Asymptotic-Preserving Neural Networks for Viscoelastic Parameter Identification in Multiscale Blood Flow Modeling

Giulia Bertaglia; Raffaella Fiamma Cabini

cs.LGApr 7, 2026

Asymptotic-Preserving Neural Networks for Viscoelastic Parameter Identification in Multiscale Blood Flow Modeling

Authors: Giulia Bertaglia, Raffaella Fiamma Cabini

Organizations: Department of Environmental and Prevention Sciences, University of Ferrara, Corso Ercole I d’Este 32, 44121, Ferrara, Italy

Abstract

Mathematical models and numerical simulations offer a non-invasive way to explore cardiovascular phenomena, providing access to quantities that cannot be measured directly. In this study, we start with a one-dimensional multiscale blood flow model that describes the viscoelastic properties of arterial walls, and we focus on improving its practical applicability by addressing a major challenge: determining, in a reliable way, the viscoelastic parameters that control how arteries deform under pulsatile pressure. To achieve this, we employ Asymptotic-Preserving Neural Networks that embed the governing physical principles of the multiscale viscoelastic blood flow model within the learning procedure. This framework allows us to infer the viscoelastic parameters while simultaneously reconstructing the time-dependent evolution of the state variables of blood vessels. With this approach, pressure waveforms are estimated from readily accessible patient-specific data, i.e., cross-sectional area and velocity measurements from Doppler ultrasound, in vascular segments where direct pressure measurements are not available. Different numerical simulations, conducted in both synthetic and patient-specific scenarios, show the effectiveness of the proposed methodology.

Figures & tables

Panel of Figure 1 — Figure 1: Asymptotic limits of the multiscale constitutive framework. By appropriately scaling the viscoelastic parameters, the SLS rheological law allows the blood flow model to capture two regimes: purely elastic behavior with hyperbolic dynamics or Kelvin-Voigt viscoelastic behavior with diffusive dynamics.

Panel of Figure 2 — Figure 2: Feedforward Neural Network architecture. The network maps space–time inputs $(\mathbf{x},t)$ to the predicted solution $\mathbf{U}$ , with parameters $\boldsymbol{\theta}$ optimized by minimizing the loss $\mathcal{L}_{d}(\boldsymbol{\theta})$ .

Panel of Figure 3 — Figure 3: PINN architecture. The network incorporates both data-driven and physics-based constraints in the loss function, including the data mismatch term $\mathcal{L}_{d}$ , the PDEs residual term $\mathcal{L}_{r}$ , and the boundary or initial condition term $\mathcal{L}_{b}$ , to ensure consistency with the governing equations.

Panel of Figure 4 — Figure 4: Schematic illustration of the concept of asymptotic-preservation (AP) property in an APNN. The PINN approximates the solution of the multiscale problem $\mathcal{F}^{\varepsilon}$ by minimizing the residual $\mathcal{L}_{r}^{\varepsilon}$ . In the asymptotic limit $\varepsilon\to 0$ , the residual converges to $\mathcal{L}_{r}^{0}$ , which is consistent with the reduced system $\mathcal{F}^{0}$ , ensuring the network preserves the correct asymptotic behavior.

Panel of Figure 5 — Figure 5: Schematic representation of the APNN framework. The neural network maps the space–time inputs $(x,t)$ to the predicted fields $(\hat{A},\hat{u},\hat{p})$ . The data loss $\mathcal{L}_{d}$ is evaluated only on the observed variables $\hat{A}$ and $\hat{u}$ (highlighted by the blue box), while the pressure $\hat{p}$ is inferred implicitly through the physics-based residual $\mathcal{L}_{r}$ and boundary and initial losses $\mathcal{L}_{b}$ , which are computed from the blood flow model ( 2 ).

