A Bi-fidelity Surrogate Modeling Approach for Uncertainty Propagation in Three-Dimensional Hemodynamic Simulations

Han Gao; Jian-Xun Wang; Xueyu Zhu

REVIEW 3 major objections 5 minor 97 references

A Bi-fidelity Surrogate Modeling Approach for Uncertainty Propagation in Three-Dimensional Hemodynamic Simulations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A bi-fidelity surrogate that combines cheap low-fidelity simulations with a few dozen high-fidelity runs can propagate high-dimensional inflow uncertainties in 3D hemodynamics to full-field wall shear stress statistics at about one to two…

desk verdict A capable applied demonstration of bi-fidelity UQ for full-field 3D hemodynamics; the numerics are convincing but the method's key transferable assumptions are under-analyzed. read the letter →

arxiv 1908.10197 v1 pith:LPPKBTHW submitted 2019-08-23 physics.flu-dyn eess.IVphysics.med-ph

Han Gao , Xueyu Zhu , Jian-Xun Wang This is my paper · ORCID

classification physics.flu-dyneess.IVphysics.med-ph

keywords uncertaintyquantificationhemodynamicsmulti-fidelitysurrogatebi-fidelitymodelingwallshearstressrandomfieldscomputationalfluiddynamicspatient-specificaneurysm

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

The paper aims to show that forward uncertainty propagation in image-based cardiovascular CFD need not require hundreds of expensive three-dimensional simulations. Its bi-fidelity surrogate runs a large set of cheap low-fidelity simulations to explore the parameter space, selects a small number of important points, runs the expensive high-fidelity solver only there, and then reuses low-fidelity projection coefficients to combine high-fidelity solution snapshots at new query points. In three test cases, including a patient-specific cerebral aneurysm with a nine-dimensional uncertain inflow represented by Gaussian random fields, the surrogate reproduces the Monte Carlo mean and standard deviation fields of velocity, pressure, and wall shear stress, as well as the location distribution of minimum wall shear stress, using only 6 to 40 high-fidelity runs. If this holds in practice, patient-specific uncertainty quantification in hemodynamics becomes computationally feasible, because the method is non-intrusive and the cost per query is dominated by one cheap low-fidelity solve.

What carries the argument

The object that carries the argument is the coefficient-sharing projection of the bi-fidelity surrogate. During offline training, $M$ low-fidelity solutions are computed on a sample set $\Gamma$, and a greedy algorithm based on the pivoted Cholesky decomposition of the low-fidelity Gramian matrix $G_{ij} = \langle v_L(z_i), v_L(z_j)\rangle$ selects $m$ important points whose low-fidelity solutions are farthest from the subspace of previously selected ones; high-fidelity solutions at exactly those points form the high-fidelity basis. At a new parameter point $z$, only a low-fidelity solve is needed: the solution is projected onto the low-fidelity basis via $c_L = G^{-1}(V_L(\gamma_m))^T v_L(z)$, and the surrogate reconstructs $v_B(z) = \sum_k c_k v_H(z_k)$ by reusing the same coefficients. The empirical error bound monitors $R_s$, the ratio of high- to low-fidelity relative distances to their respective subspaces, and $R_e$, the balance between in-plane error and distance error, to decide whether the low-fidelity model is informative and when to stop collecting high-fidelity samples.

What would settle it

Choose a low-fidelity model that is deliberately uninformative, for example a coarse-mesh or 2D model that cannot represent a vortex or secondary flow present in the 3D high-fidelity solution, and compute the bi-fidelity surrogate at validation points where the coefficient-sharing relation is known to fail. If the surrogate's relative error fails to drop below the low-fidelity baseline, or if the projection coefficients $c_L$ differ substantially from the coefficients of the high-fidelity solution projected onto the high-fidelity basis, the central claim would be refuted.

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Extended reading notes

Core claim

The central claim is that the high- and low-fidelity solutions share enough structure that the same projection coefficients can serve both: after projecting a new low-fidelity solution onto the low-fidelity basis, those coefficients are used, without modification, to form a combination of high-fidelity basis functions, producing the bi-fidelity surrogate $v_B(z) = \sum_{k=1}^m c_k v_H(z_k)$. The high-fidelity snapshots at the selected points therefore act as a full-field basis, which is why the surrogate returns three-dimensional velocity, pressure, and wall shear stress fields rather than a few scalar outputs. On the patient-specific aneurysm case the surrogate matched statistics computed from a 600-run high-fidelity Monte Carlo ensemble, with relative error dropping by an order of magnitude below the low-fidelity baseline, using only 40 high-fidelity training runs. The paper also proposes an empirical a priori error estimate, based on the model-similarity ratio $R_s$ and the in-plane-to-distance error ratio $R_e$, that is intended to tell a user when the low-fidelity model is informative enough and when adding more high-fidelity samples stops helping.

