Multifidelity Proper Orthogonal Decomposition

2026-05-27

AI Summary

This study addresses the challenges of high-fidelity Proper Orthogonal Decomposition (POD), which is prone to overfitting and computationally expensive when snapshot data are scarce. To mitigate these issues, the authors propose a multi-fidelity POD framework that integrates low-fidelity model data via a controlled-variable approach, enabling weighted coupling of high- and low-fidelity snapshots. This integration yields an unbiased estimator of the projection error, which is then leveraged to optimize the POD subspace. The method substantially reduces the variance of the error estimator under limited computational budgets, enhancing subspace robustness and effectively alleviating overfitting. Demonstrated on Pine Island Glacier velocity modeling, the approach achieves accuracy comparable to single-fidelity POD while requiring only one-tenth of the offline snapshot generation cost.

Abstract

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.

Problem

• Proper Orthogonal Decomposition
• Multifidelity
• Snapshot Data
• Computational Cost
• Overfitting

Innovation

• multifidelity
• proper orthogonal decomposition
• reduced-order modeling
• control variate
• computational efficiency