Dr. Ruperto Pedro Bonet Chaple

PhD in Computational Mechanics

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.

🎯 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

Key Results from JMLR Paper:

  • ✅ 95.8% success rate on 131 scientific equations
  • ✅ Median extrapolation error < 10⁻¹² (floating-point precision limit)
  • ✅ Complete statistical separation from neural networks (Mann-Whitney U=0, p<10⁻⁶)
  • ✅ New state-of-the-art on Feynman SR Benchmark: 96.7% exact recovery at R²>0.9999 (Hybrid DeFi, +17.4 pp over AI Feynman 2.0)
  • ✅ Mean discovery time: 390 seconds per equation

📚 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!)

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 73 test cases (66 standard) R²>0.99 at fixed n=66 §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

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%]

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

Production-ready code included!

🚀 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

📊 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

🎯 Prerequisites

Minimum requirements:

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

Optional (for LLM features):

  • Anthropic API key (for 73% speedup)
  • 8GB+ RAM for parallel execution

📦 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)

🌟 Key Advantages

Why HypatiaX?

1. Near-Perfect Extrapolation

Symbolic (HypatiaX): Median error < 10⁻¹²
Neural Networks:     Median error 1,231%
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
  • Economics, finance
  • Any domain with mathematical relationships

4. Production Ready

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

📖 Additional Resources

Paper & Code:

Community:

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

🎓 Citation

If you use HypatiaX in your research:

@article{bonetchaple2026hypatiax,
  title={Why Extrapolation Breaks Na{\"i}ve Analytical Discovery},
  author={Bonet Chaple, Ruperto Pedro},
  journal={Journal of Machine Learning Research},
  year={2026},
  volume={27},
  pages={1--47}
}

🚦 Getting Started

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

Or jump to:

💡 Support

Need help?

Let’s discover some equations! 🧪🔬✨

Updated: