Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements

Hanieh Mousavi; Jeff D. Eldredge

REVIEW 4 major objections 5 minor 34 references

Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read From 11 noisy pressure sensors, a neural network reconstructs a gust-disturbed airfoil's vorticity field and lift, with separate aleatoric and epistemic uncertainty bounds that contain the true values in the tested cases.

desk verdict Useful framework, but the central generalization claim is undermined by an unspecified train/test split and a self-referential noise model. read the letter →

arxiv 2501.03406 v1 pith:WU2WATAL submitted 2025-01-06 cs.LG physics.flu-dyn

Hanieh Mousavi , Jeff D. Eldredge This is my paper · ORCID

classification cs.LGphysics.flu-dyn

keywords gustencounteraerodynamicssparsepressuremeasurementsflowfieldreconstructionliftcoefficientestimationaleatoricuncertaintyepistemicMonteCarlodropoutlift-augmentedautoencoder

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a neural network can reconstruct the two-dimensional vorticity field and the lift coefficient of an airfoil in a gust encounter from just 11 noisy surface pressure sensors, and can say how confident it is. The key idea is to compress each high-dimensional flow field into three latent variables with a lift-augmented autoencoder, then train a small MLP to map pressure readings plus sensor coordinates to a mean and a covariance for those latent variables. Sampling from the resulting latent distribution and decoding gives vorticity and lift with separate aleatoric (measurement-noise) and epistemic (model) uncertainties; in the reported cases the true values sit inside the 95% intervals and most per-pixel log-likelihoods are positive. If the claim holds, online sensor-based flow estimation for gust encounters becomes cheap enough for real-time use, because the expensive decoder is trained once and the online estimator is a small network.

What carries the argument

The load-bearing object is the three-dimensional latent representation $\boldsymbol{\xi}$ of the flow, learned by a lift-augmented autoencoder whose decoder also predicts lift. Around that latent space the paper builds a measurement-space Gramian $C_x = \mathbb{E}[\nabla f(x)^T \nabla f(x)]$ to rank sensor directions, and a predictive distribution $\pi(y|x) = \mathcal{N}(\mu, \Sigma)$ over latent states with $\Sigma = LL^T$ parameterized by a lower-triangular Cholesky factor. Monte Carlo dropout turns the estimator into a stochastic map, so $T$ forward passes yield the aleatoric distribution (mean of the predicted covariances) and the epistemic distribution (covariance of the predicted means). This is the machinery that lets 11 sensors produce field-level reconstructions with calibrated confidence intervals.

What would settle it

Hold out complete gust cases—entire triples of gust strength $G$, position $y_o/c$, and radius $2R/c$—from training, then evaluate whether the 95% predictive intervals for lift and per-pixel vorticity contain the simulation truth on those unseen cases; if coverage falls well below 95% or any test snapshot shares a gust case with the training data, the central claim is not established.

Watch

Extended reading notes

Core claim

The central discovery is that the physics of gust-airfoil interaction survives compression to a three-dimensional latent space, and that the sensor-to-flow relationship can be learned as a distribution over that space rather than over the full field. Using 11 pressure sensors stacked with their coordinates as a 33-dimensional input, the estimator learns the mean and a Cholesky-decomposed covariance of a multivariate normal in latent space under a heteroscedastic negative log-likelihood loss. Active dropout during inference supplies the epistemic component as the covariance of the predicted means, while the averaged predicted covariances supply the aleatoric component, and decoding samples from either distribution gives vorticity and lift with uncertainty bands that contain the reference values in the reported tests. The measurement-space Gramian of the estimator Jacobian has two dominant eigenmodes carrying over 99% of the energy, and the most informative sensors migrate from suction-side mid-chord to trailing-edge sensors as vortices shed and as positive or negative gusts pass.

Load-bearing premise

The load-bearing premise is that the reported test accuracy reflects generalization to gust encounters the network has not seen, which requires that the 80/20 split be made at the level of entire gust cases rather than random snapshots.

