Nonlinear reduction strategies for data compression: a comprehensive comparison from diffusion to advection problems

TL;DR

The paper investigates nonlinear data-reduction strategies for Advection-Diffusion dynamics within a unified two-stage framework, where a pre-processing map $\Phi$ transforms the data manifold $\mathcal{M}$ into $\mathcal{T}=\Phi(\mathcal{M})$, followed by a reduction $Z_N$ to a low-dimensional space $\mathcal{Z}_N$. It systematically compares Eulerian and Lagrangian approaches, including Registration methods and Optimal Transport, with manifold-learning techniques (MDS, Isomap, Spectral Clustering, LLE) and Representation-Learning Auto-Encoders, illustrating when linear reductions suffice (diffusion-dominated) and when nonlinear pre-processing is essential (advection-dominated). Numerical results on three Advection-Diffusion test cases show that Registration can accelerate the decay of the reduced representation’s spectrum and improve reconstruction in advection scenarios, while diffusion problems are better served by linear methods; kernel and manifold methods provide varying benefits depending on the regime. The work offers a cohesive landscape linking Numerical Analysis and Representation Learning, guiding the selection and design of nonlinear reduction strategies for advection-diffusion dynamics and enabling more effective reduced-order modeling. The findings have practical impact by informing the choice of reduction pipelines for PDE data in simulations and data-driven surrogates.

Abstract

This work presents an overview of several nonlinear reduction strategies for data compression from various research fields, and a comparison of their performance when applied to problems characterized by diffusion and/or advection terms. We aim to create a common framework by unifying the notation referring to a common two-stage pipeline. At the same time, we underline their main differences and objectives by highlighting the diverse choices made for each stage. We test the considered approaches on three test cases belonging to the family of Advection-Diffusion problems, also focusing on the pure Advection and pure Diffusion benchmarks, studying their reducibility while varying the latent dimension. Finally, we interpret the numerical results under the lens of the discussed theoretical considerations, offering a comprehensive landscape for nonlinear reduction methods for general Advection-Diffusion dynamics.

Figures (10)

• Figure 1: Nonlinear reduction two-stage pipeline: considering the original manifold $\mathcal{M}$, the map $\Phi$ transforms the data into the new manifold $\mathcal{T}$, on which reduction is performed through the operator $Z_N$.
• Figure 2: Bi-directional flow comparing the properties gathered respectively with methods of Manifold Learning, Representation Learning, and Numerical Analysis in the field of nonlinear reduction.
• Figure 3: On the left, some solutions at different times of the pure Advection ($c_T=4, c_D=0$), pure Diffusion ($c_T=0, c_D=0.1$), and Advection-Diffusion systems ($c_T=4, c_D=0.1$). On the right, their eigenvalues decay for increasing values of $N\in\{1,\dots,15\}$.
• Figure 4: Eigenvalues decay for PCA and Registration methods for pure Advection, pure Diffusion, and Advection-Diffusion problems respectively on the left, in the middle, and on the right.
• Figure 5: Reconstruction of train and test data with the correspondent $l_2$ error of PCA, Registration method, and AutoEncoder for pure Advection problem with $N=2$.