Key Papers

1. Lu et al. (2019) - Original DeepONet paper
2. Goswami et al. (2023) - Physics-informed DeepONets
3. Chen & Chen (1995) - Universal approximation theorem for operators

Extensions to Explore

• Multi-output operators: Vector-valued mappings
• Higher dimensions: 2D/3D PDEs
• Physics-informed training: Incorporate governing equations
• Fourier Neural Operators: Alternative operator learning architecture

Exercises

1. Modify the derivative example to learn the second derivative operator
2. Extend to 2D by implementing the Laplacian operator \( \nabla^2 u \)
3. Add physics constraints by incorporating the differential equation into the loss
4. Compare with traditional methods on computational efficiency

The journey from function approximation to operator learning represents one of the most exciting frontiers in scientific machine learning!