Epistemic Uncertainty Quantification

• Epistemic Uncertainty Quantification is the systematic evaluation of model uncertainty stemming from incomplete knowledge, critical in high-dimensional and data-limited regimes.
• It employs Bayesian methods, variational inference, deep ensembles, and normalizing flows to capture complex, multimodal posterior distributions efficiently.
• EUQ underpins practical applications like ptychographic imaging and aerodynamic load inference by providing pixelwise confidence measures for robust experimental and reconstruction designs.

Epistemic uncertainty quantification (EUQ) is the systematic assessment of a model's uncertainty due to incomplete knowledge—e.g., parametric, structural, or data-driven ambiguities—distinct from aleatoric uncertainty arising from irreducible noise. In scientific computing, image reconstruction, and dynamical system estimation, EUQ is essential for evaluating trustworthiness and identifying failure modes of predictive solvers—especially in high-dimensional, ill-posed, or data-limited regimes. EUQ supplies pixelwise, fieldwise, or pointwise confidence measures to highlight regions of low confidence and guide further data acquisition or experimental design.

1. Mathematical and Statistical Foundations

Formally, EUQ seeks to characterize the posterior distribution $p(\theta|y)$ over unknown quantities $\theta$ (model parameters, latent fields, or predicted states) given measurements $y$. Inverse problems in imaging and flow reconstruction—such as ptychography or sparse sensor interpolation—require fundamental advances in scalable posterior approximation because the solution space is high-dimensional and often multimodal. Bayesian approaches provide a principled path for epistemic uncertainty quantification:

• The Bayesian posterior embodies all knowledge after assimilating $y$, with epistemic uncertainty tied to prior ignorance reduced by data.
• For neural surrogates, parameter uncertainty is estimated by sampling from a distribution over weights (ensembles, dropout, or explicit Bayesian inference).
• In generative modeling, EUQ requires learning $p(x|y)$ where $x$ is a complex-valued object, velocity field, or any reconstructed state (Dasgupta et al., 2021, Sun et al., 2020).

Methods for EUQ include:

• Variational inference (e.g., Stein variational gradient descent, SVGD) for Bayesian neural networks (Sun et al., 2020).
• Hamiltonian Monte Carlo (HMC) and posterior sampling for PINNs (Molnar et al., 2021).
• Normalizing flows as invertible surrogates to sample from high-dimensional posteriors without restrictive Gaussian assumptions (Dasgupta et al., 2021, Orozco et al., 2023).
• Deep ensembles and variance decomposition to disentangle model-form (epistemic) from measurement (aleatoric) uncertainties (Maulik et al., 2023).
• Monte Carlo dropout as a tractable basis for epistemic UQ via random subnet instantiations (Mousavi et al., 6 Jan 2025).
• SVGP-KAN, a sparse variational Gaussian process extension for structured and calibrated epistemic uncertainty (Ju, 27 Dec 2025).

2. Methodologies for Epistemic Uncertainty Quantification

EUQ operationalizes the posterior distribution through sampling, variational surrogates, or ensemble statistics. Key methods include:

• Normalizing flows: A bijection $x=f_\theta(z)$, $z\sim\mathcal{N}(0,I)$, induces tractable densities $q_\theta(x)$ for $p(x|y)$ (Dasgupta et al., 2021, Orozco et al., 2023). Posterior samples $x^{(i)}=f_\theta(z^{(i)})$ yield posterior mean and variance. Key computational advantages: computationally efficient O(M) sampling; non-Gaussian, multimodal distribution capture.
• Physics-informed Bayesian Neural Networks: Bayesian inference over neural network weights, with priors reflecting physical constraints (e.g., Navier–Stokes residuals as physics-informed likelihoods) (Sun et al., 2020). SVGD provides non-parametric posterior representation via an ensemble of parameter particles.
• Posterior sampling in PINNs: HMC over network weights $\theta$ yields posterior samples for each spatial/temporal query, providing mean and credible intervals. Regularization is imposed via governing physics in the Bayesian prior (physics loss), so that uncertainty reflects both data coverage and model fidelity (Molnar et al., 2021).
• Ensemble and dropout-based methods: Deep ensembles and MC dropout yield sample-based epistemic uncertainty estimates by computing the variance across independent networks or stochastic subnet realizations, respectively (Mousavi et al., 6 Jan 2025, Maulik et al., 2023). Variance decomposition cleanly separates epistemic and aleatoric sources:

