Low-Fidelity Informed Uncertainty Quantification

• Low-fidelity informed uncertainty quantification is a design strategy that uses cost-effective approximate models to improve predictions and accurately characterize uncertainty.
• It combines methods such as discrepancy correction, control variate estimators, recursive surrogates, and active learning to integrate data from different fidelity levels.
• This approach is successfully applied in fields like CFD, digital twins, and kinetic simulations, yielding substantial reductions in computational cost while maintaining reliable estimates.

Low-fidelity informed uncertainty quantification denotes a class of methods in which inexpensive approximate models, cheaper simulations, or lower-accuracy labels are used to improve the prediction and uncertainty characterization of a scarce or expensive high-fidelity target. In the cited literature, low-fidelity information enters uncertainty quantification through discrepancy correction, control-variate estimators, recursive multi-fidelity surrogates, active-learning criteria, and generative posterior approximations. The shared objective is to reduce dependence on expensive high-fidelity evaluations while retaining accurate uncertainty estimates for quantities of interest, output densities, expectations, failure probabilities, or posterior distributions (Wang et al., 2015, Nitzler et al., 2020, Giannoukou et al., 2024, Cheng et al., 22 Mar 2025, Tatsuoka et al., 2 Apr 2025).

1. Problem setting and fidelity hierarchy

A standard formulation begins with uncertain inputs and a quantity of interest generated by a deterministic or noisy high-fidelity model. In Bayesian multi-fidelity Monte Carlo, the high-fidelity output density is rewritten through the low-fidelity output as

$$p(y)=\int p(y\mid y^{\mathrm{LF}})\,p(y^{\mathrm{LF}})\,dy^{\mathrm{LF}},$$

so the problem is shifted from learning the full input-output map to learning a conditional relation between low- and high-fidelity responses (Nitzler et al., 2020). In gray-box surrogate modeling with noisy data, the same basic structure appears in correction form,

$$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$

with recursive extensions for more than two fidelities (Giannoukou et al., 2024).

Within this hierarchy, high-fidelity sources are accurate but expensive or noisy, while low-fidelity sources are cheaper and more plentiful but less accurate. Several works state this distinction explicitly: high-fidelity CFD or quantum-chemical calculations cannot usually be sampled densely, whereas low-fidelity approximations can be propagated over large ensembles, used as trends, or exploited to identify informative regions of input or configuration space (Wang et al., 2015, Vinod et al., 21 Aug 2025, Kumar, 11 Mar 2025). In transonic aerodynamics, the distinction is realized as XFoil, SU2 RANS CFD, and fine-mesh RANS data; in digital twins, as sparse accurate sensor data versus cheaper and biased sources; and in kinetic equations, as full kinetic solvers versus fluid, diffusion, or BGK limits (Vaiuso et al., 2024, Desai et al., 2023, Dimarco et al., 2021).

The uncertainty itself is not uniform across the literature. Some works focus on uncertainty propagation from random inputs to outputs, some on epistemic uncertainty induced by sparse high-fidelity data, some on aleatory measurement noise, and some on both. A particularly explicit distinction is the one between confidence intervals for the underlying noise-free high-fidelity function and prediction intervals for future noisy high-fidelity observations; the latter are necessarily wider because they include irreducible observation noise (Giannoukou et al., 2024). This separation clarifies that low-fidelity informed UQ is not only about improved mean prediction, but about which uncertainty is being quantified.

2. Discrepancy learning and probabilistic correction

A canonical discrepancy-based formulation appears in the CFD framework of "Propagation of Input Uncertainty in Presence of Model-Form Uncertainty: A Multi-fidelity Approach for CFD Applications" (Wang et al., 2015). There, a cheap low-fidelity model is evaluated on a large sample set,

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$

while a small set of high-fidelity simulations is used to infer the discrepancy

$\delta(x)\approx z^H(x)-z^L(x).$

The discrepancy is modeled as a Gaussian process,

$\delta(x)\sim \mathcal{GP}(m,k),$

with zero-mean prior $m(x)=0$ and stationary squared-exponential kernel

$$k(x,x')=\sigma_f^2 \exp\left(-\frac{|x-x'|^2}{2l^2}\right).$$

Hyperparameters are chosen by maximizing the GP marginal likelihood, and posterior conditioning supplies both a correction mean and a credible uncertainty band in unsampled regions (Wang et al., 2015).

