Uncertainty-Aware Aero Data Fusion

Updated 22 December 2025

Uncertainty-aware aerodynamic data fusion integrates multi-source data with Bayesian frameworks to quantify both aleatoric and epistemic uncertainties for robust predictions.
It leverages latent-space representations and deep learning with Ensemble Kalman Filters to efficiently reconstruct high-dimensional flow fields in real time.
Multi-fidelity fusion and sensor adaptivity enhance reliability under sparse measurements and nonlinear flow conditions, driving improved aerodynamic design.

Uncertainty-aware aerodynamic data fusion denotes a family of mathematical and algorithmic frameworks that integrate multi-source, multi-fidelity, and/or sensor-based aerodynamic data in a manner that rigorously quantifies—propagates, calibrates, and adaptively manages—both predictive and epistemic uncertainties. This paradigm spans approaches from Bayesian sensor fusion, ensemble filtering, and probabilistic deep learning, to conformal calibration, and underpins robust flow reconstruction, control, and probabilistic design in conditions marked by sparse measurements, nonlinear flow phenomena, and heterogeneous data reliability.

1. Bayesian and Sequential Inference Foundations

The Bayesian probabilistic framework forms the core of uncertainty-aware aerodynamic data fusion. For a state vector $x$ (e.g., high-dimensional flow field or compressed modal coefficients) and measurement vector $y$ (e.g., sparse or dense wall pressure, sensor output), Bayes' theorem encodes the posterior uncertainty:

$p(x \mid y) \propto p(y \mid x) p(x)$

where $p(x)$ is the prior, and $p(y \mid x)$ is the likelihood determined by the observation model, such as $y = h(x) + \epsilon$ , with sensor noise $\epsilon \sim \mathcal{N}(0, R)$ (Eldredge et al., 27 Feb 2025).

Static flow inversion problems, particularly those exhibiting nonlinear or nonunique mappings (e.g., reconstructing vortex positions from wall-pressure), often exhibit multimodal posteriors, requiring sampling approaches (MCMC, importance sampling) for full uncertainty quantification (Eldredge et al., 27 Feb 2025). For unsteady or real-time applications, sequential Bayesian filtering—especially variants of Ensemble Kalman Filter (EnKF) and its low-rank or invariant extensions—permits efficient recursive updates of the conditional distribution of $x$ given streaming sensor observations (Mousavi et al., 4 Sep 2025, Eldredge et al., 27 Feb 2025, Ye et al., 2023).

In these progressive estimation schemes, uncertainty propagation and adaptive covariance updating are central, with mechanisms to incorporate process noise, sensor noise, and forecast error to avoid filter divergence and ensure credible intervals for physical observables.

2. Low-Dimensional Latent Representations and Surrogate Modeling

Uncertainty-aware aerodynamic data fusion increasingly leverages compressed, data-driven representations of the flow state, primarily via nonlinear autoencoders or principal-mode expansions. High-dimensional physical fields ( $\sim$ 10⁴–10⁵ DOF) are encoded into low-rank latent vectors (e.g., $n=3$ –$7$), which serve as the primary variable in subsequent estimation, filtering, or regression pipelines (Mousavi et al., 4 Sep 2025, Mousavi et al., 6 Jan 2025, Nieto-Centenero et al., 15 Dec 2025).

Fusion frameworks map sensor data into this latent space through learned observation operators. For example, in real-time flow estimation from 11 pressure sensors:

The physical state $x \in \mathbb{R}^{28,800}$ is compressed via an encoder $F_\phi$ (typically 2D CNN with aggressive pooling), yielding $z \in \mathbb{R}^{7}$ .
A decoder $G_\psi$ inverts $z$ back to the full vorticity field, lift, and predicted pressure at sensor locations.
Latent dynamics $dz/dt = f_\theta(z)$ are modeled via neural ODEs or RNNs, discretized for sequential EnKF updates (Mousavi et al., 4 Sep 2025).

