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. 2020 Feb 28;367(6481):1026-1030.
doi: 10.1126/science.aaw4741. Epub 2020 Jan 30.

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

Maziar Raissi  1   2 , Alireza Yazdani  3 , George Em Karniadakis  1
Affiliations

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

Maziar Raissi et al. Science. .

Abstract

For centuries, flow visualization has been the art of making fluid motion visible in physical and biological systems. Although such flow patterns can be, in principle, described by the Navier-Stokes equations, extracting the velocity and pressure fields directly from the images is challenging. We addressed this problem by developing hidden fluid mechanics (HFM), a physics-informed deep-learning framework capable of encoding the Navier-Stokes equations into the neural networks while being agnostic to the geometry or the initial and boundary conditions. We demonstrate HFM for several physical and biomedical problems by extracting quantitative information for which direct measurements may not be possible. HFM is robust to low resolution and substantial noise in the observation data, which is important for potential applications.

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Conflict of interest statement

Competing interests: Authors declare no competing interests.

Figures

Fig. 1.
Fig. 1.. Quantifying flow visualizations.
(A) Leonardo da Vinci’s scientific artistry led him to draw accurate patterns of eddies and vortices for various flow problems. [Reprinted from figure 1.4 and figure 1.5 of (17) with permission from Elsevier.] (B to D) We used HFM to quantify the velocity and pressure fields in a geometry similar to the drawing in the lower left corner of (A); our input was a point cloud of data on the concentration field c(t, x, y) shown in (B), left. In (B) to (D), left, we plot the reference concentration, pressure, and streamlines, while at right, we plot the corresponding regressed quantities of interest produced by the algorithm. [(C) and (D)] Hidden states of the system–pressure p(t, x, y) and velocity fields–obtained by using HFM based on the data on the concentration field, randomly scattered in time and space. (D) A comparison of the reference (left) and regressed (right) instantaneous streamlines. The streamlines are computed by using the velocity fields.
Fig. 2.
Fig. 2.. Arbitrary training domain in the wake of a cylinder.
(A) Domain where the training data for concentration and reference data for the velocity and pressure are generated by using direct numerical simulation. (B) Training data on concentration c(t, x, y) in an arbitrary domain in the shape of a flower located in the wake of the cylinder. The solid black square corresponds to a very refined point cloud of data, whereas the solid black star corresponds to a low-resolution point cloud. (C) A physics-uninformed neural network (left) takes the input variables t, x, and y and outputs c, u, v, and p. By applying automatic differentiation on the output variables, we encode the transport and NS equations in the physics-informed neural networks ei, i = 1, …, 4 (right). (D) Velocity and pressure fields regressed by means of HFM. (E) Reference velocity and pressure fields obtained by cutting out the arbitrary domain in (A), used for testing the performance of HFM. (F) Relative L2 errors estimated for various spatiotemporal resolutions of observations for c. On the top line, we list the spatial resolution for each case, and on the line below, we list the corresponding temporal resolution over 2.5 vortex shedding cycles.
Fig. 3.
Fig. 3.. Inferring quantitative hemodynamics in a 3D intracranial aneurysm.
(A) Domain (right internal carotid artery with an aneurysm) where the training data for concentration and reference data for the velocity and pressure are generated by using a direct numerical simulation. (B) The training domain containing only the ICA sac is shown, in which two perpendicular planes have been used to interpolate the reference data and the predicted outputs for plotting 2D contours. (C) Schematic of the NS-informed neural networks, which take c(t, x, y, z) data as the input and infer the velocity and pressure fields. (D) Contours of instantaneous reference and regressed fields plotted on two perpendicular planes for concentration c, velocity magnitude, and pressure p in each row. The first two columns show the results interpolated on a plane perpendicular to the z axis, and the next two columns are plotted for a plane perpendicular to the y axis. (E) Flow streamlines are computed from the reference and regressed velocity fields colored by the pressure field. The range of contour levels is the same for all fields for better comparison [movies S1 and S2, which correspond to (A) and (E)].
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References

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