Parameter	Value	Parameter	Value
$L_{0}$ [cm]	24.137	$R_{1}$ [MPa s/m 3 ]	18.503
$R_{0,in}$ [mm]	15.000	$R_{2}$ [MPa s/m 3 ]	104.920
$R_{0,out}$ [mm]	10.000	C [m 3 /GPa]	10.163
$h_{0}$ [mm]	1.000	$E_{0}$ [MPa]	0.727
$c$ [m/s]	5.494	$E_{\infty}$ [MPa]	0.533
$p_{0}$ [kPa]	9.467	$\eta$ [kPa s]	23.884

Table 1: Model parameters for the tapered upper TA used to generate the dataset. Geometrical and mechanical properties, taken from [ 58 ] , include vessel length (

L_{0}

), inlet and outlet radii (

R_{0,\mathrm{in}}

R_{0,\mathrm{out}}

), wall thickness (

h_{0}

), reference wave speed (

c

), diastolic pressure (

p_{0}

), Windkessel parameters (resistances

R_{1}

R_{2}

, compliance

C

, outflow pressure

p_{\mathrm{out}}

), viscoelastic wall properties (instantaneous and asymptotic Young moduli

E_{0}

E_{\infty}

, viscosity

\eta

, relaxation time

\tau_{r}

), and blood density (

\rho

). Numerical setup:

\text{CFL}=0.9

n_{c}=12

cells, simulation time

t_{\mathrm{m}}=20.0\penalty\ \mathrm{s}

(20 cardiac cycles, 63 bpm).

Panel of Figure 6 — Figure 6: Synthetic TA test results at the vessel midpoint. The first panel compares the reference numerical solution (red) and the APNN reconstruction (blue) for cross-sectional area, velocity, and pressure over one cardiac cycle. The second panel shows the corresponding percentage relative error (PRE) over time, with the mean PRE reported for each variable. Area and velocity data sampled at this location are used for training, whereas pressure is inferred entirely by the APNN.

Panel of Figure 7 — Figure 7: Synthetic TA test results over the full spatial domain. The first panel shows the APNN reconstruction of cross-sectional area, velocity, and pressure along the vessel over one cardiac cycle. The second panel displays the corresponding spatio-temporal percentage relative error (PRE) maps, with the mean PRE reported for each variable. Although training data for area and velocity were restricted to the vessel midpoint, the APNN provides the full-field distributions. The pressure field is entirely inferred by the APNN through the governing equations, without any direct measurements.

Figure 8: Training progress of the APNN for the synthetic dataset. The evolution of the parameters $E_{0}$ and $\tau_{r}$ over the training epochs is shown in blue, while the corresponding ground truth values used to generate the data are shown in red. The legend reports the final estimated value of each parameter along with its ground truth. The mean Percentage Relative Error (PRE) for each variable is also indicated.

Parameter	CCA-A	CCA-B	CFA-C
Age [years]	28	29	44
$L_{0}$ [cm]	10.900	10.900	10.900
$R_{0,in}$ [mm]	3.267	3.514	3.653
$R_{0,out}$ [mm]	2.809	2.899	2.332
$h0$ [mm]	0.300	0.300	0.300
$c$ [m/s]	5.920	5.920	5.920

Table 2: Vessel parameters for the three studied arteries: CCA-A, CCA-B, and CFA-C. Geometrical and mechanical properties include age, vessel length (

L_{0}

), inlet and outlet radii (

R_{0,in}

R_{0,out}

), wall thickness (

h_{0}

), reference wave speed (

c

), diastolic pressure (

p_{0}

), asymptotic Young modulus (

E_{\infty}

), and blood density (

\rho

Panel of Figure 9 — Figure 9: In vivo test results at the vessel measurement point for three different carotid arteries (CCA-A, CCA-B, CCA-C). For each case, the upper panel shows the comparison between the measured waveform (red) and the APNN reconstruction (blue) for cross-sectional area, velocity, and pressure over one cardiac cycle. The lower panel reports the corresponding percentage relative error (PRE) over time, with the mean PRE indicated for each variable. Area and velocity measurements at the considered location are used for training, whereas pressure is entirely inferred by the APNN model.

