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

arXiv: https://arxiv.org/abs/2308.13840 · HTML: https://mlsanity.com/paper/2308.13840/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 63 references that are in the map.
- First posted on arXiv 26 Aug 2023.

## Builds on (most cited first, 13 of 13)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [An Introduction to Convolutional Neural Networks](https://mlsanity.com/paper/1511.08458/) (2015)
- [Variational Adversarial Active Learning](https://mlsanity.com/paper/1904.00370/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Optimal Transport: Fast Probabilistic Approximation with Exact Solvers](https://mlsanity.com/paper/1802.05570/) (2018)
- [An artificial neural network approach to bifurcating phenomena in computational fluid dynamics](https://mlsanity.com/paper/2109.10765/) (2021)
- [Wasserstein Exponential Kernels](https://mlsanity.com/paper/2002.01878/) (2020)

## Built on by (most cited first, 2 of 2)

- [GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications](https://mlsanity.com/paper/2406.03569/) (2024)
- [Optimal Transport-Based Displacement Interpolation with Data Augmentation for Reduced Order Modeling of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2411.08750/) (2024)

## Abstract (arXiv)

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.