Empirical sparse regression on quadratic manifolds

TL;DR

This work addresses the challenge of reconstructing high-dimensional field data from sparse measurements when dynamics are transport-dominated or wave-like. It introduces quadratic manifold sparse regression (QMSR), which learns a quadratic manifold via a sparse greedy procedure and performs nonlinear projections from sparse samples using a quadratic decoder and a sparse encoder. The method achieves orders of magnitude higher accuracy than linear methods (e.g., QDEIM/empirical interpolation) and attains accuracy comparable to full-data quadratic-manifold reconstructions, as demonstrated on Vlasov, acoustic-wave, and rotating-detonation datasets. The results highlight QMSR's potential as a scalable, non-intrusive model-reduction tool for large-scale PDEs and challenging data manifolds.

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.

Figures (11)

• Figure 1: Charged particles: QMSR approximations are visually indistinguishable from the original data vectors, even though only $m = 10$ sparse samples out of the $n = 360,000$ components are used. QMSR also leverages the expressivity of quadratic manifolds to achieve significantly higher accuracy than the linear approximations obtained with empirical interpolation.
• Figure 2: Charged particles: Doubling the number of sparse samples from $m = 10$ to $m = 20$ significantly increases the accuracy of QMSR approximations in this example. In contrast, doubling the number of samples has little effect on the linear approximations obtained with empirical interpolation because the linear approximation space is limiting the approximation accuracy.
• Figure 3: Charged particles: The QMSR approximations obtained with $m = 2r$ sparse samples achieve a comparable accuracy on test data vectors as reconstructing the full data vectors on the quadratic manifold.
• Figure 4: Charged particles: Computing encoding with the sparse linear encoder of QMSR achieves comparable results as the nonlinear encoding obtained by applying the Gauss-Newton method to nonlinear regression problem \ref{['eq:ERQM:OptiProb']}, which indicates that the sparse linear encoder of QMSR is sufficient, at least in this example.
• Figure 5: Hamiltonian wave: The QMSR approximations from $m = 20$ samples out of $n = 1,080,000$ components accurately capture the wave evolutions over time, similarly to the quadratic-manifold reconstructions that use all $n = 1,080,000$ components. Linear approximations with empirical interpolation lead to orders of magnitude higher errors.

Theorems & Definitions (7)

• Remark 1
• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• proof