RoePINNs: An integration of advanced CFD solvers with Physics-Informed Neural Networks and application in arterial flow modeling

Abstract

The characterization of forward and inverse problems describing blood flow dynamics plays a decisive role in numerous biomedical applications. These systems can be modeled using one-dimensional (1D) approaches leading to a hyperbolic system of equations with source terms. Their numerical discretization, associated to the spatial variation of mechanical and geometrical properties, requires advanced numerical solvers that ensure both stability and an accurate description of the dynamics of the system. In this work, we present RoePINNs, a hybrid framework for the embedding of advanced Computational Fluid Dynamics (CFD) solvers into Physics-Informed Neural Networks (PINNs), and give examples of application to Burgers’ equation as well as the propagation of nonlinear waves in elastic arteries, both under the presence of geometric-type source terms, for forward and inverse problems. We demonstrate that Augmented Riemann solvers can be incorporated into the PINN framework with straightforward adjustments to the hyperparameters, providing a promising alternative to automatic differentiation (AD), especially in cases where the solution exhibits strong nonlinearities and physical constraints are required. Benefits of the proposed RoePINN compared with the vanilla PINN based in AD are twofold: on the one hand, this hybrid approach employs numerical differentiation by means of support points in the surroundings of the collocation points, hence the robustness, generalization capacity and tunability of the PINNs are, in most cases, largely enhanced. On the other hand, the RoePINN incorporates the numerical solver, hence it is also capable of capturing sharp discontinuities with an order-of-magnitude improvement in accuracy compared with the vanilla version.