Load-bearing premise

Everything rests on the assumption that the low-fidelity and high-fidelity reconstructions share the same projection coefficients: coefficients obtained by projecting a cheap low-fidelity solution onto the low-fidelity basis can be reused unmodified to combine high-fidelity basis functions at that parameter point.

Editorial extensions

If this is right

Clinically useful uncertainty maps, such as means and standard deviations of velocity, pressure, and wall shear stress plus locations of extreme WSS, can be obtained with tens of high-fidelity simulations instead of hundreds or thousands.
The method accepts a wide range of low-fidelity models, including coarse meshes, unconverged iterations, and even 2D geometry, so it can ride on existing CFD solvers without code changes.
The selected high-fidelity snapshots carry three-dimensional information the low-fidelity model lacks; in the 2D/3D bifurcation case the surrogate recovered z-direction flow that was absent from the low-fidelity solutions.
The two error metrics $R_s$ and $R_e$ provide a practical stopping rule: keep adding high-fidelity important points as long as the low-fidelity model looks similar to the high-fidelity one and the in-plane error is not dominant, and stop when it is.

Reading between the lines

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

If coefficient sharing holds whenever a cheap model captures the dominant parameter response, the same bi-fidelity construction could transfer to other expensive PDE-based UQ settings, such as cardiac electrophysiology or patient-specific structural mechanics, where a coarser or simplified model is readily available.
The success of recovering a whole missing velocity component in the 2D/3D case suggests the high-fidelity basis stores physical features absent from the low-fidelity model; a sharper test is to check whether the surrogate can also recover a flow feature that appears in the high-fidelity snapshots only at points outside the selected subset.
The empirical error estimate is explicitly heuristic, as the paper concedes; a natural next step would be a systematic comparison of the estimated bound against true errors across many low-fidelity/high-fidelity pairs to calibrate when the $R_s\approx1$ and $R_e<10$ thresholds are safe.
Because the surrogate needs only one cheap solve per new parameter sample, it converts the dominant cost of Monte Carlo propagation from running the expensive solver into a bandwidth problem: how cheaply the low-fidelity model can be evaluated at scale.

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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

Bottom line: this is a useful applied paper. It takes a known multi-fidelity stochastic collocation scheme and shows it can propagate high-dimensional inflow uncertainty to full-field 3D hemodynamic outputs using only dozens of HF runs, with accuracy matched against 600 independent HF MC samples. The patient-specific aneurysm case, with 40 HF runs and ~1-2% error, is the headline result and it is credible.

The genuinely new pieces are the application domain and the choice of low-fidelity models. Using 2D or unconverged simulations as the LF model and still recovering 3D features like z-velocity in the idealized bifurcation case is a nice demonstration. The error analysis is honest: the authors call their bound 'empirical' and admit in the Remark that it is not rigorous. The plots of error-bound versus true error show the bound tracks the trend, which is about all you can expect from a heuristic.

The soft spots are real but not fatal. The coefficient-sharing assumption in Eqs (9)-(10) is load-bearing and never directly validated. The stress-test point about the 2D-to-3D case is fair: the LF model has no z-component, so the successful recovery of z-flow in that particular nearly planar geometry doesn't guarantee it will transfer to strongly three-dimensional patient geometries. The reported Rs≈1 condition measures distance similarity, not coefficient equality, so it doesn't close the gap. A direct check of how the LF and HF coefficients correlate across test points would have been cheap to include and would make the method much more trustworthy.

The error bound also rests on constants c1=c2=1 chosen from numerical experience. The authors say this, so it's not deceptive, but it means the 'a priori' estimate is really a post-hoc diagnostic. Minor issues: no code or data released, which limits reproducibility, and no repeated-training experiments, so we don't know how sensitive the surrogate accuracy is to the random selection of the LF sample set.

Who is this for? Anyone doing practical UQ in image-based hemodynamics, especially groups that can't afford hundreds of 3D simulations. It's a solid within-subfield contribution, not a paradigm shift. I'd send it to peer review: the numerical evidence is strong enough to warrant full referee attention, and the authors should be asked to either check the coefficient-sharing premise directly or hedge the generalization claims. If they do that, this is a publishable paper.

