Pcurry/euler newton improvements

[crewsdw]

Implement mini-Euler stepping and adaptive Newton methods

Key Improvements

1. Sequential Mini-Euler Initial Guess

• Transform Gauss-Legendre nodes from [-1,1] to [0,1]
• Step sequentially through IRK nodes with variable timesteps
• Provides physics-informed initialization for neural networks
• Works at every timestep (replaces transfer learning)

2. Adaptive Newton Solvers

• Added scipy.optimize.root with multiple methods (hybr, lm, broyden1, krylov)
• Method-specific tolerance settings for proper convergence
• Enhanced diagnostics to detect false convergence claims

3. Neural Network Enhancements

• Direct weight initialization using Euler guess
• Configurable epochs reduction based on initialization quality
• Fixed get_config() methods for proper model serialization

Key Findings

🔍 Critical Discovery: Network overfitting hurts Newton convergence!

• Low NN loss (~1e-2) → Newton fails to converge
• Medium NN loss (~3-5) → Newton converges well
• Suggests NN loss ≠ true IRK solution quality

🚀 Mini-Euler is highly effective:

• Newton-Krylov converges even with order=100, dt=0.8
• Requires ~40,000 function evaluations but succeeds
• Can converge WITHOUT any NN training (Euler guess alone!)
• Raises question: When is the NN guess MORE effective? Do we need to study a more random/chaotic system?

⚡ Performance:

• GPU training: 20-40 ms/step
• Euler initialization reduces required epochs by 20-40%

🧮 Scalability:

• Successfully tested up to order=100 (vs previous order=32)
• Larger timesteps (dt=0.8) now feasible

Implementation Notes

• Sequential Euler respects IRK node structure
• Physics-informed initialization at every timestep
• Robust convergence diagnostics prevent false positives

🤖 Generated with Claude Code

Co-Authored-By: Claude noreply@anthropic.com