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Symmetry-Constrained Language-Guided Program Synthesis for Discovering Governing Equations from Noisy and Partial Observations

Mirza Samad Ahmed Baig, Syeda Anshrah Gillani · Mar 6, 2026 · Citations: 0

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Data freshness

Extraction: Fresh

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Mar 6, 2026, 8:42 PM

Recent

Extraction refreshed

Mar 14, 2026, 6:13 AM

Fresh

Extraction source

Persisted extraction

Confidence 0.15

Abstract

Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty. Here we introduce SymLang (Symmetry-constrained Language-guided equation discovery), a unified framework that brings together three previously separate ideas: (i) typed symmetry-constrained grammars that encode dimensional analysis, group-theoretic invariance, and parity constraints as hard production rules, eliminating on average 71.3% of candidate expression trees before any fitting; (ii) language-model-guided program synthesis in which a fine-tuned 7B-parameter proposer, conditioned on interpretable data descriptors, efficiently navigates the constrained search space; and (iii) MDL-regularized Bayesian model selection coupled with block-bootstrap stability analysis that quantifies structural uncertainty rather than committing to a single best equation. Across 133 dynamical systems spanning classical mechanics, electrodynamics, thermodynamics, population dynamics, and nonlinear oscillators, SymLang achieves an exact structural recovery rate of 83.7% under 10% observational noise - a 22.4 percentage-point improvement over the next-best baseline - while reducing out-of-distribution extrapolation error by 61% and near-eliminating conservation-law violations (3.1 x 10-3 vs. 187.3 x 10-3 physical drift for the closest competitor). In all tested regimes the framework correctly identifies structural degeneracy, reporting it explicitly rather than returning a confidently wrong single equation. The framework is fully open-source and reproducible, providing a principled pathway from raw data to interpretable, physically auditable symbolic laws.

Low-signal caution for protocol decisions

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  • Extraction flags indicate low-signal or possible false-positive protocol mapping.
  • Extraction confidence is 0.15 (below strong-reference threshold).
  • No explicit evaluation mode was extracted from available metadata.
  • No benchmark/dataset or metric anchors were extracted.
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HFEPX Relevance Assessment

This paper is adjacent to HFEPX scope and is best used for background context, not as a primary protocol reference.

Best use

Background context only

Use if you need

Background context only.

Main weakness

Extraction flags indicate low-signal or possible false-positive protocol mapping.

Trust level

Low

Eval-Fit Score

0/100 • Low

Treat as adjacent context, not a core eval-method reference.

Human Feedback Signal

Not explicit in abstract metadata

Evaluation Signal

Weak / implicit signal

HFEPX Fit

Adjacent candidate

Extraction confidence: Low

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Field Provenance & Confidence

Each key protocol field shows extraction state, confidence band, and data source so you can decide whether to trust it directly or validate from full text.

Human Feedback Types

missing

None explicit

Confidence: Low Source: Persisted extraction missing

No explicit feedback protocol extracted.

Evidence snippet: Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty.

Evaluation Modes

missing

None explicit

Confidence: Low Source: Persisted extraction missing

Validate eval design from full paper text.

Evidence snippet: Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty.

Quality Controls

missing

Not reported

Confidence: Low Source: Persisted extraction missing

No explicit QC controls found.

Evidence snippet: Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty.

Benchmarks / Datasets

missing

Not extracted

Confidence: Low Source: Persisted extraction missing

No benchmark anchors detected.

Evidence snippet: Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty.

Reported Metrics

missing

Not extracted

Confidence: Low Source: Persisted extraction missing

No metric anchors detected.

Evidence snippet: Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty.

Rater Population

missing

Unknown

Confidence: Low Source: Persisted extraction missing

Rater source not explicitly reported.

Evidence snippet: Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty.

Human Data Lens

  • Uses human feedback: No
  • Feedback types: None
  • Rater population: Unknown
  • Unit of annotation: Unknown
  • Expertise required: Law
  • Extraction source: Persisted extraction

Evaluation Lens

  • Evaluation modes:
  • Agentic eval: None
  • Quality controls: Not reported
  • Confidence: 0.15
  • Flags: low_signal, possible_false_positive

Protocol And Measurement Signals

Benchmarks / Datasets

No benchmark or dataset names were extracted from the available abstract.

Reported Metrics

No metric terms were extracted from the available abstract.

Research Brief

Deterministic synthesis

Here we introduce SymLang (Symmetry-constrained Language-guided equation discovery), a unified framework that brings together three previously separate ideas: (i) typed symmetry-constrained grammars that encode dimensional analysis,… HFEPX protocol signal is limited in abstract-level metadata, so treat it as adjacent context. Updated from current HFEPX corpus.

Generated Mar 14, 2026, 6:13 AM · Grounded in abstract + metadata only

Key Takeaways

  • Here we introduce SymLang (Symmetry-constrained Language-guided equation discovery), a unified framework that brings together three previously separate ideas: (i) typed…
  • Across 133 dynamical systems spanning classical mechanics, electrodynamics, thermodynamics, population dynamics, and nonlinear oscillators, SymLang achieves an exact structural…
  • Abstract shows limited direct human-feedback or evaluation-protocol detail; use as adjacent methodological context.

Researcher Actions

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Caveats

  • Generated from title, abstract, and extracted metadata only; full-paper implementation details are not parsed.
  • Low-signal flag detected: protocol relevance may be indirect.

Research Summary

Contribution Summary

  • Here we introduce SymLang (Symmetry-constrained Language-guided equation discovery), a unified framework that brings together three previously separate ideas: (i) typed symmetry-constrained grammars that encode dimensional analysis,…
  • Across 133 dynamical systems spanning classical mechanics, electrodynamics, thermodynamics, population dynamics, and nonlinear oscillators, SymLang achieves an exact structural recovery rate of 83.7% under 10% observational noise - a 22.4…

Why It Matters For Eval

  • Abstract shows limited direct human-feedback or evaluation-protocol detail; use as adjacent methodological context.

Researcher Checklist

  • Gap: Human feedback protocol is explicit

    No explicit human feedback protocol detected.

  • Gap: Evaluation mode is explicit

    No clear evaluation mode extracted.

  • Gap: Quality control reporting appears

    No calibration/adjudication/IAA control explicitly detected.

  • Gap: Benchmark or dataset anchors are present

    No benchmark/dataset anchor extracted from abstract.

  • Gap: Metric reporting is present

    No metric terms extracted.

Category-Adjacent Papers (Broader Context)

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