The resulting propagation algorithm is explicitly probabilistic. After fitting the GP to sparse discrepancy data, multiple realizations are drawn from the GP posterior and used to correct the low-fidelity ensemble,

$$\hat f_j^{(k)} = f_j^L + \delta^{(k)}(x_j), \qquad k=1,\dots,N_c.$$

The final output uncertainty therefore includes the uncertainty from the input sampling and the uncertainty from imperfect knowledge of model discrepancy. The method is described as a probabilistic model calibration of the low-fidelity model, not a deterministic fit, and the paper notes that discrepancy reconstruction usually inflates the propagated output uncertainty, with the inflation shrinking as more high-fidelity data are added (Wang et al., 2015).

The numerical examples make the correction mechanism concrete. In the synthetic case, five high-fidelity points suffice for the corrected output PDF to nearly match the truth, including a bimodal distribution that the low-fidelity model alone fails to capture. In the NACA 0012 example, with inputs $\alpha$ and $M_\infty$ and quantity of interest $$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$0, thin airfoil theory with Prandtl–Glauert correction serves as low fidelity and RANS solutions serve as higher-fidelity reference. Using only 10 high-fidelity simulations, the corrected PDF closely matches the truth, while with 40 high-fidelity points the posterior CDF spread becomes smaller (Wang et al., 2015).

A related but noise-aware extension appears in "Uncertainty-aware multi-fidelity surrogate modeling with noisy data" (Giannoukou et al., 2024). There the target is the underlying noise-free high-fidelity function, observed only through noisy realizations, and the multi-fidelity surrogate again uses a low-fidelity trend plus discrepancy. The distinctive addition is bootstrap construction of confidence intervals for the latent high-fidelity function and prediction intervals for future noisy observations. This framework explicitly handles noisy high-fidelity and noisy low-fidelity sources and emphasizes that interval reliability can be affected by noise and by surrogate approximation errors such as PCE truncation (Giannoukou et al., 2024).

3. Monte Carlo variance reduction, control variates, and biased sampling

A second major line of work uses low-fidelity models not as corrected predictors of the full output distribution, but as devices for reducing the variance or cost of Monte Carlo estimators. In context-aware multi-fidelity Monte Carlo, the low-fidelity models themselves are trained specifically to improve the variance-reduction efficiency of the final estimator. The central point is that a low-fidelity model need not be a uniformly excellent approximation of the high-fidelity model everywhere; it only needs to be “good enough” in the particular role it plays inside a multi-fidelity control-variate estimator (Farcas et al., 2022). This reframing leads to explicit training-versus-sampling tradeoffs and to finite optimal training sizes that are bounded independently of the total budget once the low-fidelity models are “good enough” for variance reduction (Farcas et al., 2022).

The same control-variate logic appears in kinetic equations. With a cheap surrogate $$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$1, the multi-scale control-variate estimator is written as

$$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$2

with optimal coefficient

$$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$3

The reduced variance is

$$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$4

so strongly correlated surrogates such as Euler, Navier–Stokes, BGK, diffusion limits, or steady states can produce substantial variance reduction (Dimarco et al., 2021). In this literature, the low-fidelity model is chosen because it shares the same asymptotic limit as the kinetic model in the regime of interest (Dimarco et al., 2021).

The nonlocal Cahn–Hilliard study gives a concrete multi-fidelity Monte Carlo realization with one high-fidelity model and eight surrogate models derived by varying mesh width $$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$5 and horizon $$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$6. Because the reported surrogate correlations satisfy $$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$7 to $$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$8, most of the computational budget can be allocated to the cheap models while the high-fidelity model is sampled sparsely to maintain unbiasedness. For a given computational budget, the use of MFMC results in about one-order-of-magnitude reduction in the mean-squared error of the expected value of the output of interest relative to Monte Carlo (Khodabakhshi et al., 2023).

A more aggressive use of low fidelity appears in failure probability estimation. "Langevin Bi-fidelity Importance Sampling for Failure Probability Estimation" constructs a smooth biasing density

$$\hat{\psi}_{\mathrm H}(x)=\rho\,\hat{\psi}_{\mathrm L}(x)+\delta(x),$$9

so the low-fidelity model shapes the sampling distribution itself. Because the score function of $f_j=z^L(x_j), \qquad j=1,\dots,N^L,$0 is available, the method uses the Metropolis-adjusted Langevin algorithm to concentrate high-fidelity evaluations in failure-relevant regions of the input space (Cheng et al., 22 Mar 2025). This is a different use of low fidelity: not direct prediction, but efficient steering of high-fidelity sampling.

The large-scale plasma examples further show that variance-reduction formulations can produce substantial runtime reductions. The context-aware hierarchy built around the GENE code reports speedups of up to two orders of magnitude compared to standard estimators, corresponding to a runtime reduction from 72 days to about four hours on one node of the Lonestar6 supercomputer (Farcas et al., 2022). In related plasma micro-turbulence work, data-driven low-fidelity models yield up to four orders of magnitude more efficiency than standard Monte Carlo methods, translating into a runtime reduction from around eight days to one hour on 240 cores (Konrad et al., 2021).

