HypatiaX Tutorial Series

HypatiaX: LLM-Guided Symbolic Discovery

Welcome to the complete HypatiaX tutorial series! Learn how to use hybrid LLM + symbolic regression to discover scientific equations from data with near-perfect extrapolation.

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🎯 What is HypatiaX?

HypatiaX is a groundbreaking framework that combines:

• Large Language Models (LLMs) for intelligent initialization
• Symbolic Regression for mathematically rigorous discovery
• Multi-layer Validation for ensuring correctness

v2 Note (March 2026): A bug in the evaluate_llm_formula measurement harness was corrected before final paper submission. All benchmark commands require the --v2 flag. Results generated before March 2026 must be regenerated. See Tutorial 2 for full details.

Key Results from JMLR Paper:

• ✅ 89.2% near-perfect success rate (R²>0.99) on 74 DeFi tasks — +27 pp over pure LLM (62.2%)
• ✅ Median extrapolation error < 10⁻¹² (floating-point precision limit, Core-15 benchmark)
• ✅ Complete statistical separation from neural networks (Mann-Whitney U=0, p<10⁻⁶)
• ✅ Feynman SR Benchmark: 9/30 (30.0%) under aggressive PCA-directed extrapolation protocol, comparable to AI Feynman 2.0 under equivalent conditions
• ✅ 1.73× median speedup over neural-network inference (LLM-routed cases, 68 of 74 tasks)

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📚 Tutorial Series

Tutorial 1: Environment Setup and First Discovery

Time: 15 minutes | Difficulty: Beginner

Learn to install HypatiaX and discover your first equation (Ohm’s Law). Includes:

• Installation and verification
• Your first formula discovery
• Extrapolation validation
• Understanding the core advantage: symbolic vs neural

What you’ll discover:

# HypatiaX: Median error 2.34e-13 (near floating-point precision!)
# Neural Network: Median error 12.47 (1,247% error!)

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Tutorial 2: Running Benchmark Experiments

Time: 45 minutes (active) + 3–8 hours (compute) | Difficulty: Intermediate

Reproduce the three benchmark evaluations from the JMLR paper:

Benchmark	Equations	Primary metric	Section
Core 15	15 across 4 domains	Extrapolation error (%)	§6.4
DeFi Extrapolation	74 test cases	R²>0.99 at fixed n=74	§6.5
Feynman SR	30-equation subset	Recovery rate at R²>0.9999	§5.8

Key experiments:

• Run individual benchmark campaigns
• Compare discovery systems (Pure Symbolic, Hybrid, Pure LLM)
• Parallel execution for faster results
• Checkpoint/resume functionality

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Tutorial 3: Statistical Analysis and Publication Figures

Time: 45 minutes | Difficulty: Intermediate

Generate all 13 publication-quality figures and reproduce statistical analyses:

• Figure 3: Arrhenius equation extrapolation failure
• Figure 2: Success rate comparison across domains
• Figure 6: Validation cascade breakdown
• Figure 9: Five-system unified comparison (13 figures total in paper)
• Figures 11–13: DeFi extrapolation benchmark (new in v2)

Statistical validation:

Mann-Whitney U test: U=0, p<10⁻⁶
Cohen's d: 0.95 (pooled; see paper note on degenerate distributions)
95% CI for neural error: [1,087%, 1,456%]

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Tutorial 4: Custom Applications and Extensions

Time: 45 minutes | Difficulty: Advanced

Apply HypatiaX to your own scientific problems:

6 Complete Real-World Examples:

1. Materials Science: Discover Hall-Petch relationship (yield stress)
2. Environmental Science: CO₂ sequestration formulas
3. Scikit-learn Integration: Drop-in replacement for ML pipelines
4. Multi-Equation Systems: Discover coupled ODEs (predator-prey)
5. Production Deployment: REST API + Docker
6. Custom Operators: Add domain-specific mathematical functions

Advanced extensibility: Subclass DomainValidator to add custom physical constraints for any scientific domain.

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🚀 Quick Start Path

New to symbolic discovery? Follow this path:

Week 1: Tutorial 1 (15 min)
  → Install and verify
  → Discover Ohm's Law
  → Understand extrapolation

Week 2: Tutorial 2 (8-12 hours)
  → Run Core 15 and DeFi benchmarks
  → Compare with baselines
  → Understand discovery paths

Week 3: Tutorial 3 (2 hours)
  → Generate all 13 figures
  → Validate statistics (v2 corrected)
  → Create publication materials

Week 4: Tutorial 4 (varies)
  → Apply to your domain
  → Customize and extend
  → Deploy in production

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📊 What You’ll Learn

Core Concepts:

• Health factor calculations in DeFi
• Liquidation thresholds and zombie positions
• Symbolic vs neural extrapolation
• Multi-layer validation cascades

Technical Skills:

• Python symbolic regression (PySR)
• Julia backend integration
• LLM-guided initialization
• Production deployment patterns

Research Methods:

• Benchmark design and execution
• Statistical validation (Mann-Whitney U, effect sizes)
• Publication-quality visualization
• Reproducibility best practices

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🎯 Prerequisites

Minimum requirements:

• Python 3.8+
• 4GB RAM
• Basic command line knowledge
• Understanding of mathematical formulas

Optional (for LLM features):

• Anthropic API key (for 1.73× median speedup on LLM-routed cases, 68 of 74 tasks)
• 8GB+ RAM for parallel execution

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📦 What’s Included

Each tutorial provides:

• ✅ Complete working code (copy-paste ready)
• ✅ Real examples (not toy problems)
• ✅ Expected outputs (verify your results)
• ✅ Troubleshooting (common issues solved)
• ✅ Quick reference (commands at a glance)

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🌟 Key Advantages

Why HypatiaX?

1. Near-Perfect Extrapolation

Symbolic (HypatiaX): Median error < 10⁻¹²
Neural Networks:     Mean error 1,231% (95% CI: [1,087%, 1,456%], n=13)
Complete statistical separation: U=0, p<10⁻⁶

2. Mathematical Rigor

• Discovers exact symbolic formulas
• Not black-box approximations
• Interpretable and verifiable

3. Domain Agnostic

• Physics, chemistry, biology
• DeFi AMM, DeFi Risk (finance)
• Any domain with mathematical relationships

4. Production Ready

• API deployment examples
• Docker containerization
• Scikit-learn integration

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📖 Additional Resources

Paper & Code:

• 📄 JMLR Paper (2026)
• 💻 GitHub Repository
• 📚 Full Documentation

Community:

• 💬 Discussions
• 🐛 Report Issues
• 🎓 Academic Citations

Related Work:

• PySR (Python Symbolic Regression)
• SymbolicRegression.jl (Julia backend)
• AI Feynman (physics-inspired discovery)

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🎓 Citation

If you use HypatiaX in your research:

@article{bonetchaple2026hypatiax,
  title={HypatiaX: A Hybrid Symbolic-Neural Framework for Extrapolation-Reliable Analytical Discovery},
  author={Bonet Chaple, Ruperto Pedro},
  journal={Journal of Machine Learning Research},
  year={2026},
  volume={27},
  pages={1--47}
}

Ready to begin? Start with Tutorial 1: Environment Setup!

Or jump to:

• Tutorial 2: Benchmark Experiments (if already installed)
• Tutorial 3: Analysis & Figures (if experiments complete)
• Tutorial 4: Custom Applications (if ready to build)

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💡 Support

Need help?

• 📖 Check the troubleshooting sections in each tutorial
• 💬 Ask questions in GitHub Discussions
• 🐛 Report bugs via GitHub Issues

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Let’s discover some equations! 🧪🔬✨