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Register a new userUncovering near-wall blood flow from sparse data with physics-informed neural networks
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Near-wall blood flow and wall shear stress (WSS) regulate major forms of cardiovascular disease, yet they are challenging to quantify with high fidelity. Patient-specific computational and experimental measurement of WSS suffers from uncertainty, low resolution, and noise issues. Physics-informed neural networks (PINN) provide a flexible deep learning framework to integrate mathematical equations governing blood flow with measurement data. By leveraging knowledge about the governing equations (herein, Navier-Stokes), PINN overcomes the large data requirement in deep learning. In this study, it was shown how PINN could be used to improve WSS quantification in diseased arterial flows. Specifically, blood flow problems where the inlet and outlet boundary conditions were not known were solved by assimilating very few measurement points. Uncertainty in boundary conditions is a common feature in patient-specific computational fluid dynamics models. It was shown that PINN could use sparse velocity measurements away from the wall to quantify WSS with very high accuracy even without full knowledge of the boundary conditions. Examples in idealized stenosis and aneurysm models were considered demonstrating how partial knowledge about the flow physics could be combined with partial measurements to obtain accurate near-wall blood flow data. The proposed hybrid data-driven and physics-based deep learning framework has high potential in transforming high-fidelity near-wall hemodynamics modeling in cardiovascular disease.
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We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled advection-dispersion and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward advection-dispersion equations (ADEs), where its performance for various P{e}clet numbers ($Pe$) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization-based methods for $Pe$ larger than 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy of the PINN solution of the backward ADE.
A physics-informed neural network (PINN), which has been recently proposed by Raissi et al [J. Comp. Phys. 378, pp. 686-707 (2019)], is applied to the partial differential equation (PDE) of liquid film flows. The PDE considered is the time evolution of the thickness distribution $h(x,t)$ owing to the Laplace pressure, which involves 4th-order spatial derivative and 4th-order nonlinear term. Even for such a PDE, it is confirmed that the PINN can predict the solutions with sufficient accuracy. Nevertheless, some improvements are needed in training convergence and accuracy of the solutions. The precision of floating-point numbers is a critical issue for the present PDE. When the calculation is executed with a single precision floating-point number, the optimization is terminated due to the loss of significant digits. Calculation of the automatic differentiation (AD) dominates the computational time required for training, and becomes exponentially longer with increasing order of derivatives. By splitting the original 4th-order one-variable PDE into 2nd-order two-variable PDEs, the computational time for each training iteration is greatly reduced. The sampling density of training data also significantly affects training convergence. For the problem considered in this study, mproved convergence was obtained by allowing the sampling density of training data to be greater in earlier time ranges, where the rapid diffusion of the thickness occurs.
Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier-Stokes equations (NSE), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized NSE. Moreover, solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes. Here, we review flow physics-informed learning, integrating seamlessly data and mathematical models, and implementing them using physics-informed neural networks (PINNs). We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows, supersonic flows, and biomedical flows.
Data assimilation for parameter and state estimation in subsurface transport problems remains a significant challenge due to the sparsity of measurements, the heterogeneity of porous media, and the high computational cost of forward numerical models. We present a physics-informed deep neural networks (DNNs) machine learning method for estimating space-dependent hydraulic conductivity, hydraulic head, and concentration fields from sparse measurements. In this approach, we employ individual DNNs to approximate the unknown parameters (e.g., hydraulic conductivity) and states (e.g., hydraulic head and concentration) of a physical system, and jointly train these DNNs by minimizing the loss function that consists of the governing equations residuals in addition to the error with respect to measurement data. We apply this approach to assimilate conductivity, hydraulic head, and concentration measurements for joint inversion of the conductivity, hydraulic head, and concentration fields in a steady-state advection--dispersion problem. We study the accuracy of the physics-informed DNN approach with respect to data size, number of variables (conductivity and head versus conductivity, head, and concentration), DNNs size, and DNN initialization during training. We demonstrate that the physics-informed DNNs are significantly more accurate than standard data-driven DNNs when the training set consists of sparse data. We also show that the accuracy of parameter estimation increases as additional variables are inverted jointly.
Driven or active suspensions can display fascinating collective behavior, where coherent motions or structures arise on a scale much larger than that of the constituent particles. Here, we report experiments and numerical simulations revealing that red blood cells (RBCs) assemble into regular patterns in a confined shear flow. The order is of pure hydrodynamic and inertialess origin, and emerges from a subtle interplay between (i) hydrodynamic repulsion by the bounding walls which drives deformable cells towards the channel mid-plane and (ii) intercellular hydrodynamic interactions which can be attractive or repulsive depending on cell-cell separation. Various crystal-like structures arise depending on RBC concentration and confinement. Hardened RBCs in experiments and rigid particles in simulations remain disordered under the same conditions where deformable RBCs form regular patterns, highlighting the intimate link between particle deformability and the emergence of order. The difference in structuring ability of healthy (deformable) and diseased (stiff) RBCs creates a flow signature potentially exploitable for diagnosis of blood pathologies.
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