Editorial extensions

If this is right

Flow reconstruction and load estimation from sparse noisy pressure become feasible online, because the estimator is a small MLP in a three-dimensional latent space rather than a full flow solver.
Separating aleatoric from epistemic uncertainty tells an operator whether to improve sensor accuracy or collect more training data, since the two uncertainties have different dominant directions in latent space.
The measurement-space Gramian gives a quantitative sensor-placement criterion: the dominant eigenmodes identify which sensors carry the most information at each phase of vortex shedding and gust passage.
If the split generalizes to unseen gust parameters, the 95% intervals demonstrated here would support safety-relevant decisions such as gust load alleviation with calibrated confidence.
The same pretrained-decoder-plus-latent-uncertainty pipeline can be reused for other distributed measurements, since only the small sensor-to-latent map must be retrained.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A decisive check the paper leaves implicit is case-level holdout: train on some gust parameter triples and test on others, because the described 80/20 split is not stated to be stratified by gust case.
The nearly tangential aleatoric uncertainty ellipses suggest that adding a temporal filter or state-space prior over successive latent snapshots could shrink the noise-driven bounds beyond the paper's per-snapshot inference.
The reweighting of dominant sensors as the gust crosses the airfoil hints at adaptive sensing: a sensor suite whose weights track the leading eigenmode could keep information content high through the whole encounter.
The paper's own hypothesis about large uncertainty in high-gradient regions implies a testable extension: at higher Reynolds numbers, where shear layers fragment, the 95% coverage should be re-measured to see whether the latent-space assumption still holds.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: this is a serious paper on a useful problem, and the engineering is mostly careful, but the evidence for the headline claim—that sparse pressure readings reconstruct gust flows they haven't seen—is not established as written. The train/test split is described as a random 80/20 over 78,225 snapshots, not over the 105 gust cases. If test snapshots come from the same gust trajectories that appear in training, the reported log-likelihoods are interpolation within a known encounter, not generalization. The paper never says the split is stratified by case, so a reader can't verify the core claim. That's the main thing I'd want fixed before believing the numbers.

What's genuinely new and good: combining the lift-augmented autoencoder with heteroscedastic latent-space regression and MC dropout for gust-encounter flows is a sensible combination, and the time-resolved Gramian-based sensor importance analysis is a nice contribution—it actually gives physical insight into which sensors matter at different phases of a gust interaction. The latent-space visualizations are informative, and the distinction between aleatoric and epistemic uncertainty is handled correctly in the mathematics. The underlying DNS data looks solid. The citation pattern is fine; they build on the right prior work, and the self-citation for the flow solver is legitimate.

Where I'd push back: the aleatoric noise model is self-referential. They inject noise aligned with the dominant eigenvectors of the Gramian computed from the network's own Jacobian, then evaluate the network's uncertainty on that same noise. That tells you how sensitive the network is to its own most sensitive directions, but it's not a realistic sensor-noise test. The paper also doesn't report any calibration metric like coverage across the full test set; we get selected examples and per-image log-likelihoods, which can look fine while missing systematic under-coverage. And there are no baselines—no deterministic MLP, no linear regression, no Kalman filter—so we can't tell how much the UQ machinery actually buys you.

None of these are fatal to the idea. They're fixable: split by case, retrain, report coverage, and add a baseline. If those come back positive, this would be a useful paper for people working on sensor-based estimation and gust load alleviation. As it stands, I'd send it to review, because the framework is well-motivated and the sensor analysis is worth refereeing, but I'd expect major revision.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a two-stage deep-learning pipeline for reconstructing two-dimensional vorticity fields and lift coefficients from 11 noisy surface pressure measurements (a 33-dimensional input) in unsteady gust-encounter flow around a NACA 0012 airfoil at Re = 100. A lift-augmented autoencoder compresses the high-dimensional flow fields into a three-dimensional latent space, and a separate MLP maps stacked pressure readings and sensor coordinates to the mean and covariance of a multivariate Gaussian in that latent space, trained with a heteroscedastic negative log-likelihood loss and Monte Carlo dropout. The paper also analyzes sensor informativeness through eigenmodes of the measurement-space Gramian and presents aleatoric and epistemic uncertainty estimates for latent variables, lift, and vorticity fields, supported by selected log-likelihood values and qualitative comparisons of predicted and reference trajectories.

