  Nonlinear model order reduction has opened the door to parameter optimization
and uncertainty quantification in complex physics problems governed by
nonlinear equations. In particular, the computational cost of solving these
equations can be reduced by means of local reduced-order bases. This article
examines the benefits of a physics-informed cluster analysis for the
construction of cluster-specific reduced-order bases. We illustrate that the
choice of the dissimilarity measure for clustering is fundamental and highly
affects the performances of the local reduced-order bases. It is shown that
clustering with an angle-based dissimilarity on simulation data efficiently
decreases the intra-cluster Kolmogorov $N$-width. Additionally, an a priori
efficiency criterion is introduced to assess the relevance of a ROM-net, a
methodology for the reduction of nonlinear physics problems introduced in our
previous work in [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order
reduction assisted by deep neural networks (ROM-net), Advanced Modeling and
Simulation in Engineering Sciences 7 (16), 2020]. This criterion also provides
engineers with a very practical method for ROM-nets' hyperparameters
calibration under constrained computational costs for the training phase. On a
non-reducible 1D heat transfer problem, our physics-informed clustering
strategy significantly outperforms classic strategies for the construction of
local reduced-order bases in terms of projection errors.
  Keywords: local reduced-order bases, cluster analysis, dissimilarity
measures, ROM-nets.
