Loss Function

Type	From
residual	PDE/physics
regression	data
variational	physics

We will mainly focus on PDEs instead of ODEs. To fully visualize the solution, the input of \(u\) is usually of dimension=2, so the selected MLP architecure is typically

where the solution to the PDE in the context will be denoated as \(u\), and the approximation will be denoted as \(u(\cdot;\theta)\).

The convention here is, when semicolon and \(\theta\) appears in the term \(u\) it refers to the ANN approximation, and on the contrary, when \(\theta\) is missing \(u\) denotes the PDE solution.

Denote the loss function associated with the PDE problem as \(L(\theta)\), where \(\theta\in\mathbb{R}^d\) is the parameter vector in an ANN

The goal is to design \(L(\theta)\) such that when \(u(\cdot;\theta)\) well approximates \(u\), \(L\) attains its global minimum.