GradNetOT: Learning Optimal Transport Maps with GradNets
Optimal Transport (OT) is the mathematical problem of moving “mass” from one distribution to another in the most efficient way. Think of reshaping a pile of sand into a new shape with minimal effort. GradNetOT is a novel machine‑learning method that learns exactly these efficient maps using neural networks equipped with a built‑in “bias” toward physically correct solutions. What Is Optimal Transport? Classic formulation: Given two probability distributions (e.g., piles of sand and holes to fill), find a mapping that moves mass at minimal total cost. Monge’s theorem: For certain costs (like squared distance), the optimal map is the gradient of a convex function satisfying a Monge–Ampère equation. The GradNetOT Approach GradNetOT leverages a special neural network architecture called a Monotone Gradient Network (mGradNet) to represent convex functions implicitly. By enforcing convexity and monotonicity, the network’s output gradient automatically yields a valid OT map. ...