Obstacle problems: search with PDE constraints
Given an obstacle function \(v\in H^1(\Omega)\), the variational problem \[ \inf_{u\geq v, u\in H_0^1(\Omega) } \int_{\Omega} |\nabla u|^2 \mathrm{d}x. \]
The Euler-Lagrange equation is \[\begin{align} \Delta u \leq 0 & & \text{in }\Omega \\ \Delta u = 0 & & \{x: u(x)>v(x)\} \end{align}\]
The corresponding regression loss is \[ L_\text{reg}(\theta) \propto \int_{\Omega} (v(x)-u(x;\theta))^+ \mathrm{d}x. \]