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

TL;DR

This paper introduces a novel reduced order modeling framework that leverages optimal transport theory, Wasserstein distances, and Sinkhorn algorithms to better capture data geometry, especially in challenging cases with slow-decaying Kolmogorov n-width, improving accuracy and efficiency.

Contribution

The paper presents a new ROM approach integrating optimal transport and neural networks, utilizing Sinkhorn divergence for training stability and better geometric data representation.

Findings

Outperforms traditional ROMs in accuracy and efficiency.

Enhances training stability and robustness against noise.

Effective on challenging problems with slow Kolmogorov n-width decay.

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.