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

This study demonstrates that while Physics-Informed Neural Networks (PINNs) can reconstruct wall shear stress from passive scalar data only when near-wall measurements are available, a differentiable physics framework based on PDE-constrained optimization successfully recovers accurate wall shear stress across diverse measurement scenarios in both canonical and patient-specific cardiovascular flows.

The Gist

Imagine you are trying to figure out how fast the wind is blowing right against the side of a house (the "wall"), but you aren't allowed to stand next to the wall to measure it. Instead, you can only see how smoke or dye is moving in the air a few feet away from the house.

This is the core challenge of the paper: How do we calculate the "friction" of a fluid (like blood) against a vessel wall when we can only see the "smoke" (a passive tracer) moving in the middle of the flow?

Here is a breakdown of the paper's story, methods, and findings using everyday analogies.

The Problem: The Invisible Wall

In our bodies, blood flows through arteries. The force of that blood rubbing against the artery wall is called Wall Shear Stress (WSS). This force is crucial; if it's too low or weirdly shaped, it can cause heart disease or aneurysms.

However, measuring this force is like trying to guess the speed of a car by looking at the dust it kicks up, but you can't see the car itself.

• The Car: The blood flow.
• The Dust: A passive tracer (like a dye or contrast agent used in medical scans).
• The Goal: Figure out exactly how fast the car is moving right next to the curb (the wall) just by watching the dust.

The problem is tricky because the dust doesn't always tell the whole story. If the wind is blowing parallel to a line of dust, the dust doesn't move much, even if the wind is strong. This makes it hard to work backward from the dust to the wind speed.

The Two Detectives: PINN vs. Differentiable Physics

The authors tested two different "detective" methods to solve this mystery. Both try to guess the hidden flow, but they have very different rulebooks.

1. The "Soft Constraint" Detective (PINN)

The Analogy: Imagine a student trying to solve a math problem. They have the answer key (the data) and the textbook rules (physics equations).

• How they work: They guess an answer, check it against the answer key, and then check it against the textbook. If they get it wrong, they get a "penalty" (a loss score). They keep adjusting their guess to lower the penalty.
• The Catch: The rules are "soft." The student is encouraged to follow the textbook, but they can bend the rules a little if it helps them match the answer key better. They are trying to find a balance between the data and the physics.

2. The "Hard Constraint" Detective (Differentiable Physics)

The Analogy: Imagine a master engineer building a bridge.

• How they work: They don't just guess; they run a perfect simulation of physics first. They change the input (the wind at the start of the bridge), run the simulation, and see where the dust ends up. If the dust doesn't match the observation, they tweak the input and run the simulation again.
• The Catch: The rules are "hard." The simulation must obey the laws of physics perfectly every single time. They are essentially asking: "What specific wind at the entrance would cause the dust to land exactly where we see it, while strictly obeying the laws of fluid mechanics?"

The Experiments: Two Test Cases

The authors tested these detectives in two scenarios:

1. The 2D Step (A Simple Room): A flat channel with a sudden step down. They tested three ways of looking at the "dust":

   • Near the Wall: Watching the dust right next to the floor.
   • Far from the Wall: Watching the dust in the middle of the room.
   • Both: Watching everywhere.

2. The 3D Artery (A Real Patient): A complex, narrowed (stenotic) coronary artery from a real patient. They only looked at the dust near the wall.

The Results: Who Won?

In the Simple Room (2D Step)

• When the dust was near the wall: The Soft Constraint (PINN) detective did a great job. Since the dust was right next to the wall, it gave direct clues about the wall friction.
• When the dust was far away: The Soft Constraint detective failed miserably. It couldn't guess the wall friction just by looking at the middle of the room.
• The Hard Constraint (Differentiable Physics) detective won every time. Even when the dust was far away, this detective used the strict laws of physics to trace the wind all the way back to the wall. It didn't matter where the dust was; the physics simulation connected the dots perfectly.

In the Real Artery (3D Coronary)

• The Soft Constraint (PINN) detective struggled. It could guess the general shape of the friction, but the numbers were way off (about 31% error). It was like guessing the speed of a car but getting the number wrong by a huge margin.
• The Hard Constraint (Differentiable Physics) detective was a star. It reconstructed the flow with high precision (only 2.5% error). Because it forced the solution to obey the strict laws of fluid dynamics, it got the "friction" numbers right, even in the complex, narrow parts of the artery.

The Big Takeaway

The paper concludes that where you look matters, and the method you use matters even more.

• If you have data right next to the wall, a flexible, AI-based method (PINN) works well.
• If your data is far away, or if the geometry is complex (like a real artery), you need the strict, physics-ensured method (Differentiable Physics).

The authors found that simply throwing more data at the problem (looking at both near-wall and far-field) didn't always help. Sometimes, mixing different types of clues confused the flexible detective (PINN), while the strict detective (Differentiable Physics) remained steady and accurate.

In short: To find the hidden friction on a vessel wall using only dye observations, the "strict engineer" approach (Differentiable Physics) proved to be the most reliable detective, especially when the clues were hard to find or the situation was complex.

Technical Summary

Technical Summary: Wall Shear Stress Reconstruction from Concentration

Problem Statement

Wall Shear Stress (WSS) is a critical hemodynamic indicator governing near-wall transport dynamics, endothelial cell response, and vascular health. However, direct measurement of WSS is experimentally challenging because it requires resolving steep velocity gradients within thin viscous boundary layers, particularly in biological or in vivo settings. Conversely, passive scalar fields (e.g., contrast-agent concentration or temperature) can often be measured with higher spatial resolution. Since these scalars are advected by the flow, their spatiotemporal evolution contains indirect information about the underlying velocity field.