Significance. If the central claims withstand proper evaluation, the framework is a useful contribution: it combines a nonlinear lift-augmented autoencoder with heteroscedastic latent-space regression and MC dropout, which is a computationally attractive route to sensor-based flow estimation with uncertainty. The sensor-sensitivity analysis via the measurement-space Gramian is physically interpretable and adds value beyond a purely black-box regression. The paper is also transparent about the network architecture and gives explicit equations for the uncertainty model. However, the current evidence is weakened by the apparent snapshot-level train/test split, the lack of quantitative uncertainty calibration, and the self-referential choice of inference-time noise directions. The significance is therefore conditional on fixing these evaluation issues.

major comments (4)

[Sections 2.2 and 2.5] The data split is not defined at the level of gust cases. The manuscript states only that "eighty percent of the data is allocated for training, while the remaining twenty percent is used for validation and testing" (Section 2.2, repeated in Section 2.5). The dataset consists of 105 independent trajectories (5 base cases and 100 gust cases) with 745 temporally correlated snapshots each, so a random snapshot-level split places test snapshots from the same gust encounter, with the same gust strength, position, radius, and angle of attack, at nearby times in the training set. Because the pressure time series are deterministic functions of the trajectory, the test inputs are then nearly identical to training inputs, and the reported log-likelihoods and 95% intervals in Figures 8 through 13 measure interpolation within known encounters rather than generalization to unseen gust conditions. The autoencoder in Section 2.2 is trained on the same snapshot pool, so its decoder is likewise contaminated. Please re-run the entire pipeline with a leave-gust-cases-out split, state the number of cases in training, validation, and testing, and re-report all quantitative results.
[Sections 3.2.2 and 3.2.3, Figures 8, 10, 12, 13] The central claim that the true lift and vorticity fall within the 95% uncertainty intervals is not supported by a quantitative coverage statistic. The only quantitative metrics are average per-pixel log-likelihoods for a small number of selected cases (for example, "ll" values in Figures 8, 10, 12, and 13), and the "within the uncertainty bounds" statements are made qualitatively. Please report, on the case-level held-out set, the empirical fraction of time instants at which the true CL lies inside the 95% interval, the analogous pixel-level coverage for the vorticity field, and the same statistics for the aleatoric and epistemic distributions separately. A calibration curve or reliability diagram would also help. Without these numbers, the uncertainty-quantification claim in the abstract is not quantitatively established.
[Section 2.5 and Section 3.2.2] The inference-time noise used to evaluate aleatoric uncertainty is generated as η ~ ζ U_r, where U_r contains the dominant eigenvectors of the measurement-space Gramian C_x computed from the Jacobian of the same network F_p that produces the predictive covariance. This choices makes the reported aleatoric intervals a measure of the network's sensitivity in its own most sensitive directions rather than of a physical sensor-noise process. Moreover, the training-time data augmentation is described as generic Gaussian injection, so the train and test noise models are not matched. The coverage of the uncertainty intervals under independent per-sensor noise or under a physically motivated correlated-noise model is not reported. Please evaluate the same estimators under i.i.d. sensor noise and at least one correlated-noise model, and report the resulting coverage and log-likelihoods; this is important for the stated real-world online-estimation use case.
[Section 2.2] The latent dimension is fixed at l = 3 with the statement that its appropriateness "will be discussed below," but no reconstruction-error analysis, explained-variance metric, or comparison across latent dimensions appears in Section 3.1. Since the estimator and the uncertainty quantification operate entirely in this latent space, a demonstration that the 3D latent representation is adequate (for example, reconstruction error on held-out gust cases and sensitivity of results to l = 2, 3, 4) is needed to support the "low-order" and "key physics" claims.

minor comments (5)

[Equations (12) and (13)] The predictive distributions are described as Gaussian, but the distribution of the mean over dropout passes is not necessarily Gaussian; please clarify whether the reported intervals are computed from the Gaussian forms in Equations (12) and (13) or from the empirical samples, and justify the Gaussian approximation.
[Section 2.3] The text says that stacking sensor x and y positions "in a global reference frame" helps the network distinguish angles of attack, but the airfoil geometry and sensor coordinates are fixed for each case; please clarify exactly how the coordinates vary with angle of attack and why this is not just a one-hot encoding of the five cases.
[Figure 3] The latent-space axes in Figure 3 are unlabeled; please state which latent coordinate is shown on each axis and what the units are, since the paper later refers to ξ1, ξ2, and ξ3.
[Figures 8, 10, 12, 13] The captions report "ll" as an average pixel-wise log-likelihood, but it is not clear whether the average is over pixels only or also over time snapshots; please define the exact averaging procedure in the captions or in the text.
[Section 2.2] The loss-balance coefficient β = 0.05 and the regularization and dropout hyperparameters are stated without a sensitivity analysis; please report the range of values explored or state explicitly that the results are insensitive to these choices.