The training of both encoder-decoder and observation operators is conducted with multi-term loss functions penalizing reconstruction errors in physical, sensor, and mission-relevant QoIs (e.g., aerodynamic loads), as well as temporal smoothness in latent trajectories.

This latent-space formalism is essential for both computational efficiency and for enforcing physical observability constraints. Uncertainties can be modeled and propagated explicitly in this space, elucidating which state directions are constrained by available sensors and which remain ambiguous (“nullspace” to observation operator gradients).

3. Multi-Fidelity Data Fusion and Uncertainty Quantification

When fusing multi-fidelity aerodynamic data (e.g., low-fidelity panel or vortex methods, RANS, wind-tunnel/LES/DNS), uncertainty-aware frameworks must account both for bias, variance, and hierarchical information content:

Co-Kriging and Multi-Fidelity Gaussian Processes (MF-GP): Use autoregressive constructions where high-fidelity predictions are corrected versions of low/mid-fidelity surrogates, with recursive propagation of variances; posterior variance quantifies residual epistemic error and model-form uncertainty (Mukhopadhaya et al., 2019, Kumar, 11 Mar 2025).
Autoencoder-Based Transfer Learning: Low-fidelity simulations are used for pretraining, creating a physics-informed latent representation. Frozen encoders are coupled to decoders fine-tuned with scarce high-fidelity data. Uncertainty bands are placed on outputs using conformal prediction protocols, yielding rigorous empirical coverage (Nieto-Centenero et al., 15 Dec 2025).
Multi-Fidelity Bayesian Neural Networks (MF-BNNs): Deep variational inference and transfer learning over hierarchical datasets, with epistemic and aleatoric uncertainties combined by Monte Carlo sampling of NN weights and explicit likelihood variances (Vaiuso et al., 8 Jul 2024). This strategy yields highly calibrated CIs even when HF samples are scarce.

4. Sensor Informativeness, Observability, and Adaptive Sensing

Sensor placement, informativeness, and dynamic re-weighting under degraded conditions are rigorously addressed via observability metrics and Gramian analysis:

The eigenvalue spectra of observation Gramians quantify how sensor arrays resolve latent flow directions; leading eigenvectors (“observation modes”) correspond to sensor combinations most influential for uncertainty reduction (Mousavi et al., 4 Sep 2025, Mousavi et al., 6 Jan 2025).
Time-resolved analysis shows that sensor importance can be highly situation dependent (e.g., leading-edge sensors dominate during gust impingement), guiding adaptive reconfiguration for robust estimation.
Simulated sensor dropout is handled by inflating noise variance for the missing channel. Rank-deficient Kalman updates automatically suppress corrections in unobservable directions and adaptively weight neighboring sensors (Mousavi et al., 4 Sep 2025).

5. Modeling and Propagating Uncertainty: Aleatoric and Epistemic Components

Rigorous uncertainty quantification distinguishes irreducible sensor/data noise (aleatoric) from model uncertainty (epistemic):

Aleatoric UQ is commonly modeled by heteroscedastic outputs in neural networks (data-driven covariance prediction or Cholesky-factor networks), and, in GPs/MF-Kriging, via fidelity-dependent noise levels (Mousavi et al., 6 Jan 2025, Nieto-Centenero et al., 15 Dec 2025, Mukhopadhaya et al., 2019).
Epistemic UQ arises from finite/biased datasets and model-form error, estimated via ensemble methods, Monte Carlo dropout (variational Bayesian approximation), or full Bayesian neural networks (stochastic latent or weight sampling) (Mousavi et al., 6 Jan 2025, Vaiuso et al., 8 Jul 2024).
Uncertainty is propagated from sensor-level, through latent representations (e.g., sampling from $\mathcal{N}(\mu(x), \Sigma(x))$ ), to physical outputs by Monte Carlo forwarding through decoders, enabling pixelwise or integral uncertainty bands on flow and aerodynamic QoIs (Mousavi et al., 6 Jan 2025, Mousavi et al., 4 Sep 2025).

