Optimal Transport-inspired Deep Learning Framework for Slow-Decaying
  Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel
Moaad Khamlich Federico Pichi Gianluigi Rozza
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.