Related papers: Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem

Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem

URL: http://arxiv.org/abs/2501.10162v1
Date: Fri, 17 Jan 2025 12:51:25 GMT
Title: Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem
Authors: Alexandre Caboussat, Anna Peruso,
Abstract summary: Optimal transportation of raw material from suppliers to customers is an issue arising in logistics. A physics informed neuralnetwork method is advocated here for the solution of the corresponding generalized Monge-Ampere equation. A particular focus is set on the enforcement of transport boundary conditions in the loss function.
Score: 49.1574468325115
License:
Abstract: Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous model relying on optimal transport theory. A physics informed neuralnetwork method is advocated here for the solution of the corresponding generalized Monge-Amp`ere equation. Convex neural networks are advocated to enforce the convexity of the solution to the Monge-Amp\`ere equation and obtain a suitable approximation of the optimal transport map. A particular focus is set on the enforcement of transport boundary conditions in the loss function. Numerical experiments illustrate the solution to the optimal transport problem in several configurations, and sensitivity analyses are performed.

Related papers

Conditional Optimal Transport on Function Spaces [53.9025059364831]
We develop a theory of constrained optimal transport problems that describe block-triangular Monge maps. This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions. We present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.
arXiv Detail & Related papers (2023-11-09T18:44:42Z)
Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs [49.1574468325115]
We consider the problem of recovering the $Wimat$-optimal transport map T between absolutely continuous measures $mu,nuinmathcalP(mathbbRn)$ as the flow of a linear-control neural ODE.
arXiv Detail & Related papers (2023-11-02T17:17:03Z)
A Computational Framework for Solving Wasserstein Lagrangian Flows [48.87656245464521]
In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. We propose a novel deep learning based framework approaching all of these problems from a unified perspective. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference.
arXiv Detail & Related papers (2023-10-16T17:59:54Z)
Constrained Mass Optimal Transport [0.0]
This paper introduces the problem of constrained optimal transport. A family of algorithms is introduced to solve a class of constrained saddle point problems. Convergence proofs and numerical results are presented.
arXiv Detail & Related papers (2022-06-05T06:47:25Z)
Neural Optimal Transport [82.2689844201373]
We present a novel neural-networks-based algorithm to compute optimal transport maps and plans for strong and weak transport costs. We prove that neural networks are universal approximators of transport plans between probability distributions.
arXiv Detail & Related papers (2022-01-28T16:24:13Z)
Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark [133.46066694893318]
We evaluate the performance of neural network-based solvers for optimal transport. We find that existing solvers do not recover optimal transport maps even though they perform well in downstream tasks.
arXiv Detail & Related papers (2021-06-03T15:59:28Z)
Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation [13.807546494746207]
We propose a Deep Learning approach to solve the continuous Optimal Mass Transport problem. We focus on the ubiquitous density estimation and generative modeling tasks in statistics and machine learning.
arXiv Detail & Related papers (2021-04-02T18:44:11Z)
Relaxation of optimal transport problem via strictly convex functions [0.0]
An optimal transport problem on finite spaces is a linear program. Recently, a relaxation of the optimal transport problem via strictly convex functions, especially via the Kullback--Leibler divergence, sheds new light on data sciences. This paper provides the mathematical foundations and an iterative process based on a gradient descent for the relaxed optimal transport problem via Bregman divergences.
arXiv Detail & Related papers (2021-02-15T04:32:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.