Nonlinear Model Order Reduction using Diffeomorphic Transformations of a Space-Time Domain

TL;DR

This paper introduces a nonlinear model order reduction method for hyperbolic conservation laws that uses diffeomorphic transformations of space-time to align discontinuities, enabling better approximation of complex solutions.

Contribution

The proposed approach employs diffeomorphic transformations and Lie group theory to improve model reduction for hyperbolic PDEs with discontinuities, a novel technique in this context.

Findings

Effective alignment of shocks in numerical experiments

Potential for improved reduced models of hyperbolic equations

Utilizes Lie group structure for diffeomorphism reduction

Abstract

In many applications, for instance when describing dynamics of fluids or gases, hyperbolic conservation laws arise naturally in the modeling of conserved quantities of a system, like mass or energy. These types of equations exhibit highly nonlinear behaviors like shock formation or shock interaction. In the case of parametrized hyperbolic equations, where, for instance, varying transport velocities are considered, these nonlinearities and strong transport effects result in a highly nonlinear solution manifold. This solution manifold cannot be approximated properly by linear subspaces. To this end, nonlinear approaches for model order reduction of hyperbolic conservation laws are required. We propose a new method for nonlinear model order reduction that is especially well-suited for hyperbolic equations with discontinuous solutions. The approach is based on a space-time discretization and employs diffeomorphic transformations of the underlying space-time domain to align the discontinuities. To derive a reduced model for the diffeomorphisms, the Lie group structure of the diffeomorphism group is used to associate diffeomorphisms with corresponding velocity fields via the exponential map. In the linear space of velocity fields, standard model order reduction techniques, such as proper orthogonal decomposition, can be applied to extract a reduced subspace. For a parametrized Burgers' equation with two merging shocks, numerical experiments show the potential of the approach.