Shape-Informed Graph Neural Networks and Data Assimilation: Application to Velocity and Pressure Reconstruction in Aortic Blood Flow

Computational hemodynamics can enhance image-based diagnosis and provide complementary insights to predict, understand, and monitor treatments. The high computational costs and the complexity associated with handling patient-specific settings remain a major challenge toward clinical applications. In this work, we propose a novel robust shape registration method for nonparametric aortic geometries, describing different applications for projection-based reduced-order modeling for the training of graph neural networks and for data assimilation. The registration approach is based on ResNet-LDDMM, trained with a dataset of synthetic shapes, generated from real ones with statistical shape modeling. The optimization is tailored to surface meshes and does not rely on a priori assumptions on domain parameterization. We employ a multigrid strategy during the training phase that allows handling realistic mesh sizes. The registration enables the definition of geometric encoding of different blood flow solutions on a single reference shape, as well as the design of projection-based reduced-order models. We use this geometrical encoding to improve the training of graph neural networks and to present potential applications in data assimilation problems, combined with a generalized parameterized-Background Data-Weak formulation. As a particular example of data assimilation problem, we address the reconstruction of velocity fields and wall shear stresses, as well as the estimation of pressure fields and pressure-related biomarkers, such as the pressure drop, from low-resolution velocity observations. We show various numerical tests based on synthetic data, comparing the proposed strategies with state-of-the-art estimators.