Abstract

Surrogate Modelling of Fluid Dynamics within Various Vascular Geometries with Physics-Informed Neural Networks (June, 2023)

Accurate and real-time prediction of the flow dynamics in human vessels carries huge clinical significance in cardiovascular disease diagnosis, treatment, prognosis and prevention. With the upsurge of machine learning advances and ever-increasing computational power over the years, physics-informed neural networks (PINN) have become an attractive surrogate for conventional numerical solvers. By embedding physical governing equations and boundary conditions into a deep neural network, PINN can compute deterministic solutions of the Navier-Stokes equations in a mesh-free fashion.

In this project, we embarked on the data-free (surrogate) modelling of vascular fluid dynamics in 4 parameterized geometries (straight, stenotic, aneurysmal, and curved vessels) with PINN. We first demonstrated the effectiveness of 3 proposed performance-enhancing techniques, including using adaptive learning rates, hard boundary corrections, and increasing order, by showing significant improvement in loss convergence and network performance compared with the baseline. Subsequently, we trained PINNs in both single-case and multi-case environments. Within the single-case environment, we argue that neural networks may fail to capture details such as small perturbations in dynamic systems, reflected by their poor performance in the simulation of the flow dynamics in the recirculation zones in the aneurysmal vessels. On the other hand, we pioneered the simultaneous computation of fluid solutions in multiple vessel structures, unlocking the potential of predicting unseen vascular fluid dynamics with PINN in real time. Lastly, we broke down a hidden pitfall associated with gradient vanishments due to improper geometry parameterization, hoping to alert future developers to re-think the coupling between parametrization and hard boundary distance functions. Our future work involves evaluating the impact of hyperparameters and different architectures on the performance of PINN through a comprehensive sensitivity analysis, meanwhile developing a fully parameterized fluid model with PINN for the heart.

Figure: The implemented PINN architecture. A DL network (light-green) using case geometric parameters f₁,...,f_n is used to determine the weights of nodes in the PINN network (light-orange). A PINN pre-layer (yellow) increases the order of the input parameters, and a hard boundary constraint post-layer rewrites the boundary conditions to the NN output.