Referee Report

3 major / 5 minor

Summary. This paper proposes a bi-fidelity surrogate modeling framework for uncertainty propagation in three-dimensional cardiovascular CFD. The method uses many cheap low-fidelity (LF) simulations to select a small number of 'important' parameter points via pivoted Cholesky decomposition of the LF Gramian, then uses the LF projection coefficients on those points to combine high-fidelity (HF) basis functions, yielding full-field velocity/pressure/WSS predictions. The framework is demonstrated on three vascular cases: an idealized stenosis with converged/unconverged and coarse/fine mesh pairs, an idealized bifurcation aneurysm with a 3D/2D model pair, and a patient-specific middle cerebral artery aneurysm with coarse/fine meshes. The authors report that the bi-fidelity surrogate reproduces Monte Carlo benchmark statistics (means, standard deviations, extreme-value distributions) with relative errors near 1–2% using only 6–40 HF runs, while achieving large computational speedups. They also propose an 'empirical error bound estimation' procedure based on a model similarity metric Rs and an in-plane/distance error ratio Re, and they test this estimator on the three cases.

Significance. If the reported accuracy transfers to practical clinical geometries, the proposed approach would be an important step toward making full-field, high-dimensional forward UQ in image-based hemodynamics computationally feasible. The paper's strengths are its focus on full-field predictions rather than scalar quantities of interest, its use of independent HF test sets to validate the surrogate, and its exploration of several distinct LF/HF pairings, including a challenging 2D-to-3D case and a 9-dimensional patient-specific case. The reported results are specific and falsifiable, and the method is non-intrusive to the CFD solver. However, the second claimed contribution, the a priori error bound, is presented as a conjecture rather than a proven bound, and the central coefficient-sharing assumption is asserted without direct validation; these issues limit the strength of the manuscript in its current form.

major comments (3)

[Section 2.2.3, Eqs. (9)–(10)] The method's core premise is that the LF and HF reconstructions share the same projection coefficients, but this assumption is never checked against data. The authors state 'Since we assume the HF and LF reconstructions share the same reconstruction coefficients' and proceed to Eq. (10), but no numerical comparison between c_L(z) and the actual HF coefficients c_H(z) = [V_H(γ_m)^T V_H(γ_m)]^{-1} V_H(γ_m)^T v_H(z) is reported. The heuristic Rs≈1 condition in Eq. (12) compares relative distances to the respective spans, which does not imply coordinate-coefficient equality. This is load-bearing for the central claim: in the 2D-to-3D case (Section 3.2), the LF model has no z-velocity component, so the successful recovery of z-direction flow (Figs. 6c and 6f) is not guaranteed by Eq. (10); the test geometry is nearly planar, and the result may not transfer to strongly three-dimensional patient geometries. I request a direct diagnostic of coefficient-sharing on the test sets (e.g., scatter plots of c_L versus c_H at a few parameter points, or a reported norm of their difference), or a theoretical condition under which the sharing is justified.
[Section 2.2.4, Theorem 1 and Eq. (14)] The paper's second contribution, the 'empirical error bound estimation', is not actually a bound after the step where constants are introduced. Theorem 1 is stated with the proof 'rather trivial and omitted here', which is not acceptable for a formal theorem in a methods paper; at minimum the triangle-inequality steps should be shown. More importantly, Eq. (14) is labeled a conjecture, and the constants c1 and c2 are set to 1 'based on numerical experience' from the same experiments used to evaluate the estimator. This makes the reported 'bound' an a posteriori heuristic indicator rather than an a priori bound. Numerical evidence in Section 4 and Fig. 15(d)–(f) shows only that the estimator captures the qualitative trend of the true error; in Cases 1 and 2 the estimated curve is not consistently above the true error for all m, which contradicts the term 'bound'. The authors do add a Remark acknowledging the estimate is 'not rigorous', but the abstract and Section 1 still describe it as an 'empirical error bound estimation approach'. I recommend either reframing the contribution as an empirical error indicator and removing the word 'bound', or providing a rigorous derivation with explicit, verifiable constants and validating the inequality pointwise on the test sets.
[Section 3.3 and Fig. 14] The headline claim that 40 HF runs suffice for the patient-specific 9-dimensional case is presented against an HF-MC benchmark of 600 samples, but no uncertainty quantification of that benchmark is given. Since the ground truth itself is a Monte Carlo estimate, the reported 1–2% relative errors are only meaningful if the MC statistics are sufficiently converged; a convergence check or bootstrap confidence intervals for the MC means/standard deviations would strengthen the comparison. This is not a fatal issue, but it affects the precision with which the central claim can be stated.