The aggregated predictive variance combines these components:

$\mathrm{Var}[\hat{y}] = \mathbb{E}_{w}[ \mathrm{Var}(\hat{y} \mid w) ] + \mathrm{Var}_{w}[ \mathbb{E}(\hat{y} \mid w) ]$

with the first term (aleatoric/data) and second (epistemic/model) separable by dropout-ensemble or variational sampling.

6. Algorithmic and Computational Aspects

Efficient real-time operation is enabled through multiple algorithmic mechanisms:

Fully latent-space assimilation steps (forecast and Kalman update) restrict updates to observable subspaces, accelerating computation to $\sim$ 10 ms per update (50 Hz) for moderate latent dimensions ( $n=7$ ) and sensor counts ( $d=11$ ) (Mousavi et al., 4 Sep 2025).
Low-rank truncation of covariances and Kalman gains, equivariant ensemble design (e.g., right-invariant filtering for state groups on $SE_2(3)\times SO(3)$ ), and adaptive ensemble tuning are adopted for scalability in high-dimensional settings and robustness to model/sensor failures (Mousavi et al., 4 Sep 2025, Ye et al., 2023).
Deep surrogate models (autoencoders, MF-DNNs, MF-BNNs) drastically reduce inference cost relative to high-fidelity CFD or full MCMC sampling, supporting kHz-range embedded deployment for flight monitoring, active control, and adaptive sensing (Mousavi et al., 6 Jan 2025, Nieto-Centenero et al., 15 Dec 2025, Mousavi et al., 4 Sep 2025).

7. Applications, Validation, and Limitations

Applications span disturbed flow reconstruction under gusts, sensor-limited CFD database acceleration, digital twin field fusion, and full-state UAV navigation under GNSS denial (Mousavi et al., 4 Sep 2025, Nieto-Centenero et al., 15 Dec 2025, Renganathan et al., 2019, Ye et al., 2023).

Quantitative results from these frameworks include:

Vorticity reconstruction errors of $\epsilon \lesssim 0.15$ –$0.20$ after a few assimilation steps, lift RMSE $\sim 0.1$ –$0.2$ for strong unsteadiness (Mousavi et al., 4 Sep 2025).
Surrogate Monte Carlo matches analytical QoI PDFs in high-dim (32D, 100D) benchmarks; MF-DNNs outperform co-kriging in accuracy and scalability (Kumar, 11 Mar 2025).
MSCP-calibrated pressure intervals achieve empirical coverage $\geq$ 95% with tens of HF samples (Nieto-Centenero et al., 15 Dec 2025).

Key limitations include unobservable or weakly constrained directions, failure modes with qualitatively mismatched LF/HF data, and computational costs at extreme database dimensionality; ongoing research targets incorporation of physics-informed priors and advanced calibration for simultaneous multivariate coverage (Mousavi et al., 4 Sep 2025, Nieto-Centenero et al., 15 Dec 2025, Mukhopadhaya et al., 2019).

References

“Sequential estimation of disturbed aerodynamic flows from sparse measurements via a reduced latent space” (Mousavi et al., 4 Sep 2025)
“Multi-fidelity aerodynamic data fusion by autoencoder transfer learning” (Nieto-Centenero et al., 15 Dec 2025)
“Uncertainty Quantification for Multi-fidelity Simulations” (Kumar, 11 Mar 2025)
“Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements” (Mousavi et al., 6 Jan 2025)
“A review of Bayesian sensor-based estimation and uncertainty quantification of aerodynamic flows” (Eldredge et al., 27 Feb 2025)
“Multi-Fidelity Bayesian Neural Network for Uncertainty Quantification in Transonic Aerodynamic Loads” (Vaiuso et al., 8 Jul 2024)
“Aerodynamic Data Fusion Towards the Digital Twin Paradigm” (Renganathan et al., 2019)
“Multi-Fidelity modeling of Probabilistic Aerodynamic Databases for Use in Aerospace Engineering” (Mukhopadhaya et al., 2019)
“Semi-Aerodynamic Model Aided Invariant Kalman Filtering for UAV Full-State Estimation” (Ye et al., 2023)