Data-Driven Model Order Reduction for Problems with Parameter-Dependent Jump-Discontinuities

TL;DR

This paper introduces a data-driven model order reduction method for parametrized PDEs with parameter-dependent jump-discontinuities, using spatial transforms and regression to improve approximation quality.

Contribution

It presents a novel MOR approach that handles discontinuities by transforming solutions and applying regression, enabling efficient online approximation.

Findings

Effective for hyperbolic and parabolic PDEs with discontinuities

Decoupled online and offline stages improve computational efficiency

Transform-based approach enhances low-dimensional approximation quality

Abstract

We propose a data-driven model order reduction (MOR) technique for parametrized partial differential equations that exhibit parameter-dependent jump-discontinuities. Such problems have poor-approximability in a linear space and therefore, are challenging for standard MOR techniques. We build upon the methodology of approximating the map between the parameter domain and the expansion coefficients of the reduced basis via regression. The online stage queries the regression model for the expansion coefficients and recovers a reduced approximation for the solution. We propose to apply this technique to a transformed solution that results from composing the solution with a spatial transform. Unlike the (untransformed) solution, it is sufficiently regular along the parameter domain and thus, is well-approximable in a low-dimensional linear space. To recover an approximation for the (untransformed) solution, we propose an online efficient regression-based technique that approximates the inverse of the spatial transform. Our method features a decoupled online and offline stage, and benchmark problems involving hyperbolic and parabolic equations demonstrate its effectiveness.