizon reasoning loop with an experience buffer and a reinforcement-learning (RL) fine-tuning stage based on GRPO. Experiments on the LSR-Synth benchmark across four scientific domains show solid empirical gains over existing SR and LLM baselines, supported by ablations and robustness analyses.", "questions": "- Why does RL underperform? Was training unstable or reward scaling flawed?\n- How is data contamination avoided in LSR-Synth?\n- Try to do an apples-to-apples comparison. Would SR-Scientist still outperform if all methods were given the same compute budget or wall-clock time?\n- Why restrict the framework to two tools instead of adding others, like dimensional checks or simplification modules?\n- What causes the observed drop in performance beyond 25 turns?\n- Can you provide qualitative examples of reasoning trajectories that refine equations step by step?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ The paper is well motivated and clearly framed, presenting the idea of turning LLMs into autonomous “scientists” that iteratively refine hypotheses through data analysis and reasoning.\n+ The evaluation covers multiple domains, five different LLM backbones, and includes analyses for noise robustness, out-of-domain generalization, and symbolic accuracy.\n+ The modular two-tool framework with memory buffering is versatile and adaptable to other discovery tasks beyond symbolic regression.\n+ The empirical results are consistently strong across settings and supported by detailed ablation studies.\n+ The framework provides a practical system contribution that can serve as a foundation for other research works in scientific AI agents.", "weaknesses": "- The work mainly integrates existing components such as tool use, GRPO-based RL, and memory buffering, without introducing new learning principles. The proposed mechanisms are largely heuristic and not deeply analyzed.\n- The RL fine-tuning stage performs worse than inference-only variants and is limited to a single backbone. Its “one-iteration” setup contradicts the long-horizon reasoning concept and lacks convergence or stability analysis.\n- The comparison with baselines is not compute-normalized. SR-Scientist executes up to 25 turns × 40 iterations, while non-LLM baselines are capped at 100k equations, leading to possible resource imbalance.\n- The metric choice emphasizes Accτ with a 5% trimming rule and omits standard regression metrics like RMSE or MAE. Results are averaged over only three runs, which weakens statistical confidence.\n- Symbolic accuracy remains around 7%–8%, much lower than numeric accuracy (>60%), suggesting improved curve fitting but not genuine symbolic recovery.\n- The paper lacks qualitative examples illustrating how the agent refines equations through multi-step reasoning, making the “scientific inspiration” claim speculative.\n- Several parameters and mechanisms, such as K in the experience buffer, the MAPE goal threshold, and the exclusive use of BFGS for optimization, are presented without sensitivity analysis or justification.\n- The dataset evaluation omits LSR-Transform due to contamination concerns, but does not demonstrate that LSR-Synth is free of memorization. There are no experiments on real-world scientific data.\n- Figures are dense, tables inconsistent, and citations incomplete, which affects readability.\n- In summary, the lack of algorithmic novelty, limited RL justification, resource imbalance, and weak interpretability analysis reduce the overall impact. The current version of the paper stands as a well-executed system but needs stronger methodological contributions, fairer comparisons, and deeper reasoning analysis to meet the bar for ICLR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "39tZkF4l5j", "reviewer": "Reviewer_ewF1", "summary": "The paper introduces SR-Scientist, an agentic framework for scientific equation discovery through data-analysis and equation-evaluation tool calling. The authors also develop an RL training pipeline that improves agent performance. The method outperforms baselines on LLM-SRBench [1].\n\n[1] Shojaee et al. \"LLM-SRBench: A new benchmark for scientific equation discovery with large language models.\" (2025).", "questions": "1. Have the authors considered the use of broader tools for data analysis, such data visualization or graphical analysis? Do you think that expanding the framework's tool capabilities could be useful?\n2. Could the authors please elaborate on the mechanisms that allow the framework to obtain a long-horizon behavior?\n3. Do the authors plan to evaluate their method on real-world datasets?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The method is simple and clear. The task of scientific equation discovery is important, and the problem formulation is concise.\n- SR-Scientist consistently outperforms state-of-the-art baselines on all models and across all tested scientific fields. The selected baselines make sense.\n- The paper successfully demonstrates a path forward towards self improvement of the agent through an end-to-end RL pipeline.", "weaknesses": "- The work appears to be heavily inspired by SR-LLM [2]. Several key components, such as the equation-evaluator tool or the experience buffer, closely resemble those introduced in [2]. While the reframing of the problem as an agentic framework is probably useful, it is not a novel idea and has been widely used in many recent works.\n- Results are evaluated solely on synthetic data (LSR-Synth, a synthetic subset of [1]). This limits the assessment of the method for practical applications. Adding experiments on real-world datasets or use cases where data often includes noise, outliers, etc., could provide a stronger evidence. Also, designing new tools to handle such challenges would strengthen the paper.\n- The current design of the experience buffer ablation (Table 3) only confirms that accumulating knowledge over long horizons is better than constantly starting from scratch. Instead, this ablation should isolate the contribution of the suggested fetch-top-K buffer (for example, against a baseline that randomly selects equations).\n\n[2] Shojaee et al. \"LLM-SR: Scientific equation discovery via programming with large language models.\" (2024).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)