feat: extend PDE discovery system to support Burgers equation

[devin-ai-integration]

feat: extend PDE discovery system to support Burgers equation

Summary

This PR extends the existing PDE discovery system to handle nonlinear partial differential equations, specifically the Burgers equation (∂u/∂t + u×∂u/∂x = ν×∂²u/∂x²). The implementation adds:

• BurgersFDM class: Nonlinear finite difference solver with upwind differencing for convection terms
• Burgers equation discovery: New symbolic regression method with nonlinear candidate formulas
• Enhanced Streamlit UI: Equation type selection between heat conduction and Burgers equations
• Data generation & testing: Complete pipeline for synthetic Burgers equation data and validation
• Updated documentation: Comprehensive coverage of both linear and nonlinear PDE discovery

The system now supports discovering both linear diffusion equations and nonlinear convection-diffusion equations from numerical data, significantly expanding its capabilities for physics-informed machine learning applications.

Review & Testing Checklist for Human

• Critical: Verify upwind differencing implementation - The BurgersFDM.solve() method uses upwind differencing for the nonlinear convection term u×∂u/∂x. Review lines 122-125 in pde_discovery.py to ensure the conditional logic and finite difference stencils are mathematically correct.

• Critical: Investigate PDE discovery accuracy - Current tests show 50% error in viscosity coefficient discovery (test_burgers_system.py output). The structural discovery works (finds correct u×∂u/∂x + ν×∂²u/∂x² form) but parameter optimization needs improvement.

• Test end-to-end functionality - Run streamlit run pde_discovery.py --server.port 8504 and verify both equation types work correctly: heat conduction with Gaussian initial conditions and Burgers with step function initial conditions producing visible shock waves.

• Verify stability conditions - The combined CFL (≤ 1.0) and diffusion (r ≤ 0.5) stability checks in BurgersFDM.init should prevent numerical instability. Test with various parameter combinations to ensure warnings appear appropriately.

Recommended test plan:

1. Generate data with python burgers_data_generation.py
2. Run both test suites: python test_pde_system.py and python test_burgers_system.py
3. Launch Streamlit app and test discovery for both equation types with different parameter settings
4. Verify shock wave formation is visible in Burgers equation visualization

---

Diagram

%%{ init : { "theme" : "default" }}%%
graph TD
    UI["pde_discovery.py<br/>Streamlit Interface"]:::major-edit
    HeatFDM["HeatConductionFDM<br/>Linear Solver"]:::context
    BurgersFDM["BurgersFDM<br/>Nonlinear Solver"]:::major-edit
    PDERegression["PDESymbolicRegression<br/>Discovery Engine"]:::major-edit
    
    DataGen["burgers_data_generation.py<br/>Synthetic Data"]:::major-edit
    TestSuite["test_burgers_system.py<br/>Validation Tests"]:::major-edit
    
    UI --> HeatFDM
    UI --> BurgersFDM
    UI --> PDERegression
    
    BurgersFDM --> DataGen
    PDERegression --> TestSuite
    
    PDERegression --> |"discover_heat_equation()"| HeatFDM
    PDERegression --> |"discover_burgers_equation()"| BurgersFDM
    
    subgraph Legend
        L1["Major Edit"]:::major-edit
        L2["Minor Edit"]:::minor-edit  
        L3["Context/No Edit"]:::context
    end

classDef major-edit fill:#90EE90
classDef minor-edit fill:#87CEEB  
classDef context fill:#FFFFFF

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Notes

• Session Info: Requested by satoshi minamoto (minimumtone@gmail.com) - Devin Session
• Current Test Status: Heat conduction tests pass, but Burgers equation discovery shows 50% viscosity error - this needs investigation before merge
• Mathematical Complexity: The upwind differencing scheme for nonlinear convection is numerically challenging and requires careful verification
• Backward Compatibility: Existing heat conduction functionality appears preserved based on test results

[devin-ai-integration]

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