🤖 AI Summary
Traditional reduced-order models (ROMs) struggle to accurately capture the geometric structure of high-dimensional solution manifolds due to slow decay of the Kolmogorov *n*-width. To address this, we propose a novel nonlinear ROM framework integrating optimal transport (OT) with deep learning. Our method introduces a Wasserstein kernel by embedding the Wasserstein distance into the kernel function of kernel proper orthogonal decomposition (kPOD), and employs the Sinkhorn divergence as the loss function for neural network training—enabling OT-aware nonlinear dimensionality reduction. This approach significantly enhances geometric fidelity to slowly decaying manifolds. Compared to conventional ROMs, it achieves superior accuracy, improved training stability, enhanced robustness to noise, faster convergence, and higher computational efficiency.
📝 Abstract
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.