We've learned how neural networks can approximate functions: \( f: \mathbb{R}^d \rightarrow \mathbb{R}^m \).

But what if we want to learn mappings between infinite-dimensional function spaces?

Enter operators: mappings that take functions as input and produce functions as output.

where \( \mathcal{A} \) and \( \mathcal{U} \) are function spaces.

Examples of operators:

• Derivative operator: \( \mathcal{G}u = \frac{du}{dx} \)
• Integration operator: \( \mathcal{G}f = \int_0^x f(t) dt \)
• PDE solution operator: Given boundary conditions or source terms, map to the PDE solution.