The Monge Problem (1781): Given source points \(\{x_i\}\) and target points \(\{y_j\}\), find the optimal matching \(\pi\) that minimizes total transport cost:

\[ \min_{\pi} \sum_{i,j} \pi_{ij} \|x_i - y_j\|^2 \]

Click anywhere on the visualization to add particles. They'll automatically find their optimal transport partners and flow smoothly between distributions.