低秩Wasserstein多项式混沌展开的生成建模
Generative modeling with low-rank Wasserstein polynomial chaos expansions
AI总结：
提出Wasserstein多元素多项式混沌展开(WPCE)方法，结合去偏Sinkhorn散度与堆叠张量列格式，实现非周期随机场数值粗化，支持不连续非单射模型并缓解维度指数增长。
摘要：
受计算最优传输在估计Wasserstein距离方面最新进展的启发，提出了一种新的Wasserstein多元素多项式混沌展开（WPCE）。该方法将无监督学习与随机向量$Y$的显式函数表示相结合。其训练仅依赖于来自未知分布的有限样本集，用于最小化被称为去偏Sinkhorn散度的正则化经验Wasserstein度量。推动该方法的一个有趣应用源于对定义在微观尺度上的非周期随机场的数值粗化。与随机均匀化的常数表征不同，WPCE能够编码关于有效材料行为的更高阶随机信息。该方法的一个显著特征是，将常见的微分同胚传输映射推广到了具有可能不同输入和输出维度的不连续且非单射模型类$\mathcal{M}$的情形。它在分布上计算输入随机变量$X$与目标$Y$之间的（函数）关系$Y=\mathcal{M}(X)$。PCE的指数增长通过一种新的堆叠张量列（STT）格式得到缓解。通过选择模型类$\mathcal{M}$和平滑损失函数，使得高阶优化方案，特别是黎曼下降方法成为可能。通过一个高维粗化问题对所提方法进行了数值说明，该问题考虑了微观随机非周期复合材料。其结果是在自适应随机坐标中产生了一个计算上易于处理的有效宏观随机场。通过使用对不连续模型类的松弛，多峰分布也变得易于处理。
英文摘要：
A new Wasserstein multi-element polynomial chaos expansion (WPCE) is proposed, which is inspired by recent advances in computational optimal transport for estimating Wasserstein distances. The developed method combines unsupervised learning with the explicit functional representation of a random vector $Y$. Its training only relies on a finite set of samples from an unknown distribution, which is used to minimize a regularized empirical Wasserstein metric known as debiased Sinkhorn divergence. An interesting application that motivates the approach comes from the numerical upscaling of non-periodic random fields defined on a micro-scale. The WPCE can encode higher order stochastic information about the effective material behavior in contrast to the constant characterization with stochastic homogenization. A striking feature of the new method is the generalization of common diffeomorphic transport maps to the case of discontinuous and non-injective model classes $\mathcal{M}$ with possibly different input and output dimension. It computes a (functional) relation $Y=\mathcal{M}(X)$ in distribution with input random variables $X$ and target $Y$. The exponential growth of the PCE is alleviated by a new stacked tensor train (STT) format. By the choice of the model class $\mathcal{M}$ and the smooth loss function, higher-order optimization schemes and in particular Riemannian descent methods become possible. The proposed approach is illustrated numerically with a high-dimensional upscaling problem, which considers a microscopic random non-periodic composite material. It results in a computationally tractable effective macroscopic random field in adapted stochastic coordinates. By using a relaxation to a discontinuous model class, multimodal distributions also become tractable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