minor comments (5)

[Eq. (11)] The typesetting of Eq. (11) appears garbled: after the second equality the denominator '||vH(z*)||' is repeated without a clear operator, and the grouping of the factor (1 + ...) is hard to parse. Please rewrite the equation with unambiguous parentheses or split it into two lines.
[Eq. (14) and surrounding text] There are notation inconsistencies: the first factor on the right-hand side uses 'vL(z)' instead of 'vL(z*)', and the text after Eq. (14) refers to 'Re(z) = (12)' while Re is first defined in Eq. (15); please correct these cross-references.
[Fig. 8 caption] The caption reads 'relative root mean squared error (RMSE)' but the label in the figure says 'RSME'; please use the correct acronym consistently.
[Section 5 and Acknowledgments] The heading 'Aknowledgement' is misspelled; it should be 'Acknowledgments'.
[Fig. 15 caption] The caption says 'error bound estimation (d-e)' but there are three subpanels (d), (e), and (f); please change to '(d–f)'.

Circularity Check

1 steps flagged · score 4.0 of 10

The main surrogate-accuracy claim is validated against independent HF test samples and is not circular, but the claimed a priori error bound is calibrated on the same numerical experiments it is then used to describe.

fitted input called prediction [Section 2.2.4 (Eqs. 14-16), validated in Section 4 (Fig. 15d-f)]
"Numerical experiments we have conducted in the Section 4 support our conjectures above and indicate that when c1 and c2 are set to be 1, if Rs≈ 1 and Re < 10, the BF approximation can usually deliver good results (better than the low-ﬁdelity solutions)."

The conjectured error bound in Eqs. (14)-(16) contains undetermined constants c1 and c2. The paper fixes these constants as c1 = c2 = 1 based on the numerical experiments reported in Section 4, and then presents those same Section 4 results as confirmation: 'the empirical error bound estimations can basically capture the trends of the true errors in terms of the number of HF training samples in all cases.' Therefore the agreement between the 'error bound' and the true errors in Fig. 15 is not an independent a priori prediction; the free constants were chosen after inspecting the same true-error curves. The paper's own caveat that 'the error bound estimation is not rigorous' confirms that this is a calibrated empirical description rather than a derived bound.

full rationale

The central claim of the paper, that the bi-fidelity surrogate can propagate inflow uncertainties to full-field hemodynamic quantities using few high-fidelity runs, is not circular. In each test case the surrogate is evaluated on independently sampled HF Monte Carlo test sets (e.g., 600 HF simulations in the patient-specific case), and the reported accuracy is an empirical comparison against those ground-truth samples. The coefficient-sharing assumption in Eq. (10) is an unproved heuristic, but it is not a tautology: the surrogate could fail and the paper checks it against data, so the main accuracy result has independent content. The method is imported from prior works [76-78], including coauthored references, but those are published methodological papers and the present manuscript supplies its own implementations, geometries, and independent validation; self-citation alone is not circular. The one genuine circular element is the 'empirical error bound estimation' advertised as a novel contribution: the constants c1 and c2 in the conjectured bound are set to 1 by inspecting the same numerical experiments that are then used to demonstrate that the bound captures the true-error trends. That makes the error-bound demonstration a calibration exercise rather than a prediction. Because this affects only the secondary a priori-error contribution and not the main surrogate-accuracy claim, the appropriate score is 4.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The main free parameters are the empirical constants in the error bound conjecture. The key axioms are the coefficient-sharing assumption between HF and LF reconstructions, the informativeness of the LF model for point selection, and the unproved error-bound conjecture. No new physical entities are introduced.

free parameters (2)

c1, c2 (error bound constants) = 1, 1
In Eqs. (14)-(16), the conjectured error bound depends on constants c1 and c2, which are set to 1 in the remark after Eq. (16) based on the numerical experiments in Section 4, not derived from theory.
Re threshold (<10) = 10
The paper suggests stopping when the in-plane-to-distance error ratio Re is less than 10, a heuristic threshold chosen to flag when additional HF samples are not helpful. No theoretical justification is given.

