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arxiv: 2606.06313 · v1 · pith:RY2K5KMInew · submitted 2026-06-04 · ⚛️ physics.flu-dyn · cs.LG· physics.comp-ph

Wall Shear Stress Reconstruction from Concentration: Differentiable Physics and Physics-Informed Neural Networks

Mahmoud Elhadidy , Siva Viknesh , Roshan M. D'Souza , Amirhossein Arzani This is my paper

Pith reviewed 2026-06-27 23:27 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LGphysics.comp-ph
keywords wall shear stresspassive scalar transportdifferentiable physicsphysics-informed neural networksinverse problemshemodynamicsbackward-facing step
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The pith

Differentiable physics recovers accurate wall shear stress from concentration observations even with far-field measurements, while PINNs require near-wall data.

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

The paper shows that wall shear stress can be reconstructed from passive scalar concentration fields using inverse methods that enforce the flow equations. A differentiable physics approach based on discrete adjoint optimization treats the governing equations as hard constraints and recovers accurate WSS in a 2D backward-facing step flow under near-wall, far-field, and combined measurement scenarios. Physics-informed neural networks, which use soft constraints, only succeed when near-wall data are available and fail on far-field data alone. In a 3D patient-specific stenotic coronary artery, the differentiable physics method also produces better WSS estimates than PINNs. The results indicate that measurement location and the choice of how constraints are imposed together control whether scalar data can reveal near-wall velocity gradients.

Core claim

For the 2D backward-facing step, the differentiable physics framework recovers accurate wall shear stress under near-wall, far-field, and combined measurement scenarios, while PINNs achieve high accuracy only with near-wall data and fail with far-field measurements. In the 3D patient-specific stenotic coronary artery, the differentiable physics framework yields more accurate WSS reconstruction than PINNs.

What carries the argument

Discrete adjoint PDE-constrained optimization that enforces the advection-diffusion equation for the passive scalar as a hard constraint to recover the underlying velocity field and its near-wall gradients from concentration observations.

If this is right

  • WSS can be inferred from scalar transport data without requiring velocity measurements near the wall.
  • Hard enforcement of the advection-diffusion equation enables recovery of near-wall gradients from distant observations.
  • The same reconstruction approach applies to both simple 2D benchmark flows and complex 3D patient-specific geometries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend to temperature or other passive scalars in additional flow configurations where direct velocity data are sparse.
  • Non-invasive imaging of contrast agents could supply the concentration fields needed for hemodynamic WSS estimation in clinical settings.
  • Hybrid schemes that combine hard-constraint optimization with neural-network representations might improve robustness on noisy or incomplete data.

Load-bearing premise

The observed concentration field is produced by advection and diffusion under exactly the same velocity field that sets the wall shear stress.

What would settle it

In the 2D backward-facing step or 3D artery simulations, if the wall shear stress reconstructed from far-field concentration data shows large errors relative to the ground-truth values computed directly from the known velocity field, the claim does not hold.

Figures

Figures reproduced from arXiv: 2606.06313 by Amirhossein Arzani, Mahmoud Elhadidy, Roshan M. D'Souza, Siva Viknesh.

Figure 1
Figure 1. Figure 1: Comparison between physics-informed neural networks (PINNs) and differentiable physics (DP) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the test cases. (a) 2D backward-facing step (2D-BFS), showing geometry, velocity [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: PINN results for the 2D-BFS case. (a) Comparison of predicted and ground-truth (GT) WSS. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ensemble differentiable physics results for the 2D-BFS case. (a) Governing formulation and [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: PINN results for the 3D coronary artery case. Comparison of the predicted and ground-truth [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Coronary artery differentiable physics results. (a) Ensemble training procedure is summarized. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Coronary artery WSS results. (a) Isometric views of the ground-truth (GT) WSS field, together [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Surface Transport Model (STM) setup and results. (a) 3D constricted channel with a [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
read the original abstract

