Abstract: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.
| Subjects: | Numerical Analysis (math.NA); Machine Learning (cs.LG); Optimization and Control (math.OC) |
| Cite as: | arXiv:2503.15105 [math.NA] |
| (or arXiv:2503.15105v4 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2503.15105 arXiv-issued DOI via DataCite |
From: Minh-Binh Tran [view email]
[v1]
Wed, 19 Mar 2025 11:04:36 UTC (31 KB)
[v2]
Wed, 26 Mar 2025 17:56:07 UTC (31 KB)
[v3]
Mon, 19 May 2025 10:04:15 UTC (45 KB)
[v4]
Tue, 19 May 2026 19:16:21 UTC (49 KB)
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