We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs -- which are dominated by snapshot generation -- of model reduction techniques for nonlinear advection-dominated problems.

翻译：我们提出了一种针对参数化场的非线性插值技术，通过利用解中相干结构的最优传输实现精确性能。该方法将[Iollo, Taddei, J. Comput. Phys., 2022]中引入的非线性插值过程推广至多维参数域及多快照数据集。基于高保真仿真库，我们借助标量测试函数与点集配准方法，以排序点云形式识别解场中的相干结构。针对新参数值，采用回归方法预测新点云；随后通过边界感知配准技术定义双射映射，在保持域边界的同时将新点云变形为数据集中邻近元素的点云；最终将估计值定义为由邻近快照与预先构建的映射复合所得模态的加权组合。我们针对可压缩与不可压缩、有黏与无黏流动给出了多个数值算例，验证了方法的准确性。此外，将非线性插值过程用于增强基于线性子空间投影的模型降阶仿真数据集：该数据增强策略旨在降低非线性对流主导问题的模型降阶技术离线成本（主要由快照生成主导）。