Wall shear stress (WSS) governs near-wall transport dynamics and is a key hemodynamic indicator in cardiovascular flows, yet remains difficult to infer accurately due to the need for precise computation of near-wall velocity gradients. Passive scalar fields, such as concentration or temperature, are advected by the same underlying velocity field and have the potential to uncover hidden flow physics metrics such as WSS. In this work, we demonstrate such reconstruction from spatially limited passive scalar observations using two fundamentally different inverse frameworks: a differentiable physics framework based on discrete adjoint, PDE-constrained optimization, which enforces the governing equations as hard constraints, and physics-informed neural networks (PINNs), which treat them as soft constraints. Benchmark problems include a 2D canonical backward-facing step (2D-BFS) and a 3D patient-specific stenotic coronary artery. For the 2D-BFS case, evaluated under three measurement scenarios (near-wall, far-field, and combined), PINN achieves high accuracy when near-wall data are available but fails when restricted to far-field measurements, whereas the differentiable physics approach recovers accurate WSS across all scenarios. In the 3D patient-specific case, the differentiable physics framework outperforms PINNs, yielding accurate WSS reconstruction. These results establish that measurement location and inverse formulation jointly determine reconstruction fidelity in scalar-based near-wall flow inference. The proposed framework opens a path toward estimation of near-wall hemodynamics from scalar transport data, with broader applicability to fluid flow problems where passive scalars can be observed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes reconstructing wall shear stress (WSS) from passive scalar concentration observations using two inverse approaches: a differentiable physics method based on discrete adjoint PDE-constrained optimization (hard constraints) and physics-informed neural networks (soft constraints). It benchmarks both on a 2D backward-facing step (BFS) under near-wall, far-field, and combined measurement scenarios, plus a 3D patient-specific stenotic coronary artery. The central claim is that the differentiable physics framework recovers accurate WSS even from far-field data alone, while PINNs succeed only when near-wall data are available; the 3D case similarly favors the differentiable physics approach.

Significance. If the far-field reconstruction result holds with supporting quantitative evidence, the work would be significant for enabling inference of near-wall hemodynamics from scalar transport data in applications such as cardiovascular flows. The explicit comparison of hard versus soft enforcement of the advection-diffusion equations is a useful contribution to inverse problem methodology in fluid dynamics.

major comments (2)
  1. [Abstract] Abstract: the claim that the differentiable physics approach 'recovers accurate WSS across all scenarios' (including far-field only) is not supported by any quantitative error metrics, relative errors, or validation details against ground-truth WSS; without these, the performance difference versus PINNs cannot be assessed.
  2. [Results (2D-BFS far-field case)] The 2D-BFS far-field reconstruction result (central to the paper's novelty) lacks any sensitivity analysis, uniqueness discussion, or regularization study. In recirculating advection-dominated flow, far-field scalar observations may be insensitive to near-wall gradients; the manuscript provides no test (e.g., adjoint-based sensitivity or multiple initializations) confirming that the recovered WSS is determined by the data rather than being a non-unique solution consistent with far-field measurements alone.
minor comments (2)
  1. [Methods] Notation for the discrete adjoint and the concentration transport equation should be introduced with explicit equation numbers for reproducibility.
  2. [Figures] Figure captions for the 2D-BFS and 3D cases should include the specific error norms or visual comparison metrics used to support the 'high accuracy' statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment point-by-point below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the differentiable physics approach 'recovers accurate WSS across all scenarios' (including far-field only) is not supported by any quantitative error metrics, relative errors, or validation details against ground-truth WSS; without these, the performance difference versus PINNs cannot be assessed.

    Authors: We agree that the abstract would benefit from explicit quantitative support. While the results section contains visual and comparative evidence, the abstract itself summarizes findings at a high level. We will revise the abstract to include relative L2 error metrics for WSS reconstruction under each measurement scenario (near-wall, far-field, combined) and note the corresponding errors for the PINN baseline, enabling direct quantitative assessment of the performance difference. revision: yes

  2. Referee: [Results (2D-BFS far-field case)] The 2D-BFS far-field reconstruction result (central to the paper's novelty) lacks any sensitivity analysis, uniqueness discussion, or regularization study. In recirculating advection-dominated flow, far-field scalar observations may be insensitive to near-wall gradients; the manuscript provides no test (e.g., adjoint-based sensitivity or multiple initializations) confirming that the recovered WSS is determined by the data rather than being a non-unique solution consistent with far-field measurements alone.

    Authors: We recognize that additional analysis would strengthen confidence in the far-field result. The differentiable physics formulation already employs the discrete adjoint for gradient computation within the PDE-constrained optimization; we will augment the manuscript with an explicit adjoint-based sensitivity study, results from multiple random initializations to demonstrate convergence to the same WSS field, and a brief regularization study. These additions will directly test whether the far-field data determine a unique near-wall solution consistent with the ground truth. revision: yes

Circularity Check

0 steps flagged

No circularity: independent numerical comparison of two inverse solvers

full rationale

The paper formulates and compares two distinct inverse frameworks (discrete-adjoint PDE-constrained optimization versus PINN soft-constraint enforcement) for recovering WSS from scalar transport data. Performance claims rest on direct numerical experiments across measurement scenarios on 2D-BFS and 3D artery geometries; neither framework is algebraically derived from the other, nor are any reported quantities obtained by fitting a parameter to a subset and then relabeling the fit as a prediction. No self-citation chain, uniqueness theorem, or ansatz smuggling appears in the provided text. The derivation chain is therefore self-contained and externally falsifiable via the reported benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that the advection-diffusion equation for the scalar is known and that the inverse problem is well-posed under the stated measurement scenarios.

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