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

arXiv: https://arxiv.org/abs/2511.03756 · HTML: https://mlsanity.com/paper/2511.03756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 50 references that are in the map.
- First posted on arXiv 5 Nov 2025.

## Builds on (most cited first, 8 of 8)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Batch Bayesian Optimization via Local Penalization](https://mlsanity.com/paper/1505.08052/) (2015)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Unbounded Bayesian Optimization via Regularization](https://mlsanity.com/paper/1508.03666/) (2015)
- [Bayesian Optimization for Probabilistic Programs](https://mlsanity.com/paper/1707.04314/) (2017)
- [Polynomial Chaos Surrogate Construction for Random Fields with Parametric Uncertainty](https://mlsanity.com/paper/2311.00553/) (2023)

## Abstract (arXiv)

We present a bifidelity Karhunen--Lo`eve expansion (KLE) surrogate model for field-valued quantities of interest (QoIs) under uncertain inputs. The QoIs considered here are scalar fields. 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 can enable accurate and computationally affordable surrogate construction. To further improve surrogate accuracy, we develop 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). The experiments show that bifidelity gains depend on LF accuracy, discrepancy approximation, and the allocation of simulation cost. Active learning improves prediction over random sampling in several settings, while the cost-matched comparisons identify both favorable regimes and cases where an HF-only surrogate is more accurate.