Physics-Constrained Coupled Neural Differential Equations: Application to 1D Blood Flow Modeling

Coupled partial differential equations (PDEs) are crucial for modeling complex multi-physics phenomena, such as cardiovascular flows, which have significant real-world implications. Yet, accurately and efficiently modeling such phenomena poses considerable challenges. Here, we tackle the problem of blood flow in a stenosed artery with deformable walls, where the geometry is idealized but the problem is made reasonably complex by a realistic pulsatile inlet flow rate waveform and wall movement. We develop a low-dimensional model using physics-constrained coupled neural differential equations (PCNDEs), inspired by the 1D blood flow equations, to bridge the gap between 1D and 3D finite element simulations. Although data-driven approaches have shown promise in cardiovascular applications, neural PDE methods have not yet been applied to blood flow problems. Our innovative approach reformulates neural PDEs in space rather than the traditional temporal formulation, significantly improving the stability of the trained coupled PDEs. Temporal periodicity is explicitly enforced in the continuity equation using Fourier-series. This novel framework accurately captures flow rate and area variations, even when extrapolating to unseen inlet flow waveforms and stenosis blockage ratios. Our results offer a new perspective on using neural PDEs to model coupled PDEs with time-periodic boundary conditions.