Numerical investigations of reduced-order models with time-varying parameters generated by hybrid training with automatic hyper-reduction in nonlinear diffusion–reaction systems

In this paper, we investigate the reduced-order models (ROM) with time-varying parameters for nonlinear diffusion–reaction systems generated by a new hybrid training framework with automatic hyper-reduction. Due to limited computational acceleration in ROM where the system matrices and Jacobians need to be updated at scale per time state, this framework is designed with alternations between full-order modeling (FOM) ⇔ ROM with automatic hyper-reduction (ROM-hyper) to tackle the speedup in training process for nonlinear problems with time-varying parameters (the parameters are time-vary values). In the whole training period, the total time span is divided into multiple intervals with equal length, and each of them is simulated on-the-fly with either FOM or local ROM-hyper, and the model switch criteria and algorithms are discussed in detail. During the hybrid training, the numbers of basis vectors in both state-space data and residuals are automatically updated at the end of each FOM period, which indicates the POD numbers for training data and residuals do not need to be determined in a priori. In numerical studies, several nonlinear diffusion–reaction problems with time-varying parameters are examined for validation and verification. It is found that the POD basis vectors are updated on-the-fly, and the numbers of basis vectors in both state-space training data and residuals increase with the number of update procedures. Computational acceleration in training process can be achieved by controlling the tolerances of model switch criteria, and the computational acceleration reaches about from 33% to 50%.