Operator Networks

Recall that a typical MLP \(\boldsymbol{y}= \boldsymbol{\mathcal {F}}(\boldsymbol{x};\boldsymbol{\theta })\) is a function that takes as input \(\boldsymbol{x}\in \mathbb {R}^d\) and gives an output \(\boldsymbol{y}\in \mathbb {R}^d\) with trainable weights \(\boldsymbol{\theta }\) . Also, as we discussed in Chap.  5 , a PINN is a network of the form \(\boldsymbol{u}(\boldsymbol{x};\boldsymbol{\theta }) = \boldsymbol{\mathcal {F}}(\boldsymbol{x};\boldsymbol{\theta })\) taking as input the independent variable \(\boldsymbol{x}\) of the underlying PDE and giving the approximate solution \(\boldsymbol{u}(\boldsymbol{x};\boldsymbol{\theta })\) (of the PDE) as output. The network is trained by minimizing the weighted sum of the PDE and boundary residual.