assumptions (4)

domain assumption HF and LF reconstructions share the same coefficients (Eqs. 9-10).
The online reconstruction projects the LF solution onto the LF basis and reuses the coefficients to combine HF basis. The entire surrogate hinges on this unverified equivalence.
domain assumption The LF model is informative for important-point selection (Rs≈1, Eq. 12).
The greedy point selection uses only LF solutions via pivoted Cholesky on the LF Gramian. If the LF solution manifold is structurally different from the HF one, the chosen points may not be important for HF reconstruction.
ad hoc to paper The error bound conjecture (Eq. 14) with bounded constants.
The authors explicitly state the bound is not rigorous and that numerical experiments support it. It is an empirical assumption, not a theorem.
domain assumption Steady, rigid-wall, Newtonian incompressible Navier-Stokes equations (Eq. 1).
The hemodynamic model treats vessels as rigid and blood as Newtonian, standard assumptions for this class of studies but not valid for all clinical flows.

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Cite this review

Pith. "Pith review of A Bi-fidelity Surrogate Modeling Approach for Uncertainty Propagation in Three-Dimensional Hemodynamic Simulations." pith.science (2026). https://pith.science/paper/LPPKBTHW

@misc{pith2026190810197,
  author       = {Pith},
  title        = {Pith review of: A Bi-fidelity Surrogate Modeling Approach for Uncertainty Propagation in Three-Dimensional Hemodynamic Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPPKBTHW}},
  note         = {Machine review of arXiv:1908.10197}
}

read the original abstract

Image-based computational fluid dynamics (CFD) modeling enables derivation of hemodynamic information, which has become a paradigm in cardiovascular research and healthcare. Nonetheless, the predictive accuracy largely depends on precisely specified boundary conditions and model parameters, which, however, are usually uncertain in most patient-specific cases. Quantifying the uncertainties in model predictions due to input randomness can provide predictive confidence and is critical to promote the transition of CFD modeling in clinical applications. In the meantime, forward propagation of input uncertainties often involves numerous expensive CFD simulations, which is computationally prohibitive in most practical scenarios. This paper presents an efficient bi-fidelity surrogate modeling framework for uncertainty quantification (UQ) in cardiovascular simulations, by leveraging the accuracy of high-fidelity models and efficiency of low-fidelity models. Contrary to most data-fit surrogate models with several scalar quantities of interest, this work aims to provide high-resolution, full-field predictions. Moreover, a novel empirical error bound estimation approach is introduced to evaluate the performance of the surrogate a priori. The proposed framework is tested on a number of vascular flows with both standardized and patient-specific vessel geometries, and different combinations of high- and low-fidelity models are investigated. The results show that the bi-fidelity approach can achieve high predictive accuracy with a significant reduction of computational cost, exhibiting its merit and effectiveness. Particularly, the uncertainties from a high-dimensional input space can be accurately propagated to clinically relevant quantities of interest in the patient-specific case using only a limited number of high-fidelity simulations, suggesting a good potential in practical clinical applications.

Figures

Figures reproduced from arXiv: 1908.10197 by the authors.

**Figure 2.** The 3-D asymmetric nozzle with (a) coarse and (b) fine computational meshes. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** The (a) pressure and (b) velocity magnitude profiles along the center line with 1 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: The relative root mean squared error (RMSE) of the BF model (black) over 600 test parameter [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: The idealized bifurcation aneurysm geometry with (a) 2-D and (b) 3-D computational meshes. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: The (a-c) pressure and (d-f) velocity magnitude profiles along [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: The probability density functions (PDF) of maximum velocity magnitude, vorticity magnitude [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: The relative root mean squared error (RMSE) of the BF model (black) over 250 test samples with [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: The patient-specific MCA aneurysm geometry with (a) coarse and (b) fine computational meshes [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: Volume rendering of the (a-c) mean and (d-f) standard deviation of the internal velocity fields [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]

**Figure 11.** Figure 11: Contours of the (a-c) mean and (d-f) standard deviation of the pressures on the vessel wall over [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]

**Figure 12.** Figure 12: Contours of the (a-c) mean and (d-f) standard deviation of the wall shear stress (WSS) magnitudes [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]

**Figure 13.** Figure 13: Probability density distribution of the spatial locations of minimum WSS. [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]

**Figure 14.** Figure 14: The relative root mean squared error (RMSE) of the BF model (black) over 600 test parameter [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]

**Figure 15.** Figure 15: The relative distance (a-c) and error bound estimation (d-e) of the BF models with respect to [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]

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