Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics encompass the underlying structure, relationships, and essential features crucial for accurate modeling. To overcome this limitation, we propose a novel ROM framework that integrates optimal transport (OT) theory and neural network-based methods. Specifically, we investigate the Kernel Proper Orthogonal Decomposition (kPOD) method exploiting the Wasserstein distance as the custom kernel, and we efficiently train the resulting neural network (NN) employing the Sinkhorn algorithm. By leveraging an OT-based nonlinear reduction, the presented framework can capture the geometric structure of the data, which is crucial for accurate learning of the reduced solution manifold. When compared with traditional metrics such as mean squared error or cross-entropy, exploiting the Sinkhorn divergence as the loss function enhances stability during training, robustness against overfitting and noise, and accelerates convergence. To showcase the approach's effectiveness, we conduct experiments on a set of challenging test cases exhibiting a slow decay of the Kolmogorov n-width. The results show that our framework outperforms traditional ROM methods in terms of accuracy and computational efficiency.

翻译：降阶模型(ROMs)广泛应用于科学计算中处理高维系统。然而传统ROM方法可能无法完全捕捉数据的本征几何特征，这些特征涵盖对精确建模至关重要的底层结构、关系与核心要素。为突破这一局限，我们提出一种融合最优传输(OT)理论与神经网络方法的新型ROM框架。具体而言，我们探索了采用Wasserstein距离作为自定义核的核本征正交分解(kPOD)方法，并利用Sinkhorn算法高效训练生成的神经网络(NN)。通过基于OT的非线性降维，该框架能够捕捉数据的几何结构，这对准确学习降阶解流形至关重要。相较均方误差或交叉熵等传统度量，采用Sinkhorn散度作为损失函数可增强训练稳定性、提升对过拟合与噪声的鲁棒性，并加速收敛。为验证该方法的有效性，我们在系列呈现Kolmogorov n-宽度慢衰减的挑战性测试用例上开展实验。结果表明，本框架在精度与计算效率方面均优于传统ROM方法。