A data-driven model can be built to accurately accelerate computationally expensive physical simulations, which is essential in multi-query problems, such as uncertainty quantification, design optimization, optimal control, and inverse problems. It is important to build interpretable and explainable models when data is integrated to predict physics calculations for unseen parameters to guarantee desirable accuracy and preserve the underlying structure. For such data-driven approaches, the procedural formalism from data compression to latent space dynamics discovery will be discussed, e.g., nonlinear and linear compression and discovery of latent space dynamics from black-box to physics-constrained approaches. Several recent developments of generalizable and robust data-driven reduced order models with different level of interpretability will be demonstrated for various physical simulations. For example, a hyper-reduced time-windowing physics-constrained reduced order model overcomes the difficulty of advection-dominated shock propagation phenomenon, achieving a speed-up of O(20~100) with a relative error much less than 1% for Lagrangian hydrodynamics problems, such as 3D Sedov blast problem, 3D triple point problem, 3D Taylor–Green vortex problem, 2D Gresho vortex problem, and 2D Rayleigh–Taylor instability problem. The nonlinear manifold reduced order model also overcomes the challenges posed by the problems with slowly decaying Kolmogorov’s width through deep compressibility of neural network without introducing windowing approach. The space–time reduced order model accelerates a large-scale particle Boltzmann transport simulation by a factor of 2,700 with a relative error less than 1%. Furthermore, successful application of these reduced order models for meta-material lattice–structure design optimization problems will be presented. Finally, the library for reduced order models, i.e., libROM (www.librom.net), and its webpage and several YouTube tutorial videos about data-driven approaches will be introduced.

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