Bifidelity Karhunen-Loève Expansion Surrogate with Active Learning for Random Fields

TL;DR

This work tackles uncertainty quantification for field-valued QoIs with costly high-fidelity simulations. It introduces BF-KLE-AL, a surrogate framework that fuses bifidelity modeling with Karhunen–Loève expansions and polynomial chaos to maintain explicit input–output mappings while leveraging cheap low-fidelity simulations. An active learning loop uses cross-validated surrogate error modeled by Gaussian processes to selectively acquire new high-fidelity data via an expected improvement criterion, balancing accuracy and cost. The method is demonstrated on a 1D analytical benchmark, a 2D convection–diffusion problem, and a 3D turbulent jet simulation, consistently improving predictive accuracy and sample efficiency over single-fidelity and random-sampling approaches. These results highlight the potential of BF-KLE-AL for efficient forward UQ in complex, field-valued systems with heterogeneous fidelity data.

Abstract

We present a bifidelity Karhunen-Loève 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.

Figures (17)

• Figure 1: Evaluations for $y_{\textrm{LF}}$ and $y_{\textrm{HF}}$ at select values of $a$ and $b$ for C1 (top row) and C2 (bottom row)
• Figure 2: Pearson correlation coefficient over the spatial domain between $y_{\textrm{HF}}$ and $y_{\textrm{LF}}$ over $x$ for C1 (left) and C2 (right), calculated using 1000 samples of $\theta$. C2 exhibits a more consistent correlation throughout the domain.
• Figure 3: Pilot design for the 1D pulse problem with $(N_{\mathrm{LF}}^P, N_{\Delta}^P)=(200, 5)$. The design is mapped from the non-dimensional $(\xi_1,\xi_2)$ space to the corresponding $\theta$ parameters for C1 and C2 via affine scaling.
• Figure 4: Comparison of integrated relative error for BF-KLE-AL, BF-KLE-RS, and anti-informative sampling using 60 additional HF points for C1 and C2. Results are averaged over 20 replicates.
• Figure 5: Pilot HF samples and additional points acquired via BF-KLE-AL (left), anti-informative sampling (middle), and BF-KLE-RS (right), aggregated over 5 replicates for C2. Points are colored by acquisition order.