Panel of Figure 10 — Figure 10: In vivo test results on the full spatial domain for three carotid arteries (CCA-A, CCA-B, CCA-C). For each case, the panels show the APNN reconstruction of cross-sectional area, velocity, and pressure over one cardiac cycle along the vessel. Measurements were available only at a single spatial location ( $x_{m}$ ) for area and velocity, while the pressure field is entirely inferred by the APNN through the governing equations.

Panel of Figure 11 — Figure 11: Training progress of the APNN for the three carotid arteries (CCA-A, CCA-B, CCA-C). The evolution of the parameters $E_{0}$ and $\tau_{r}$ over the training epochs is shown in blue, while reference values from standard calibration procedures are shown in green. These reference values are approximate estimates and do not represent exact ground truth. The legend reports the final estimated value of each parameter along with its corresponding reference value.

Parameter Estimation

Parameter estimation studies how to infer unknown quantities in statistical, physical, and machine-learning models from observed data.

See all 144 papers

Physics Informed Neural Network model for the dynamical study of Abdominal Aortic Aneurysm
Sep 14, 2026Adrián Robles Arques, Martín Ruiz Fernandez, Javier Sanchis +2Neural PDE Solvers Physics-Informed ML
Lucentia Research; Instituto Universitario de Investigación en Informática, Universidad de Alicante, Carr. San Vicent del Raspeig, Alicante, 03690, Comunidad Valenciana, España
A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations
Sep 19, 2025Simon Klaes, Axel Klawonn, Natalie Kubicki +4PDE Solving PDE Surrogate Modeling
Department of Mathematics and Computer Science, University of Cologne, Germany · Center for Data and Simulation Science, University of Cologne, Germany · Institute for Advanced Simulation (IAS-2), Forschungszentrum Jülich, Germany +1
Real-Time Surrogate Modeling for Personalized Blood Flow Prediction and Hemodynamic Analysis
Apr 3, 2026Sokratis J. Anagnostopoulos, George Rovas, Vasiliki Bikia +3Neural Surrogate Modeling Clinical Prediction
Laboratory of Hemodynamics and Cardiovascular Technology, EPFL, Lausanne, Switzerland · Stanford DBDS and HAI, Stanford University, Palo Alto, USA · Medical School, National and Kapodistrian University of Athens, Athens, Greece

Sep 14, 2026J/K move · Enter open · S save

Physics Informed Neural Network model for the dynamical study of Abdominal Aortic Aneurysm

Adrián Robles Arques, Martín Ruiz Fernandez, Javier Sanchis, Miguel A. Teruel+1

Lucentia Research; Instituto Universitario de Investigación en Informática, Universidad de Alicante, Carr. San Vicent del Raspeig, Alicante, 03690, Comunidad Valenciana, España

Neural PDE Solvers Physics-Informed ML Computational Fluid Dynamics

Open paper AskPDF arXiv

We present the development and application of a three-dimensional Physics-Informed Neural Network (PINN) framework for the investigation of haemodynamic behaviour in the human aorta. The model incorporates a time-resolved simulation of pulsatile blood flow over a two-minute interval, enabling the extraction of pressure and velocity fields with high temporal fidelity. The mechanical stress exerted on the aortic wall was quantified through Laplace's law, with temporal averaging applied to derive representative stress distributions. This approach circumvents the computational overhead associated with conventional computational fluid dynamics (CFD) methods by eliminating mesh generation and exploiting the automatic differentiation capabilities inherent to neural networks. The proposed methodology demonstrates that PINNs can serve as an efficient and accurate alternative for modelling complex vascular flow phenomena, offering significant advantages in scalability and computational cost reduction while maintaining physical consistency.

Abstract

Parameter Estimation

Explore similar work

Physics Informed Neural Network model for the dynamical study of Abdominal Aortic Aneurysm

A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations

Real-Time Surrogate Modeling for Personalized Blood Flow Prediction and Hemodynamic Analysis