[Paper Review] Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

Virginie Ehrlacher, Damiano Lombardi+2 more|arXiv (Cornell University)|Sep 14, 2019

Model Reduction and Neural Networks73 references38 citations

TL;DR

The paper develops nonlinear model reduction for parametrized PDEs in metric spaces, focusing on Wasserstein space, and introduces two data-driven approaches (tangent-space PCA and barycenter-based) to efficiently approximate solution sets of 1D conservative PDEs.

ABSTRACT

We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. As a consequence of working in metric spaces, both approaches are automatically nonlinear. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the $L^2$-Wasserstein space.

Motivation & Objective

• Motivate model reduction beyond Banach/Hilbert spaces for transport-dominated problems.
• Propose nonlinear reduction methods in general metric spaces to capture translations and shifts.
• Specialize the framework to the L2-Wasserstein space in 1D and analyze theoretical benefits.
• Provide numerical evidence on simple PDEs showing improved approximation properties.

Proposed method

• Introduce two nonlinear reduction strategies in metric spaces: one based on tangents (tPCA) and another using barycenters (gBar).
• Use the exponential/logarithmic maps on Wasserstein space to transform the solution set for reduction.
• Define and compute barycenters of snapshot sets via optimal weights in Wasserstein space.
• Show that transforming the solution set with Log_w yields faster decay of Kolmogorov widths in transport problems.
• Utilize the isometry between (P2(Omega), W2) and (I, ||·||_L2) to perform reduction in a Euclidean setting after transformation.
• Provide purely data-driven online phase that does not solve the original PDE in the reduced space.

Experimental results

Research questions

• RQ1 Can model reduction in metric spaces outperform linear methods for transport-dominated PDEs?
• RQ2 Do tangent-space PCA and barycenter-based reductions provide effective low-complexity representations in Wasserstein spaces?
• RQ3 How do logarithmic/logarithmic transformations affect Kolmogorov widths of solution manifolds in 1D conservative PDEs?
• RQ4 What is the impact of these nonlinear reductions on computational cost and accuracy for Burgers, Camassa-Holm, and KdV-type equations?

Key findings

• For a pure transport problem, the transformed width d_n(T, L2([0,1])) vanishes for n>1, while the original width d_n(M, L2(Omega)) decays no faster than n^-1/2.
• Inviscid Burgers’ equation examples illustrate a priori width decay behaviors suggesting nonlinear metric-based reduction can outperform linear methods.
• Numerical experiments on Burgers’, Camassa-Holm, and KdV demonstrate the ability of tPCA and gBar to capture transport phenomena at much reduced online cost (viscous Burgers’: ~100x speedup).
• The framework enables purely data-driven online emulation that avoids solving the original PDE online in reduced spaces.