$$\sigma^2_\mathcal{E}(\mathbf{x}) = \overbrace{\frac1{K}\sum_{j=1}^K \sigma^2_{\theta_j}(\mathbf{x})}^{\text{aleatoric}} + \overbrace{\frac1{K-1}\sum_{j=1}^K\left(\mu_{\theta_j}(\mathbf{x})-\mu_\mathcal{E}(\mathbf{x})\right)^2}^{\text{epistemic}}$$

(Maulik et al., 2023).

• Diffusion and flow-matching generative models: Diffusion-based refinement and conditional flow-matching offer epistemic UQ by direct sampling from stochastic evolution processes conditioned on measurement features, yielding ensemble statistics (Liu et al., 2023, Parikh et al., 20 Apr 2025).
• Gaussian process surrogates: SVGP-KAN topology combines interpretability (KAN) with structured, calibrated epistemic UQ via the GP predictive posterior, facilitating direct propagation of epistemic variance to reconstructed physical fields (Ju, 27 Dec 2025).

3. Role of EUQ in Inverse and Reconstruction Problems

EUQ is critical where direct measurement of the ground truth is infeasible or the inverse problem is ill-posed/non-convex:

• Ptychographic imaging: EUQ detects spurious artifacts, quantifies resolution limits in regions with poor measurement, and guides adaptive scanning for improved experiment design (Dasgupta et al., 2021).
• Transcranial ultrasound: Per-pixel posterior variance correlates strongly with reconstruction error and enables reliability diagrams, confirming uncertainty calibration. Posterior contraction with increased observation coverage provides diagnostic confidence (Orozco et al., 2023).
• Tomographic flow field reconstruction: Bayesian PINNs and GLS-based solvers with propagated variance/covariance yield best-linear unbiased estimators, enabling robust uncertainty bounds even under sparse/noisy data regimes (Molnar et al., 2021, Zhang et al., 2019).
• Near-wall turbulence: Conditional generative models with explicit epistemic UQ preserve intermittency and structure in stochastic reconstructions and reliably detect undersampled or unobservable flow regimes (Parikh et al., 20 Apr 2025).
• Aerodynamic load inference from sparse sensors: MC dropout-based epistemic UQ identifies model-form errors as a function of gust strength and flow regime, essential for flight safety (Mousavi et al., 6 Jan 2025).

4. Quantitative Metrics and Assessment of Uncertainty Calibration

Robust EUQ frameworks report and validate uncertainty through multiple metrics:

• Posterior standard deviation and credible intervals: Quantify spatially resolved epistemic uncertainty. Empirical coverage probability (observed fraction of ground truth within predicted intervals) is a direct calibration metric (Dasgupta et al., 2021, Orozco et al., 2023).
• Fieldwise error–uncertainty correlation: Slope of error vs. predicted uncertainty, and reliability diagrams, test calibration (Ju, 27 Dec 2025).
• Uncertainty ratio ($p$): Ratio of mean predicted uncertainty to empirical error, ideal close to 1 (Ju, 27 Dec 2025).
• Expected Coverage Error (ECE): Area between observed coverage and nominal confidence across interval bins (Ju, 27 Dec 2025).
• Variance decomposition: Explicitly separating epistemic from aleatoric contributions (ensemble variance vs. mean model prediction) (Maulik et al., 2023, Mousavi et al., 6 Jan 2025).

Observed trends include: posterior credible intervals expanding in poorly observed regions or under model misspecification, contraction as data coverage increases, and spatial alignment of uncertainty “hotspots” with measurement gaps or known physical boundaries.

