Model Order Reduction of Cerebrovascular Hemodynamics Using POD_Galerkin and Reservoir Computing_based Approach

Rahul Halder; Arash Hajisharifi; Kabir Bakhshaei; Gianluigi Rozza

math.NAMar 19, 2026

Model Order Reduction of Cerebrovascular Hemodynamics Using POD_Galerkin and Reservoir Computing_based Approach

Authors: Rahul Halder, Arash Hajisharifi, Kabir Bakhshaei, Gianluigi Rozza

Organizations: Mathematics Area, mathLab, SISSA, via Bonomea 265, I-34136 Trieste, Italy · Biorobotics Institute, SMART Lab, Sant’Anna School of Advanced Studies, Pisa, Italy · Department of Computer Science, University of Pisa, Pisa, Italy

Abstract

We investigate model order reduction (MOR) strategies for simulating unsteady hemodynamics within cerebrovascular systems, contrasting a physics-based intrusive approach with a data-driven non-intrusive framework. High-fidelity 3D Computational Fluid Dynamics (CFD) snapshots of an idealised basilar artery bifurcation are first compressed into a low-dimensional latent space using Proper Orthogonal Decomposition (POD). We evaluate the performance of a POD-Galerkin (POD-G) model, which projects the Navier-Stokes equations onto the reduced basis, against a POD-Reservoir Computing (POD-RC) model that learns the temporal evolution of coefficients through a recurrent architecture. A multi-harmonic and multi-amplitude training signal is introduced to improve training efficiency. Both methodologies achieve computational speed-ups on the order of 10^2 to 10^3 compared to full-order simulations, demonstrating their potential as efficient and accurate surrogates for predicting flow quantities such as wall shear stress.

Figures & tables

Figure 1: Full 3D cerebrovascular network and zoomed view of the basilar artery bifurcation. The highlighted bifurcation region corresponds to the anatomical segment used to construct the idealized 3D computational model employed in this study. Both the basilar artery mesh and the cerebrovascular network geometry were generated using the meshing framework of Decroocq et al. [ 35 ] and subsequently merged for visualization purposes.

Figure 2: Synthetic inlet velocity boundary conditions used in this work. The training waveform is a multi-harmonic signal, while the test waveform is a single-frequency sinusoid. These signals drive the unsteady inflow in the FOM and are used to evaluate the temporal learning and generalization capability of the surrogate models.

Panel of Figure 3 — Figure 3: Schematic of the surrogate workflow. The Offline Phase (gray) consists of FOM data generation and POD basis extraction. In the Online Phase (blue), the temporal coefficients are predicted either by an intrusive POD–Galerkin ROM or by a non-intrusive POD–RC surrogate; reconstructed fields are then evaluated against the FOM reference.

Figure 4: Layers of Reservoir Computing Architecture, first Layer is the input layer, second one is fixed and randomly initialized, and only the final output layer is trainable.

Panel of Figure 6 — Figure 6: First six POD modes of the velocity magnitude field for the cerebrovascular bifurcation geometry. The modes are ordered according to decreasing energetic contribution with the first mode representing the dominant flow structure.

Panel of Figure 8 — Figure 8: First six POD modes of the pressure field for the cerebrovascular bifurcation geometry. The modes are ordered according to decreasing energetic contribution, with the first mode representing the dominant pressure distribution.

Figure 9: Time evolution of the

L^{2}

error ( 24 ) between FOM and POD-Galerkin projection ROM solutions for pressure (blue curve), velocity magnitude (orange curve) and wall shear stress (green curve).

Figure 10: Comparison of the evolution of $p$ given by the FOM (first column) and the POD–Galerkin ROM (second column) and their differences in absolute value (third column).

Figure 11: Comparison of the evolution of $U$ given by the FOM (first column) and the POD–Galerkin ROM (second column) and their differences in absolute value (third column).

Figure 12: Comparison of the evolution of $\mathrm{WSS}$ given by the FOM (first column) and the POD–Galerkin ROM (second column) and their differences in absolute value (third column).

Figure 13: Time evolution of the

L^{2}

error ( 24 ) between FOM and POD-RC ROM solutions for pressure (blue curve), velocity magnitude (orange curve) and wall shear stress (green curve).

Figure 14: Comparison of the evolution of p given by the FOM (first column) and the POD–Reservoir Computing ROM (second column) and their differences in absolute value (third column).

Figure 15: Comparison of the evolution of U given by the FOM (first column) and the POD–Reservoir Computing ROM (second column) and their differences in absolute value (third column).

Figure 16: Comparison of the evolution of $WSS$ given by the FOM (first column) and the POD–Reservoir Computing ROM (second column) and their differences in absolute value (third column).

Panel of Figure 17 — Figure 17: Relative L2 error comparison over time

Method	Training Time (s)	Online Time (s)	Speedup vs FOM
High-Fidelity CFD	$\sim$ 12,000	$\sim$ 12,000	1.0
POD–Galerkin	$\sim$ 15	$\sim$ 10	$\sim$ 1,000
Reservoir Computing	$\sim$ 2	$\sim$ 0.1	$\sim$ 100,000

Table 1: Comparison of computational costs. The data-driven surrogates (RNN and RC) offer orders-of-magnitude speedups compared to the full-order solver.

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