Mathematics

    In recent years, numerical simulations of hæmodynamics have gained considerable attention in the medical community. Indeed, they offer healthcare professionals valuable insights into blood flow patterns and vascular dynamics, ultimately aiding in the diagnosis and treatment of cardiovascular diseases. However, conducting accurate computational fluid dynamics simulations demands extensive computing resources and this hinders their application in standard clinical practice. To address these limitations, reduced order models (ROMs) have proven to be highly advantageous, due to their ability of providing precise approximations of flow conditions at low computational costs. In this work, we focus on the reduced basis (RB) method, an intrusive ROM based on the projection (in space) of the problem at hand onto a suitably defined low–dimensional subspace. The RB method comes with its own set of challenges; here we tackle temporal complexity and geometrical variability. Temporal complexity is particularly relevant in problems where either the simulation interval should be very large or the timestep size should be very small, in order to properly capture some relevant behaviours. We propose to extend the reduction paradigm typical of the RB method to the temporal dimension, introducing space–time reduced basis (ST-RB) methods. We discuss their application to the Navier–Stokes equations and investigate their stability. Numerical results, obtained in a reduced fluid–structure interaction (RFSI) context, empirically demonstrate the method’s potential. Furthermore, we introduce a physics–based deep learning model — named ST–RB–DNN — which leverages a tensorial ST–RB solver as a deterministic decoder for solving either inverse problems or parameter estimation tasks. On the other end, addressing geometrical variability is crucial when performing patient–specific hæmodynamic simulations, since the peculiar shapes of the vessels significantly affect blood flow patterns. Traditional RB approaches face challenges in this context, as they require computational meshes with the same topology for all the available geometries. To overcome this issue, we envision the use of a data–driven non–intrusive mesh–free solver (USM–Net), which relies on geometrical latent encodings and diffeomorphic mappings to a reference shape (Atlas), in turn obtained through an auto–decoder structure. We present early numerical results, achieved on a cohort of healthy patient–specific aortic geometries.    References[1] N. Dal Santo, S. Deparis, and L. Pegolotti. Data–driven approximation ofparametrized PDEs by reduced basis and neural networks. Journal of Computational Physics, 416:109550, 2020.[2] R. Tenderini, S. Pagani, A. Quarteroni, S. Deparis. PDE-aware deep learning for inverse problems in cardiac electrophysiology. SIAM Journal on Scientific Computing. 44(3):B605-39. 2022.[3] F. Regazzoni, S. Pagani, and A. Quarteroni. Universal Solution Manifold Networks (USM–Nets): non–intrusive mesh–free surrogate models for problems in variable domains. Journal of Biomechanical Engineering. 144(12), p.121004. 2022.[4] R. Tenderini, N. Mueller, and S. Deparis. Space–time reduced basis methods for parametrized unsteady Stokes equations. arXiv preprint arXiv:2206.12198. 2023. [To appear on SIAM Journal on Scientific Computing.][5] R. Tenderini, and S. Deparis. Model order reduction of hæmodynamics by space–time reduced basis and reduced fluid–structure interaction.