Circularity Check

1 steps flagged · score 4.0 of 10

Aleatoric UQ evaluation is self-referential: test noise is drawn from the trained estimator's own Jacobian Gramian, while the flow-reconstruction claim itself remains an independent supervised fit.

fitted input called prediction [Section 2.5, paragraph after Eq. (13); evaluated in Section 3.2.2]
"To quantify the uncertainty in the output most influenced by variations in the input during inference, we introduce noise ηηη aligned with the principal directions of measurement variation. These directions are identified by the matrix UUU r, which contains the eigenvectors associated with the largest eigenvalues of the measurement space Gramian, as discussed in Section 2.4."

The 'measurement noise' used for the aleatoric evaluation is generated from U_r, the eigenvectors of C_x = E[∇f^T ∇f], where f is the same trained estimator Fp whose uncertainty is being quantified (Section 2.4, Eqs. (4)-(5)). Therefore the reported 95% intervals and log-likelihood values in Figures 8 and 10 measure the model's sensitivity to the input directions that this same model identifies as most informative; they are not an independent test of the effect of sensor noise. The abstract claims aleatoric uncertainty 'arising from noisy input measurements,' but the injected noise is derived from the fitted network rather than from a physical or external noise model.

full rationale

The central reconstruction pipeline is not circular: the lift-augmented autoencoder and the pressure-to-latent MLP are trained by supervised regression against DNS-generated vorticity and lift, and the decoder is reused from the autoencoder, so the mapping from pressure to flow is an empirical fit, not a derivation from its own outputs. The paper's self-citations (Eldredge 2022 solver, Le Provost and Eldredge filters) are not load-bearing for the ML claims; the architecture and MC-dropout equivalence are cited from Fukami and Taira and Gal and Ghahramani, which are external. The one genuine circular element is the aleatoric uncertainty evaluation: the noise injected at inference is not an independent sensor-noise model but is constructed from the eigenvalues and eigenvectors of the Gramian of the trained estimator's own Jacobian (Section 2.4), so the resulting intervals and log-likelihoods measure the model's sensitivity to directions it already regards as most informative. This is a partial, evaluation-level circularity; it does not make the reconstruction claim itself circular. Separately, the paper does not state whether the 80/20 split is at the case level, so generalization to unseen gust parameters is not demonstrated; this is a missing-support risk rather than a demonstrated circularity. The conclusion's caveat that validation on real sensor data remains future work is appropriately stated.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The framework rests on a small set of hyperparameters and standard domain assumptions. The latent dimension and loss-balance coefficient are adopted from a prior paper without independent validation. The noise model, dropout rate, and energy threshold are chosen by hand. No new physical entities are introduced.

free parameters (7)

Latent dimension l = 3
Taken from Fukami and Taira (2023); the paper promises a discussion of its appropriateness but no reconstruction-error analysis is shown.
Loss balance coefficient beta = 0.05
Fixed as in Fukami and Taira (2023); no sensitivity study is reported.
Dropout rate = 0.05 (drop probability)
Chosen after experimentation; the authors note it had minimal impact on log-likelihood.
Weight regularization constant = 1e-7
Selected after experimentation to maximize average likelihood during training.
Sensor noise variance sigma_x^2 = 2.5e-5
Synthetic noise corresponding to 0.15% of the maximum pressure reading; not derived from real sensor data.
Energy threshold gamma for Gramian rank = 0.99
Used to select the number of dominant eigenmodes rx of the measurement-space Gramian; typically rx=1 or 2 in this study.
Number of MC passes T and samples M = 100 and 100
Chosen for inference; no convergence study with respect to T or M is reported.

