Gaussian-process-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility

Following the promising results of a previous work using artificial neural networks (PROM-ANN) (Barnet et al. 2023 JCP) and the realization that one single hidden layer leads to same order-of-magnitude errors, a combination of a projection-based reduced-order model (PROM) and a Gaussian process (GP) is proposed to mitigate the Kolmogorov barrier to reducibility of parametric and/or highly nonlinear, high-dimensional, physics-based models. The main objective of our PROM-GP concept is to reduce the dimensionality of the online approximation of the solution beyond what is achievable using affine and quadratic approximation manifolds, while maintaining accuracy. As well as for its PROM-ANN counterpart, the training of the GP part does not involve data whose dimension scales with that of the high-dimensional model; and the resulting PROM-GP can be efficiently hyperreduced using any well-established hyperreduction method. The added value of using GPs is in the ability to derive mathematical bounds for the errors, not possible in the case of ANNs. All these features make the present concept particularly well-suited for industry-relevant computational problems. Finally, we demonstrate the computational tractability of its offline stage and the superior wall clock time performance of its online stage for a large-scale, parametric, two-dimensional, model problem that is representative of shock-dominated unsteady flow problems.