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LLM-SR: Scientific Equation Discovery via Programming with Large Language Models
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Introduces LLM-SR, which treats equations as Python programs proposed by LLMs and refined via an islands-based evolutionary buffer and parameter optimization, evaluated on four new benchmarks designed to avoid LLM recitation.
Problem
Symbolic regression / data-driven equation discovery is combinatorially hard and existing methods ignore domain prior knowledge and use restricted representations such as expression trees; Feynman benchmarks may be memorized by LLMs.
Approach
Represent equations as Python program skeletons with a params vector, have an LLM propose hypotheses using scientific problem descriptions and in-context experience from an islands-based evolutionary buffer, and fit parameters with BFGS or Adam against data.
Key results
- On Oscillation 2, LLM-SR (GPT-3.5) reaches ID NMSE 2.12e-7 and OOD NMSE 3.81e-5, versus PySR ID 0.0002 and OOD 0.0098.
- On E. coli growth, LLM-SR (Mixtral) achieves OOD NMSE 0.0037 while all baselines except E2E-level ones exceed OOD NMSE 1 (PySR 1.0141, uDSR 5.4584).
- LLM-SR converges in roughly 2.5K iterations while search baselines run for 2M+ iterations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LLM-SR with Mixtral instead of the strongest reported SR baseline | Symbolic regression on Oscillation 1 · Oscillation 1 ID | NMSE: 0.0003 -> 7.89e-8 | Helped |
| Use LLM-SR with Mixtral instead of the strongest reported SR baseline | Symbolic regression on Oscillation 1 · Oscillation 1 OOD | NMSE: 0.0007 -> 0.0002 | Helped |
| Use LLM-SR with GPT-3.5 instead of the strongest reported SR baseline | Symbolic regression on Oscillation 2 · Oscillation 2 ID | NMSE: 0.0002 -> 2.12e-7 | Helped |
| Use LLM-SR with GPT-3.5 instead of the strongest reported SR baseline | Symbolic regression on Oscillation 2 · Oscillation 2 OOD | NMSE: 0.0015 -> 3.81e-5 | Helped |
| Use LLM-SR with Mixtral instead of the strongest reported SR baseline | Symbolic regression on E. coli growth · E. coli growth ID | NMSE: 0.0376 -> 0.0026 | Helped |
| Use LLM-SR with Mixtral instead of the strongest reported SR baseline | Symbolic regression on E. coli growth · E. coli growth OOD | NMSE: 1.0141 -> 0.0037 | Helped |
| Use LLM-SR with Mixtral instead of the strongest reported SR baseline | Symbolic regression on Stress-Strain · Stress-Strain ID | NMSE: 0.0331 -> 0.0162 | Helped |
| Use LLM-SR with GPT-3.5 instead of the strongest reported SR baseline | Symbolic regression on Stress-Strain · Stress-Strain OOD | NMSE: 0.1304 -> 0.0516 | Helped |
| Add the natural-language scientific problem and variable description (ablation) | LLM-SR on Oscillation 2 · Oscillation 2 | NMSE | Helped |
| Use flexible program representation rather than single-line mathematical expressions (ablation) | LLM-SR on Oscillation 2 · Oscillation 2 | NMSE | Helped |
| Add iterative refinement and the optimization loop (ablation) | LLM-SR on Oscillation 2 · Oscillation 2 ID | NMSE: 1.01e-1 -> 2.12e-7 | Helped |
| Add iterative refinement and the optimization loop (ablation) | LLM-SR on Oscillation 2 · Oscillation 2 OOD | NMSE: 1.81e-1 -> 3.81e-5 | Helped |
Surprising findings
- Feynman benchmark problems reach low error within very few LLM iterations, and Feynman equations show lower perplexity, suggesting LLM recitation.
- Removing the natural language problem specification or program representation hurts performance, and numpy+BFGS slightly beats torch+Adam, attributed to LLMs writing numpy better.
Limitations
- Performance depends on the LLM's training data quality and may carry biases or gaps in scientific domains.
- Iterative LLM queries and parameter optimization may be computationally costly for large-scale problems.
Experimental setup
Datasets: Feynman benchmark Aluminum 6061-T651 stress-strain data (Mendeley)
Benchmarks: Oscillation 1 Oscillation 2 E. coli growth Stress-Strain
Metrics: NMSE perplexity
Baselines: GPlearn PySR DSR uDSR NeSymReS E2E
Builds on: FunSearch PySR DSR uDSR
Closest papers by meaning
- LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models (ICML 2025 · similarity 0.95)
- FunctionEvolve: Structure-Guided Symbolic Regression with LLMs (NeurIPS 2026 · similarity 0.90)
- MOOSE-Chem2: Exploring LLM Limits in Fine-Grained Scientific Hypothesis Discovery via Hierarchical Search (NeurIPS 2025 · similarity 0.89)
- FIND: A Framework for Discovering Formulas in Data (AAAI 2025 · similarity 0.88)
- SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training (ICLR 2024 · similarity 0.87)
- Symmetry-Informed Governing Equation Discovery (NeurIPS 2024 · similarity 0.86)
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