Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential's gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact Hölder regularity of the potential, the flux and the flow generating is derived on the basis of Hölder regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable'') generative learning, we provide sufficient conditions for separate and joint Hölder continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-Hölder norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.

翻译：最优运输中的Beckmann问题旨在最小化从源分布到目标分布的连续运输问题中的总平方通量。本文利用无约束拉格朗日公式并求解变分一阶最优性条件，发展了最优运输中Beckmann问题解的正则性理论。结果表明，强制Beckmann散度约束的拉格朗日乘子满足泊松方程，而通量向量场可作为势函数的梯度获得。利用椭圆正则性理论中的Schauder估计，基于有界正则域上源密度和目标密度的Hölder正则性，推导了势函数、通量及流生成的精确Hölder正则性。当目标分布依赖于参数（如条件（“可提示”）生成学习中的情形）时，我们提供了向量场在参数和数据维度上分别及联合Hölder连续的充分条件。根据Belomnestny等人近期结果，此类向量场可通过深度ReQu神经网络在C^(k,α)-Hölder范数下逼近。我们还证明该方法可推广至其他概率路径，例如Fisher-Rao梯度流。