The core inverse problem addressed in this work is the reconstruction of WSS from spatially limited passive scalar observations without any direct velocity measurements or prior knowledge of inlet flow boundary conditions. The study specifically investigates whether accurate WSS can be inferred when scalar data is confined to a small region of the flow domain (e.g., only near the wall or only in the far-field) and compares two fundamentally different inverse frameworks to solve this ill-posed problem.

Methodology

The authors compare two distinct physics-constrained data-driven approaches:

1. Physics-Informed Neural Networks (PINNs)

• Formulation: PINNs treat the governing equations (Navier-Stokes and advection-diffusion) as soft constraints. The solution fields (velocity, pressure, concentration) are represented by neural networks optimized to minimize a loss function comprising data mismatch and physics residuals.
• Constraints: The governing PDEs are enforced via automatic differentiation within the loss function. Boundary conditions (specifically no-slip at the wall) are imposed as soft penalties. No concentration boundary conditions were prescribed in the final formulation, as preliminary tests showed they degraded WSS accuracy.
• Optimization: The neural network parameters are optimized using the AdamW optimizer. The study investigates the effect of the weighting parameter $\alpha$, which controls whether physics residuals are enforced only within the observation region or across the entire domain.

2. Differentiable Physics (DP)

• Formulation: This approach employs a hard-constraint formulation. A numerical PDE solver (based on FEniCS and the discrete adjoint method via dolfin-adjoint) is embedded directly within the optimization loop.
• Control Variable: Instead of optimizing the velocity field at every grid point (which is highly ill-posed), the optimization variable is restricted to the inlet velocity profile ($u_{in}$). The governing equations are solved exactly at every step, ensuring mass and momentum conservation are strictly satisfied.
• Gradient Computation: Gradients of the objective functional with respect to the inlet profile are computed via the discrete adjoint method, ensuring numerical exactness and consistency with the discretized forward solver.
• Strategy: To handle convergence sensitivity, an ensemble optimization strategy was employed, utilizing multiple scaled initial guesses for the inlet profile to avoid poor local minima.

Benchmark Problems

Two test cases were utilized to evaluate the methods under different measurement scenarios:

1. 2D Backward-Facing Step (2D-BFS): A canonical flow with a Reynolds number ($Re$) of 53. Three measurement scenarios were tested:
   • Near-wall observations only.
   • Far-field observations only.
   • Combined near-wall and far-field observations.
2. 3D Patient-Specific Stenotic Coronary Artery: A complex geometry with $Re \approx 211$. Measurements were restricted to the near-wall region downstream of the stenosis.

Key Results

2D Backward-Facing Step

• Near-Wall Data: PINN achieved high accuracy (Rel-L2 error of 2.7%) when restricted to near-wall data, outperforming the differentiable physics approach (6% error). However, PINN performance degraded significantly as the physics loss was enforced outside the observation region ($\alpha > 0$).
• Far-Field Data: PINN failed to reconstruct accurate WSS (Rel-L2 error $\approx$ 100%), producing a nearly flat prediction. In contrast, the differentiable physics approach successfully reconstructed accurate WSS (2.4% error), demonstrating robustness even when data was far from the wall.
• Combined Data: Combining regions did not consistently improve results. PINN showed degraded accuracy compared to the near-wall-only case, while differentiable physics maintained high accuracy (2.5% error).

3D Coronary Artery

• Velocity Reconstruction: PINN exhibited high velocity errors (84–90%) in the region of interest. The differentiable physics approach achieved significantly lower velocity errors (2.6%).
• WSS Reconstruction: Differentiable physics yielded an accurate WSS reconstruction with a Rel-L2 error of 2.5%. PINN resulted in a significantly higher error of 31%, capturing the general topology but failing to accurately predict the magnitude and precise location of fixed points.

Key Contributions and Claims

The paper claims the following contributions and findings:

1. Systematic Comparison: This work provides the first direct quantitative comparison between differentiable physics (hard constraints) and PINNs (soft constraints) for reconstructing WSS from passive scalar data confined to limited spatial regions.
2. Dependence on Measurement Location: The study establishes that reconstruction fidelity is jointly determined by the measurement location and the inverse formulation. PINNs are highly sensitive to the spatial distribution of data, performing well only when near-wall data is available. Differentiable physics demonstrates greater robustness, recovering accurate WSS even from far-field measurements.
3. Inverse Formulation Impact: The results suggest that for inverse problems where the goal is to recover boundary-driven flow fields (like inlet profiles) to infer wall quantities, hard-constraint formulations (Differentiable Physics) may outperform soft-constraint neural approaches, particularly when data is sparse or distant from the region of interest.
4. Limitations of Combined Data: The study notes that adding more scalar observations from regions with different sensitivities (e.g., combining near-wall and far-field) does not always improve reconstruction and can sometimes hinder convergence due to conflicting optimization objectives.

Significance

The authors posit that the proposed framework opens a path toward estimating near-wall hemodynamics from scalar transport data (such as contrast-agent dynamics in medical imaging) without requiring direct velocity measurements. The findings highlight that while PINNs are powerful, their application to inverse hemodynamic problems must be carefully matched to the availability and location of observational data. The differentiable physics approach offers a robust alternative for scenarios where data is limited or spatially restricted, ensuring physical consistency through exact enforcement of governing equations.