Nicole Aretz

Nicole Aretz, Karen Willcox

This paper introduces a multifidelity formulation that reduces the computational cost of the proper orthogonal decomposition (POD) of a high-fidelity model by leveraging data from cheaper, lower-fidelity models. POD is a prevalent technique for extracting a low-dimensional basis from training data to achieve subsequent dimension reduction or reduced-order modeling. In scientific and engineering applications, the training data are typically numerical snapshot solutions of a high-fidelity model, and computation of a sufficiently rich snapshot set can be prohibitively expensive, especially when sampling over a high-dimensional parameter space. Insufficient snapshot training data risks overfitting and poor generalizability of the POD basis to outside the training regime. Our multifidelity POD (MFPOD) formulation reallocates computational budget to cheaper, low-fidelity models that can be sampled more extensively. MFPOD then weights high- and low-fidelity snapshot data via a control-variate formulation to guarantee an unbiased estimate of the expected high-fidelity least-squares projection error. The MFPOD subspace is chosen to minimize the estimate of this projection error, and converges in probability to the same subspace as single-fidelity POD in the limit of an arbitrarily large budget. For restrictive computational budgets, the MFPOD cost function has (under some assumptions) lower variance than the POD cost function, which makes the MFPOD subspace more robust against variations in the training data and thus less prone to overfitting. For a numerical example modeling the velocity of the Pine Island glacier, MFPOD achieves the same accuracy as single-fidelity POD with an order of magnitude reduction in the offline computational cost of snapshot generation.

Nicole Aretz, Karen Willcox

This paper presents a data-driven, nested Operator Inference (OpInf) approach for learning physics-informed reduced-order models (ROMs) from snapshot data of high-dimensional dynamical systems. The approach exploits the inherent hierarchy within the reduced space to iteratively construct initial guesses for the OpInf learning problem that prioritize the interactions of the dominant modes. The initial guess computed for any target reduced dimension corresponds to a ROM with provably smaller or equal snapshot reconstruction error than with standard OpInf. Moreover, our nested OpInf algorithm can be warm-started from previously learned models, enabling versatile application scenarios involving dynamic basis and model form updates. We demonstrate the performance of our algorithm on a cubic heat conduction problem, with nested OpInf achieving a four times smaller error than standard OpInf at a comparable offline time. Further, we apply nested OpInf to a large-scale, parameterized model of the Greenland ice sheet where, despite model form approximation errors, it learns a ROM with, on average, 3% error and computational speed-up factor above 19,000.