assumptions (5)

domain assumption The lattice Green's function / immersed-layers simulation provides ground-truth vorticity, pressure, and lift.
The reference data are generated by the in-house solver (Eldredge 2022); there is no experimental validation of the simulation itself.
domain assumption A three-dimensional latent manifold captures the essential gust-encounter flow dynamics.
The lift-augmented autoencoder with l=3 is assumed to be sufficient; no reconstruction error or latent-dimension sensitivity analysis is presented.
domain assumption Eleven surface pressure readings plus sensor coordinates are sufficient to determine the latent state.
The mapping Fp assumes observability of the low-order flow from the chosen sparse sensors; no identifiability or sensor-set comparison is provided.
domain assumption The predictive distributions in latent space, after MC dropout, are multivariate Gaussian.
Both aleatoric and epistemic predictive distributions are modeled as normal (Eqs. 12-13) and propagated through the nonlinear decoder.
standard math Monte Carlo dropout provides a valid approximation to variational inference in Bayesian neural networks.
The paper relies on the equivalence proven by Gal and Ghahramani (2016a,b), which is a standard result in the literature.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements." pith.science (2026). https://pith.science/paper/WU2WATAL

@misc{pith2026250103406,
  author       = {Pith},
  title        = {Pith review of: Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WU2WATAL}},
  note         = {Machine review of arXiv:2501.03406}
}

read the original abstract

This paper presents a novel machine-learning framework for reconstructing low-order gust-encounter flow field and lift coefficients from sparse, noisy surface pressure measurements. Our study thoroughly investigates the time-varying response of sensors to gust-airfoil interactions, uncovering valuable insights into optimal sensor placement. To address uncertainties in deep learning predictions, we implement probabilistic regression strategies to model both epistemic and aleatoric uncertainties. Epistemic uncertainty, reflecting the model's confidence in its predictions, is modeled using Monte Carlo dropout, as an approximation to the variational inference in the Bayesian framework, treating the neural network as a stochastic entity. On the other hand, aleatoric uncertainty, arising from noisy input measurements, is captured via learned statistical parameters, which propagates measurement noise through the network into the final predictions. Our results showcase the efficacy of this dual uncertainty quantification strategy in accurately predicting aerodynamic behavior under extreme conditions while maintaining computational efficiency, underscoring its potential to improve online sensor-based flow estimation in real-world applications.

Figures

Figures reproduced from arXiv: 2501.03406 by the authors.

**Figure 2.** Overview of the network architecture in the present study. The flow field data is compressed [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Low-order representation of flow data is presented with undisturbed cases highlighted in color [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Periodic variation of lift and the first mode of surface pressure measurements over time for [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: The figure illustrates the primary mode of pressure measurements at six different snapshots [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: The figure illustrates the primary mode of pressure measurements at six different time [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Predicted mean with 95% confidence ellipses of latent variables at a couple of instants for five [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Aleatoric (data) uncertainty of five undisturbed cases due to measurement noise, represented [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Impact of a random gust on aerodynamics. The top panel displays the behavior of the true [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: Aleatoric uncertainty in gust-airfoil aerodynamics is illustrated with the predicted mean [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: Epistemic (model) uncertainty of estimation of five undisturbed cases. Predicted mean with [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: Epistemic (model) uncertainty of five undisturbed cases, represented by the predicted mean [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

**Figure 13.** Figure 13: Epistemic uncertainty in gust-airfoil aerodynamics is illustrated with the predicted mean [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 26 canonical work pages

[1]

and Hess, H

Akay, A. and Hess, H. (2019). Deep learning: current and emerging applications in medicine and technology. IEEE journal of biomedical and health informatics , 23(3):906--920

work page 2019
[2]

M., Kucukelbir, A., and McAuliffe, J

Blei, D. M., Kucukelbir, A., and McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American statistical Association , 112(518):859--877

2017
[3]

Brunton, S. L. and Kutz, J. N. (2022). Data-driven science and engineering: Machine learning, dynamical systems, and control . Cambridge University Press

work page 2022
[4]

Che, C., Liu, B., Li, S., Huang, J., and Hu, H. (2023). Deep learning for precise robot position prediction in logistics. Journal of Theory and Practice of Engineering Science , 3(10):36--41

work page 2023
[5]

E., Fukami, K., and Taira, K

Chen, D., Kaiser, F., Hu, J., Rival, D. E., Fukami, K., and Taira, K. (2024). Sparse pressure-based machine learning approach for aerodynamic loads estimation during gust encounters. AIAA Journal , 62(1):275--290

work page 2024
[6]

