We examine the infinite-dimensional optimization problem of finding a
decomposition of a probability measure into K probability sub-measures to
minimize specific loss functions inspired by applications in clustering and
user grouping. We analytically explore the structures of the support of optimal
sub-measures and introduce algorithms based on Wasserstein gradient flow,
demonstrating their convergence. Numerical results illustrate the
implementability of our algorithms and provide further insights.

我们研究无限维优化问题，即查找将概率测度分解为 K 个概率子测度以最小化受聚类和用户分组应用启发的特定损失函数。我们分析了最优子测度支撑集的结构，并介绍了基于 Wasserstein 梯度流的算法，证明了它们的收敛性。数值结果说明了我们算法的可实施性并提供了进一步的见解。

最优概率测度分解的 Wasserstein 梯度流

Wasserstein gradient flow for optimal probability measure decomposition

We study an infinite-dimensional optimization problem that aims to identify
the Nemytskii operator in the nonlinear part of a prototypical semilinear
elliptic partial differential equation (PDE) which minimizes the distance
between the PDE-solution and a given desired state. In contrast to previous
works, we consider this identification problem in a low-regularity regime in
which the function inducing the Nemytskii operator is a-priori only known to be
an element of $H^1_{loc}(\mathbb{R})$. This makes the studied problem class a
suitable point of departure for the rigorous analysis of training problems for
learning-informed PDEs in which an unknown superposition operator is
approximated by means of a neural network with nonsmooth activation functions
(ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the
controls, it is possible to derive a classical stationarity system for local
minimizers and to solve the considered problem by means of a gradient
projection method. The convergence of the resulting algorithm is proven in the
function space setting. It is also shown that the established first-order
necessary optimality conditions imply that locally optimal superposition
operators share various characteristic properties with commonly used activation
functions: They are always sigmoidal, continuously differentiable away from the
origin, and typically possess a distinct kink at zero. The paper concludes with
numerical experiments which confirm the theoretical findings.

本论文研究了一个无限维优化问题，旨在识别带有 Nemytskii 算子的半线性椭圆偏微分方程中的非线性部分，该算子使得 PDE 解与给定状态的距离最小化，并探讨了在低正则性情况下解决此类识别问题的方法，为学习驱动的 PDEs 的训练问题进行了严格的分析。最优超级位置算子与常见的激活函数具有共同的特征。