4. Surrogate architectures, operator learning, and probabilistic neural models

Many low-fidelity informed UQ methods are realized as surrogate-learning architectures in which low-fidelity information enters as an additional input, a residual baseline, or a recursive prior. H-PCFE combines polynomial correlated function expansion with a Gaussian process term that captures local residual variations and provides uncertainty quantification, while deep-H-PCFE extends this into a cascading arrangement of models with different fidelities using nonlinear auto-regression schemes and space-dependent cross-correlations among the models (Desai et al., 2023). The stated goal is to fuse low- and high-fidelity data, eliminate erroneous low-fidelity effects, provide predictive uncertainty, and work for both time-domain and frequency-domain digital twin updates (Desai et al., 2023).

Operator-learning formulations follow the same residual logic. The multi-fidelity wavelet neural operator trains an LF-WNO on a large low-fidelity dataset and an HF-WNO on a much smaller high-fidelity dataset, with the HF network learning the residual field

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$1

The final prediction is reconstructed as low fidelity plus predicted residual. In the stochastic heat-equation example, uncertainty is quantified through the randomness in the PDE input field and propagated to the output solution field by the surrogate; the paper does not introduce a separate epistemic-uncertainty estimator, but uses low-fidelity information to improve the fidelity of predicted solution distributions in the low-data regime (Thakur et al., 2022).

The multi-fidelity Laplace Neural Operator extends this idea with a low-fidelity base model, parallel linear and nonlinear high-fidelity correctors, and dynamic inter-fidelity weighting,

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$2

Uncertainty quantification is provided by a modified replica exchange stochastic gradient Langevin algorithm that samples an approximate posterior over model parameters; the ensemble predictions are then used to compute mean predictions and 95% confidence intervals. Across the Lorenz system, Duffing oscillator, Burgers equation, and Brusselator reaction-diffusion system, the paper reports testing losses reduced by 40% to 80% compared to traditional approaches (Zheng et al., 1 Feb 2025).

In transonic aerodynamics, MF-BayNet uses transfer learning across low-, mid-, and high-fidelity datasets while placing probability distributions over network weights. On the Benchmark Super Critical Wing, trained first on low-fidelity data, then transferred to mid-fidelity data, and finally fine-tuned on only 7 high-fidelity points, MF-BayNet reports $f_j=z^L(x_j), \qquad j=1,\dots,N^L,$3 total error on the high-fidelity test set versus $f_j=z^L(x_j), \qquad j=1,\dots,N^L,$4 for Co-Kriging, together with tighter uncertainty bounds (Vaiuso et al., 2024). In CFD surrogate modeling for the NACA0012 airfoil, co-kriging is paired with a more scalable multi-fidelity deep neural network; the paper states that co-kriging works well for low-dimensional problems but exhibits limitations when addressing 32-Dimension problems due to the limitation of memory capacity for storage and manipulation, whereas the MF-DNN efficiently predicts probability density distributions and statistical moments in 1-, 32-, and 100-dimensional tests (Kumar, 11 Mar 2025).

Generative models provide another route. BF-VAE uses an LF encoder, a latent auto-regressive model

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$5

and an HF decoder to synthesize HF realizations from abundant LF data plus a small number of HF samples. The paper introduces the bi-fidelity information bottleneck and reports improved accuracy, compared to a VAE trained using only HF data, when limited HF data is available (Cheng et al., 2023). A reduced polynomial-chaos analogue, BF-SMR, uses LF samples to form a stochastic reduced basis and a small number of HF samples to fit the final HF approximation, while deriving error bounds that assess whether a given LF/HF pair is suitable for bi-fidelity estimation (Newberry et al., 2021).

5. Active learning and Bayesian inverse problems

Low-fidelity informed uncertainty quantification is not limited to forward propagation. In active learning for quantum chemistry, LoUQAL replaces geometry-space uncertainty heuristics with a lower-fidelity-informed score,

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$6

which is the absolute prediction error at the lower fidelity. The method uses this as a proxy for where the target model is likely to be uncertain or wrong, then selects the argmax over the unlabeled pool (Vinod et al., 21 Aug 2025). On QM7b atomization energies, VIB5 potential energy surfaces, and QeMFi excitation energies, the paper states that LoUQAL consistently beats standard uncertainty measures and random sampling, and approaches the ideal greedy scheme. It also reports that common UQ measures like GPR variance or ensemble disagreement can focus too much on the edges of configuration space and can be poorly calibrated, sometimes performing worse than random sampling (Vinod et al., 21 Aug 2025).