S., Beaulieu-Jones, B

Ching, T., Himmelstein, D. S., Beaulieu-Jones, B. K., Kalinin, A. A., Do, B. T., Way, G. P., Ferrero, E., Agapow, P.-M., Zietz, M., Hoffman, M. M., et al. (2018). Opportunities and obstacles for deep learning in biology and medicine. Journal of the royal society interface , 15(141):20170387

work page 2018
[7]

Dubois, P., Gomez, T., Planckaert, L., and Perret, L. (2022). Machine learning for fluid flow reconstruction from limited measurements. Journal of Computational Physics , 448:110733

work page 2022
[8]

Eldredge, J. D. (2022). A method of immersed layers on cartesian grids, with application to incompressible flows. Journal of Computational Physics , 448:110716

work page 2022

Show all 34 references

[9]

Frey, B. J. and Hinton, G. E. (1999). Variational learning in nonlinear gaussian belief networks. Neural Computation , 11(1):193--213

1999
[10]

and Taira, K

Fukami, K. and Taira, K. (2023). Grasping extreme aerodynamics on a low-dimensional manifold. Nature Communications , 14(1):6480

2023
[11]

and Ghahramani, Z

Gal, Y. and Ghahramani, Z. (2016a). Dropout as a bayesian approximation: Appendix

2016
[12]

and Ghahramani, Z

Gal, Y. and Ghahramani, Z. (2016b). Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning , pages 1050--1059. PMLR

2016
[13]

Gundersen, K., Oleynik, A., Blaser, N., and Alendal, G. (2021). Semi-conditional variational auto-encoder for flow reconstruction and uncertainty quantification from limited observations. Physics of Fluids , 33(1)

2021
[14]

and Waegeman, W

H \"u llermeier, E. and Waegeman, W. (2021). Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine learning , 110(3):457--506

2021
[15]

R., Cetiner, O., and Smith, M

Jones, A. R., Cetiner, O., and Smith, M. J. (2022). Physics and modeling of large flow disturbances: discrete gust encounters for modern air vehicles. Annual Review of Fluid Mechanics , 54(1):469--493

2022
[16]

V., Laga, H., Boussaid, F., Buntine, W., and Bennamoun, M

Jospin, L. V., Laga, H., Boussaid, F., Buntine, W., and Bennamoun, M. (2022). Hands-on bayesian neural networks—a tutorial for deep learning users. IEEE Computational Intelligence Magazine , 17(2):29--48

2022
[17]

and Gal, Y

Kendall, A. and Gal, Y. (2017). What uncertainties do we need in bayesian deep learning for computer vision? Advances in neural information processing systems , 30

2017
[18]

Laves, M.-H., Ihler, S., Ortmaier, T., and Kahrs, L. A. (2019). Quantifying the uncertainty of deep learning-based computer-aided diagnosis for patient safety. Current Directions in Biomedical Engineering , 5(1):223--226

2019
[19]

T., Diehl, F., Brunner, T., and Knoll, A

Le, M. T., Diehl, F., Brunner, T., and Knoll, A. (2018). Uncertainty estimation for deep neural object detectors in safety-critical applications. In 2018 21st International Conference on Intelligent Transportation Systems (ITSC) , pages 3873--3878. IEEE

2018
[20]

Le Provost, M., Baptista, R., Marzouk, Y., and Eldredge, J. D. (2022). A low-rank ensemble kalman filter for elliptic observations. Proceedings of the Royal Society A , 478(2266):20220182

2022
[21]

and Eldredge, J

Le Provost, M. and Eldredge, J. D. (2021). Ensemble kalman filter for vortex models of disturbed aerodynamic flows. Physical Review Fluids , 6(5):050506

2021
[22]

Liu, M., Grana, D., and de Figueiredo, L. P. (2022). Uncertainty quantification in stochastic inversion with dimensionality reduction using variational autoencoder. Geophysics , 87(2):M43--M58

2022
[23]

Petersen, F., Mishra, A., Kuehne, H., Borgelt, C., Deussen, O., and Yurochkin, M. (2024). Uncertainty quantification via stable distribution propagation. arXiv preprint arXiv:2402.08324