5. Computational and Practical Considerations

Implementing EUQ at scale raises specific computational and methodological challenges:

• Scalability: Non-parametric inference (SVGD, HMC) and flow-based/GP surrogates can be expensive; amortized inference, model compression, and parallelization are employed for efficiency (Orozco et al., 2023, Ju, 27 Dec 2025).
• Generalization: Methods leveraging physics-informed priors and uncertainty-driven adaptivity generalize across data regimes and measurement layouts without retraining (Orozco et al., 2023, Dasgupta et al., 2021).
• Integration into classical pipelines: EUQ modules—e.g., diffusion-based refinement, normalizing flow surrogates, GP-KAN calibrators—can be inserted into or built atop existing iterative solvers or neural surrogates with minimal overhead (Liu et al., 2023, Ju, 27 Dec 2025).
• Automated discovery: Neural architecture search and hyperparameter optimization further support the construction of diverse, high-performing deep ensembles for epistemic uncertainty quantification (Maulik et al., 2023).
• Limitations: High computational cost (training, sampling, and forward/inverse solves); residual miscalibration in extreme extrapolation or under severe data paucity; sensitivity to prior and noise model specification (Dasgupta et al., 2021, Ju, 27 Dec 2025).

6. Comparative Performance and Research Directions

Recent experiments demonstrate that modern EUQ methodologies yield calibrated, structure-preserving, and informative uncertainty estimates:

• In ptychography (Dasgupta et al., 2021):
  • Posterior multimodality is captured, with uncertainty maps showing correct correlation with problem conditioning and data overlap.
  • Uncertainty-aware reconstructions are as accurate as point-estimate-based algorithms, but uniquely provide pixelwise confidence.
• In turbulent flow and sensor-driven aerodynamic estimation (Parikh et al., 20 Apr 2025, Mousavi et al., 6 Jan 2025):
  • EUQ methods reveal when and where predictions degrade as sensors are sparsified or dynamics become extreme.
  • Coverage probabilities and fieldwise uncertainty ratios are close to optimal, indicating reliable epistemic quantification.
• In generalized least-squares pressure reconstruction (Zhang et al., 2019):
  • Uncertainty-aware integration yields up to 250% error reduction over classical methods, with robust performance even under spatially correlated input errors.

Active research directions include enhanced likelihood models (e.g., Poisson for photon statistics), physics-enriched priors, uncertainty-driven experimental design (active learning, adaptive sensing), hybrid GP–deep learning surrogates, and uncertainty quantification for time-dependent or turbulent multiphysics phenomena.

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References:

• (Dasgupta et al., 2021) "Uncertainty quantification for ptychography using normalizing flows"
• (Sun et al., 2020) "Physics-Constrained Bayesian Neural Network for Fluid Flow Reconstruction with Sparse and Noisy Data"
• (Orozco et al., 2023) "Amortized Normalizing Flows for Transcranial Ultrasound with Uncertainty Quantification"
• (Molnar et al., 2021) "Flow field tomography with uncertainty quantification using a Bayesian physics-informed neural network"
• (Liu et al., 2023) "DifFlow3D: Toward Robust Uncertainty-Aware Scene Flow Estimation with Diffusion Model"
• (Parikh et al., 20 Apr 2025) "Conditional flow matching for generative modeling of near-wall turbulence with quantified uncertainty"
• (Ju, 27 Dec 2025) "Uncertainty-Aware Flow Field Reconstruction Using SVGP Kolmogorov-Arnold Networks"
• (Maulik et al., 2023) "Quantifying uncertainty for deep learning based forecasting and flow-reconstruction using neural architecture search ensembles"
• (Mousavi et al., 6 Jan 2025) "Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements"
• (Zhang et al., 2019) "Uncertainty-based pressure field reconstruction from PIV/PTV flow measurements with generalized least-squares"
• (Gundersen et al., 2020) "Semi Conditional Variational Auto-Encoder for Flow Reconstruction and Uncertainty Quantification from Limited Observations"
• (Chen et al., 2021) "A twin-decoder structure for incompressible laminar flow reconstruction with uncertainty estimation around 2D obstacles"