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 five different physics problems, our physics-informed clustering strategy significantly outperforms classic strategies for the construction of local reduced-order bases in terms of projection errors.
A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations
SandGold AI Research（SandGold AI研究院）；University of Macau（澳门大学）；Xiamen University（厦门大学）；Sichuan University（四川大学）；South China Agricultural University（华南农业大学）；Macau University of Science and Technology（澳门科技大学）
Covariance-Informed Subspace: an Adaptive Gradient-Free Input Dimension Reduction Method for Bayesian Inference
CEA DAM DIF（法国原子能及替代能源委员会达姆达夫研究所）；Geosciences center, Mines Paris PSL（巴黎高等矿业学校PSL地球科学中心）；CMAP, CNRS, Ecole Polytechnique（法兰西学院国家科学研究中心数学与计算物理中心）；Autorité de Radioprotection et de Sureté Nucléaire (ASNR), PSE-ENV/SCAN（核安全与辐射防护管理局）
opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation
Korea Institute for Advanced Study (KIAS)（韩国高等研究院）；University of Southern California（南加利福尼亚大学）；Pohang University of Science and Technology (POSTECH)（浦项科技大学）