The same paper emphasizes that fidelity choice matters substantially. In composite-fidelity analysis for QM7b, the target is fixed at CCSD(T)-cc-pVDZ while the lower fidelity varies across combinations of method and basis. The reported observation is that basis-set choice seems to matter more than the electronic-structure method in determining LoUQAL quality; for example, MP2-ccpVDZ performs much better than CCSD(T)-STO3G as the informing fidelity (Vinod et al., 21 Aug 2025). PCA visualizations show that LoUQAL selects nearly the same points as greedy adaptive sampling, while variance-based UQ selects much more peripheral points (Vinod et al., 21 Aug 2025).

Bayesian inverse problems provide a complementary setting. The conditional-diffusion framework for multi-fidelity parameter estimation first learns a low-fidelity conditional generator

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$7

which supplies amortized Bayesian inference over a wide range of observations, and then, for a specific observation, uses the low-fidelity approximation to narrow the parameter region in which expensive high-fidelity simulations are performed. A refined high-fidelity generator

$f_j=z^L(x_j), \qquad j=1,\dots,N^L,$8

is then trained on the focused high-fidelity dataset (Tatsuoka et al., 2 Apr 2025). The paper presents this as a way to avoid repeated MCMC reruns while preserving the ability to represent multimodal posterior densities (Tatsuoka et al., 2 Apr 2025).

Taken together, these two directions suggest that low-fidelity informed UQ can act either as a selection rule over candidate data points or as a proposal mechanism over parameter space. In both cases, the low-fidelity object is not the final answer; it is an inexpensive map of where high-fidelity effort is likely to matter most.

6. Assumptions, diagnostics, and limitations

The assumptions behind these methods are explicit and problem-dependent. In the CFD discrepancy framework, input uncertainty is assumed known, the low-fidelity model is assumed pre-calibrated enough to justify a zero-mean discrepancy prior, the discrepancy is assumed smooth enough for a stationary GP with squared-exponential kernel, and the posterior over discrepancy is approximated by a modular Bayesian approach in which hyperparameters are first fit by MLE and then held fixed (Wang et al., 2015). LoUQAL assumes an ordered fidelity hierarchy and access to cheaper low-fidelity labels for the same molecular configurations (Vinod et al., 21 Aug 2025). Deep-H-PCFE assumes that lower-fidelity outputs are informative enough to aid higher-fidelity correction, and notes that if fidelities are unrelated or extremely inconsistent, performance may degrade (Desai et al., 2023).

Several works stress that the value of low fidelity depends on correlation rather than on standalone accuracy. The perspective on multi-fidelity machine learning states that low-fidelity information is valuable only when it is correlated with the quantity of interest and can be used either to build a better surrogate or to reduce Monte Carlo-like estimation variance (Zhang et al., 2024). The context-aware Monte Carlo paper makes the point more sharply: the low-fidelity model need not be a uniformly excellent approximation of the high-fidelity model everywhere; it only needs to be “good enough” for variance reduction (Farcas et al., 2022). This directly counters the common misconception that low-fidelity informed UQ requires low-fidelity models to be unbiased estimators of the truth.

Diagnostics and computational bottlenecks also recur. In BF-SMR, new error bounds and practical a posteriori estimators are developed to assess the appropriateness of a given pair of LF and HF models for bi-fidelity estimation (Newberry et al., 2021). In BMFMC, informative features computed at no extra cost can reduce the complexity of the conditional high-fidelity model, but the paper notes that if too many are added, the feature space becomes too large relative to the available high-fidelity data and epistemic uncertainty increases again (Nitzler et al., 2020). In the noisy multi-fidelity surrogate framework, bootstrap-based interval construction is computationally expensive because it requires many retrained surrogate models (Giannoukou et al., 2024). In LoUQAL, repeated prediction over large pools can become computationally expensive, especially for millions of points, and low-rank approximations are suggested as a possible remedy (Vinod et al., 21 Aug 2025). In the CFD MF-DNN study, co-kriging is reported to suffer from memory limitations in 32D and especially 100D cases (Kumar, 11 Mar 2025).

Taken together, these works suggest that low-fidelity informed uncertainty quantification is best understood not as a single algorithm, but as a design principle. The low-fidelity source may act as a trend, a discrepancy scaffold, a control variate, a recursive prior, an active-learning score, a reduced basis, or a proposal distribution. Its usefulness depends on informativeness, correlation, and cost structure; its limitations arise when those conditions fail, when noise is not modeled appropriately, or when the multi-fidelity mechanism itself becomes computationally dominant.