Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Kolmogorov n-width Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel

TL;DR

A novel ROM framework that integrates optimal transport (OT) theory and neural network-based methods is proposed that outperforms traditional ROM methods in terms of accuracy and 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.

Figures (15)

• Figure 1: Schematic illustration of the different steps of the proposed methodology
• Figure 2: Wasserstein barycenters between the solution to the Burgers equation at two different time instants. The parameter $\alpha$ represents the interpolation parameter between the two fields $\mu$ and $\nu$, which are normalized to treat them as probability distributions.
• Figure 3: Optimal transport plans between two Gaussian distributions with different regularization parameters. The blue and red dots represent samples drawn from the two distributions, and the black lines indicate the transport plans between them, with thicker lines indicating higher plan values. As the regularization parameter increases, the transport plans become smoother and more diffuse.
• Figure 4: Decay of the normalized singular values.
• Figure 5: First (left) and second (right) kPOD coefficients w.r.t. the source location. Training points represented in blue.

Theorems & Definitions (1)

• Remark 1