Data-Driven Snapshot Calibration via Monotonic Feature Matching

TL;DR

This paper introduces a data-driven snapshot calibration method for hyperbolic equations that uses monotonic feature matching to improve the decay of singular values, enabling more efficient reduced-order modeling.

Contribution

We propose a novel, realizable algorithm for spatial transformation based on monotonic feature matching, improving snapshot efficiency without requiring PDE knowledge.

Findings

Faster m-width decay of calibrated manifold.

Calibration reduces dependence on full model accuracy.

Method effectively handles shock-related phenomena.

Abstract

Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast m-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the m-width does not only depend on m but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method.