KernelNeuralOperator#

class KernelNeuralOperator(lifting_operator, integral_kernels, projection_operator)[source]#

Bases: Module

Base class for composing Neural Operators with integral kernels.

This is a base class for composing neural operators with multiple integral kernels. All neural operator models defined in PINA inherit from this class. The structure is inspired by the work of Kovachki, N. et al. see Figure 2 of the reference for extra details. The Neural Operators inheriting from this class can be written as:

\[G_\theta := P \circ K_m \circ \cdot \circ K_1 \circ L\]

where:

\(G_\theta: \mathcal{A}\subset \mathbb{R}^{\rm{in}} \rightarrow \mathcal{D}\subset \mathbb{R}^{\rm{out}}\) is the neural operator approximation of the unknown real operator \(G\), that is \(G \approx G_\theta\)
\(L: \mathcal{A}\subset \mathbb{R}^{\rm{in}} \rightarrow \mathbb{R}^{\rm{emb}}\) is a lifting operator mapping the input from its domain \(\mathcal{A}\subset \mathbb{R}^{\rm{in}}\) to its embedding dimension \(\mathbb{R}^{\rm{emb}}\)
\(\{K_i : \mathbb{R}^{\rm{emb}} \rightarrow \mathbb{R}^{\rm{emb}} \}_{i=1}^m\) are \(m\) integral kernels mapping each hidden representation to the next one.
\(P : \mathbb{R}^{\rm{emb}} \rightarrow \mathcal{D}\subset \mathbb{R}^{\rm{out}}\) is a projection operator mapping the hidden representation to the output function.