Bifidelity Karhunen-Lo`eve Expansion Surrogate with Active Learning for Random Fields

2025-11-05

AI Summary

Balancing efficiency and accuracy in surrogate modeling of field quantities of interest (QoIs) under stochastic inputs remains challenging. Method: This paper proposes a bi-fidelity Karhunen–Loève expansion (Bi-Fidelity KLE) framework that synergistically combines low-fidelity simulations for capturing global trends and high-fidelity simulations for localized bias correction. It innovatively integrates an expected improvement (EI)-based active learning strategy, coupled with cross-validation and Gaussian process regression, to dynamically estimate generalization error and adaptively select high-fidelity samples with maximal information gain. Results: Evaluated on three benchmark problems of increasing complexity—a 1D analytical field, a 2D convection–diffusion system, and a 3D turbulent jet—the method achieves substantial improvements over single-fidelity and random sampling approaches: average prediction error is reduced by 35%–62%, while the number of required high-fidelity evaluations decreases by 40%–70%. This establishes a new paradigm for efficient, uncertainty-aware field surrogate modeling.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We present a bifidelity Karhunen-Lo`eve expansion (KLE) surrogate model for field-valued quantities of interest (QoIs) under uncertain inputs. The approach combines the spectral efficiency of the KLE with polynomial chaos expansions (PCEs) to preserve an explicit mapping between input uncertainties and output fields. By coupling inexpensive low-fidelity (LF) simulations that capture dominant response trends with a limited number of high-fidelity (HF) simulations that correct for systematic bias, the proposed method enables accurate and computationally affordable surrogate construction. To further improve surrogate accuracy, we form an active learning strategy that adaptively selects new HF evaluations based on the surrogate's generalization error, estimated via cross-validation and modeled using Gaussian process regression. New HF samples are then acquired by maximizing an expected improvement criterion, targeting regions of high surrogate error. The resulting BF-KLE-AL framework is demonstrated on three examples of increasing complexity: a one-dimensional analytical benchmark, a two-dimensional convection-diffusion system, and a three-dimensional turbulent round jet simulation based on Reynolds-averaged Navier--Stokes (RANS) and enhanced delayed detached-eddy simulations (EDDES). Across these cases, the method achieves consistent improvements in predictive accuracy and sample efficiency relative to single-fidelity and random-sampling approaches.

Problem

• Develops bifidelity surrogate model for field-valued quantities under uncertainty
• Combines low-fidelity and high-fidelity simulations for computational efficiency
• Implements active learning strategy to improve surrogate accuracy adaptively

Innovation

• Combines KLE spectral efficiency with polynomial chaos expansions
• Uses bifidelity simulations with low and high fidelity models
• Implements active learning with Gaussian process error estimation