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 Kolmogorov $m$-width decay of a 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.

双曲方程的快照矩阵具有缓慢的奇异值衰减，导致低效的降阶模型。我们开发了通过在变换的空间域上计算快照来引发更快的奇异值衰减的想法，或者所谓的快照校准/变换。我们对涉及激波碰撞、激波稀疏扇形碰撞、激波形成等问题特别感兴趣。对于此类问题，我们提出了一种可实现的算法，使用单调特征匹配来计算空间变换。我们将不连续性和扭结视为特征，并通过仔细划分参数域，确保空间变换具有从理论和实现角度来看都理想的属性。我们使用这些属性来证明我们的方法可以产生快速的柯尔莫哥洛夫算法$米$-校准流形的宽度衰减。我们做出的一个重要观察是，由于校准，$米$-宽度不仅取决于$米$还在于全阶模型的准确性，这与不需要校准的椭圆和抛物线问题形成对比。我们提出的方法只需要解快照，而不需要底层的偏微分方程（PDE），因此是数据驱动的。我们进行了几次数值实验来证明我们方法的有效性。