Empirical sparse regression on quadratic manifolds

TL;DR

This paper introduces quadratic manifold sparse regression, a nonlinear approach that significantly improves approximation accuracy for transport-dominated data compared to traditional linear methods.

Contribution

The paper proposes a novel quadratic manifold sparse regression method with a greedy training algorithm and nonlinear projection for better data approximation.

Findings

Achieves orders of magnitude higher accuracy than linear methods.

Effectively models transport-dominated and wave-like dynamics.

Demonstrated through numerical experiments.

Abstract

Approximating field variables and data vectors from sparse samples is a key challenge in computational science. Widely used methods such as gappy proper orthogonal decomposition and empirical interpolation rely on linear approximation spaces, limiting their effectiveness for data representing transport-dominated and wave-like dynamics. To address this limitation, we introduce quadratic manifold sparse regression, which trains quadratic manifolds with a sparse greedy method and computes approximations on the manifold through novel nonlinear projections of sparse samples. The nonlinear approximations obtained with quadratic manifold sparse regression achieve orders of magnitude higher accuracies than linear methods on data describing transport-dominated dynamics in numerical experiments.