用凸神经网络求解最优传输问题，确保解的凸性并精准满足边界条件。
Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem
- 采用凸神经网络强制解的凸性，保证最优传输映射的有效性
- 通过损失函数显式约束边界条件，提升解的精度与物理一致性
- 适用于物流配送等需精确路径规划的连续优化场景
将原始材料从供应商到客户的运输问题视为连续优化问题，基于最优传输理论建模。提出一种物理信息神经网络方法求解广义Monge-Ampère方程。通过引入凸神经网络，强制保证解的凸性，从而获得合适的最优传输映射。重点在于将运输边界条件纳入损失函数中进行显式约束。数值实验展示了在多种配置下的求解效果，并进行了敏感性分析，验证了方法的鲁棒性与有效性。
原文摘要 · Abstract (English)
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-Ampère 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.
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