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

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.

💡 Research Summary

This paper presents a systematic comparison of two model order reduction (MOR) strategies for unsteady cerebrovascular hemodynamics, using an idealised basilar artery bifurcation as a test case. High‑fidelity three‑dimensional computational fluid dynamics (CFD) simulations generate velocity and pressure snapshots that are first compressed by Proper Orthogonal Decomposition (POD). The POD yields a low‑dimensional spatial basis and associated temporal coefficients, which serve as the common input for both reduced‑order models.

The first approach, termed POD‑Galerkin (POD‑G), is intrusive and physics‑based. The incompressible Navier‑Stokes equations are projected onto the POD subspace, resulting in a reduced set of ordinary differential equations (or differential‑algebraic equations) governing the temporal coefficients. Special attention is given to pressure stabilization, inf‑sup compatibility, and the treatment of nonlinear convective terms. This method preserves the governing equations and thus guarantees physical consistency, but its accuracy can degrade when the flow exhibits strong non‑linearity or when operating outside the training parameter range.

The second approach, POD‑Reservoir Computing (POD‑RC), is non‑intrusive and data‑driven. A fixed, randomly connected recurrent reservoir processes the POD coefficient time series, while only a linear read‑out layer is trained via ridge regression. Because the reservoir dynamics are not updated during training, the computational cost is low, yet the architecture can capture complex temporal dependencies. To improve training efficiency and generalization, the authors introduce a single multi‑harmonic, multi‑amplitude inlet waveform that replaces the need for multiple separate training signals.

Both models are evaluated on their ability to reconstruct velocity, pressure, and wall shear stress (WSS) fields, and on computational speed. Results show speed‑ups of two to three orders of magnitude relative to the full CFD solver. POD‑G typically yields slightly lower reconstruction error for the baseline multi‑harmonic training signal, reflecting its adherence to the underlying physics. However, when tested on a different inlet condition—a single‑frequency sinusoid not seen during training—POD‑RC demonstrates superior robustness, with error growth that is markedly smaller than that of POD‑G. In terms of absolute accuracy, both methods achieve relative errors in the range of 1–3 % for WSS, indicating that the reduced models can serve as reliable surrogates for clinically relevant hemodynamic quantities.

The study highlights practical trade‑offs: POD‑Galerkin offers interpretability and strict conservation properties, making it attractive for scientific investigations that require physical insight. POD‑Reservoir Computing, by contrast, requires no access to the underlying CFD solver, avoids intrusive code modifications, and adapts more readily to unseen boundary conditions, positioning it as a promising tool for real‑time monitoring, patient‑specific simulations, or integration into larger multi‑physics pipelines. The authors suggest that hybrid schemes—combining physics‑based reduced equations with data‑driven closure models—could further enhance accuracy and robustness, pointing to fruitful directions for future research in cerebrovascular flow modeling.