Control, Optimal Transport and Neural Differential Equations in Supervised Learning
This work addresses a fundamental computational problem in optimal transport and machine learning, offering a method to approximate UOT in the continuum, which is incremental in advancing mathematical foundations.
The paper tackles the problem of approximating unbalanced optimal transport (UOT) dynamics by developing a novel framework using neural differential equations (Neural ODEs), achieving convergence to true UOT dynamics with explicit error estimates through a numerical scheme inspired by the Sinkhorn algorithm.
We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal transport (UOT) in the continuum using Neural ODEs. By generalizing a discrete UOT problem with Pearson divergence, we constructively design vector fields for Neural ODEs that converge to the true UOT dynamics, thereby advancing the mathematical foundations of computational transport and machine learning. To this end, we design a numerical scheme inspired by the Sinkhorn algorithm to solve the corresponding minimization problem and rigorously prove its convergence, providing explicit error estimates. From the obtained numerical solutions, we derive vector fields defining the transport dynamics and construct the corresponding transport equation. Finally, from the numerically obtained transport equation, we construct a neural differential equation whose flow converges to the true transport dynamics in an appropriate limiting regime.