Standard neural networks learn point-wise mappings: \( \mathbb{R}^n \rightarrow \mathbb{R}^m \). But operators map functions to functions. How do we represent infinite-dimensional functions with finite data?

Traditional approach limitations:

• Fixed discretization: Networks trained on specific grids can't generalize to different resolutions.
• Curse of dimensionality: High-dimensional function spaces are computationally intractable.
• No theoretical foundation: No guarantee that standard networks can approximate operators.