Poisson Equation

\subsection{Poisson Equation} The Poisson equation with Dirichlet boundary conditions is studied: \begin{align} -\Delta u &= f, \quad \text{in } \Omega = [0,1]^2, \ u &= 0, \quad \text{on } \partial \Omega, \end{align} where (f) consists of a Gaussian superposition, with parameters (\mu_{x,i}, \mu_{y,i} \sim \text{U}(0,1)) and (\sigma_i \sim \text{U}(0.025, 0.1)). The dataset includes 4000 training, 500 validation, and 500 test samples.