What we've demonstrated:

1. Physics-Based Model: Realistic wave equation solver with finite differences
2. Automatic Differentiation: JAX computed exact gradients through entire simulation
3. Scalable Optimization: From 1 parameter (constant) to 1000 parameters (linear profile)
4. Algorithm Comparison: Adam vs L-BFGS trade-offs in practice

Key Results:

• Constant velocity: Perfect recovery with gradient descent
• Linear profile: High-fidelity reconstruction of spatially-varying parameters
• L-BFGS advantage: Superior convergence for smooth optimization landscapes

The Revolution: Physics simulations are now learnable components that can be optimized end-to-end with gradient descent, enabling inverse problems that were previously intractable.

Applications

Geophysics: Subsurface imaging, earthquake location, Earth structure

Medical imaging: Ultrasound tomography, photoacoustic imaging

Materials science: Non-destructive testing, property characterization

Engineering: Structural health monitoring, design optimization

Next Steps

• Explore the interactive demo: Projectile Control and JAX Roll
• Try different velocity models (step functions, Gaussian anomalies)
• Experiment with other PDEs (heat, elasticity, Maxwell)
• Implement multi-objective optimization with regularization

The Future: Differentiable simulation bridges physics and machine learning, enabling scientific discovery through optimization.