2024 arXiv
[24]

and Rey, V

Quinton, P. and Rey, V. (2024). Jacobian descent for multi-objective optimization. arXiv preprint arXiv:2406.16232

2024 arXiv
[25]

Raffoul, Y. N. (2022). Advanced differential equations . Academic Press

2022
[26]

and Mnih, A

Salakhutdinov, R. and Mnih, A. (2008). Bayesian probabilistic matrix factorization using markov chain monte carlo. In Proceedings of the 25th international conference on Machine learning , pages 880--887

2008
[27]

H., and Knoll, A

Shafaei, S., Kugele, S., Osman, M. H., and Knoll, A. (2018). Uncertainty in machine learning: A safety perspective on autonomous driving. In Computer Safety, Reliability, and Security: SAFECOMP 2018 Workshops, ASSURE, DECSoS, SASSUR, STRIVE, and WAISE, V \"a ster s, Sweden, Se...

2018
[28]

Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., and Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research , 15(1):1929--1958

2014
[29]

Swiatkowski, J., Roth, K., Veeling, B., Tran, L., Dillon, J., Snoek, J., Mandt, S., Salimans, T., Jenatton, R., and Nowozin, S. (2020). The k-tied normal distribution: A compact parameterization of gaussian mean field posteriors in bayesian neural networks. In International co...

2020
[30]

Tanaka, A., Tomiya, A., and Hashimoto, K. (2021). Deep learning and physics , volume 1. Springer

2021
[31]

Taylor, G. (1918). On the dissipation of eddies. Meteorology, Oceanography and Turbulent Flow , pages 96--101

1918
[32]

A., and Rival, D

Tu, H., Marzanek, M., Green, M. A., and Rival, D. E. (2022). Ftle and surface-pressure signature of dynamic flow reattachment during delta-wing axial acceleration. AIAA Journal , 60(4):2178--2194

2022
[33]

Williams, C. K. and Rasmussen, C. E. (2006). Gaussian processes for machine learning , volume 2. MIT press Cambridge, MA

2006
[34]

Zhong, Y., Fukami, K., An, B., and Taira, K. (2023). Sparse sensor reconstruction of vortex-impinged airfoil wake with machine learning. Theoretical and Computational Fluid Dynamics , 37(2):269--287

2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

and Hess, H

Akay, A. and Hess, H. (2019). Deep learning: current and emerging applications in medicine and technology. IEEE journal of biomedical and health informatics , 23(3):906--920

work page 2019

[2] [2]

M., Kucukelbir, A., and McAuliffe, J

Blei, D. M., Kucukelbir, A., and McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American statistical Association , 112(518):859--877

2017

[3] [3]

Brunton, S. L. and Kutz, J. N. (2022). Data-driven science and engineering: Machine learning, dynamical systems, and control . Cambridge University Press

work page 2022

[4] [4]

Che, C., Liu, B., Li, S., Huang, J., and Hu, H. (2023). Deep learning for precise robot position prediction in logistics. Journal of Theory and Practice of Engineering Science , 3(10):36--41

work page 2023

[5] [5]

E., Fukami, K., and Taira, K

Chen, D., Kaiser, F., Hu, J., Rival, D. E., Fukami, K., and Taira, K. (2024). Sparse pressure-based machine learning approach for aerodynamic loads estimation during gust encounters. AIAA Journal , 62(1):275--290

work page 2024

[6] [6]

S., Beaulieu-Jones, B

Ching, T., Himmelstein, D. S., Beaulieu-Jones, B. K., Kalinin, A. A., Do, B. T., Way, G. P., Ferrero, E., Agapow, P.-M., Zietz, M., Hoffman, M. M., et al. (2018). Opportunities and obstacles for deep learning in biology and medicine. Journal of the royal society interface , 15(141):20170387

work page 2018

[7] [7]

Dubois, P., Gomez, T., Planckaert, L., and Perret, L. (2022). Machine learning for fluid flow reconstruction from limited measurements. Journal of Computational Physics , 448:110733

work page 2022

[8] [8]

Eldredge, J. D. (2022). A method of immersed layers on cartesian grids, with application to incompressible flows. Journal of Computational Physics , 448:110716

work page 2022