Tensor network reduced order models for wall-bounded flows

We introduce a widely applicable tensor network-based framework for developing reduced order models describing wall-bounded fluid flows. As a paradigmatic example, we consider the incompressible Navier-Stokes equations and the lid-driven cavity in two spatial dimensions. We benchmark our solution against published reference data for low Reynolds numbers and find excellent agreement. In addition, we investigate the short-time dynamics of the flow at high Reynolds numbers for the lid-driven and doubly-driven cavities. We represent the velocity components by matrix product states and find that the bond dimension grows logarithmically with simulation time. The tensor network algorithm requires at most a few percent of the number of variables parametrizing the solution obtained by direct numerical simulation, and approximately improves the runtime by an order of magnitude compared to direct numerical simulation on similar hardware. Our approach is readily transferable to other flows, and paves the way towards quantum computational fluid dynamics in complex geometries.

Figure 1

(a) Setup of the square lid-driven cavity with edge length $L$. The upper lid moves at constant velocity $u_0$ in $x$-direction. ${\mathcal{C}}_t, {\mathcal{C}}_r, {\mathcal{C}}_b, {\mathcal{C}}_l$ are the top, right, bottom and left boundaries, respectively. ${\mathcal{L}}_v$ (${\mathcal{L}}_h$) denotes a vertical (horizontal) line through the center of the cavity. (b) Contour plot of the velocity magnitude $s=\sqrt{u^2+v^2}$ at $t=50$ for Re = 1000 and evaluated with the tensor network algorithm. (c) Comparison of the tensor network solution for the $x$ component $u$ of the velocity along ${\mathcal{L}}_v$ (black solid line) with the reference values in Ref. [12] (red dots). (d) Comparison of the tensor network solution for the $y$ component $v$ of the velocity along ${\mathcal{L}}_h$ (black solid line) with the reference values in Ref. [12] (red dots).

Figure 2

(a) Bond dimension $\chi$ versus time on a logarithmic scale and for the flow in Fig. 1. (b) The ratio between the NVPS and the total number of grid points $K^2$ in percent and as a function of time. Solid lines are a guide to the eye.

Figure 3

(a) Contour plot of the velocity magnitude $s=\sqrt{u^2+v^2}$ at $t/t_0=3$ for the flow configuration shown in Fig. 1. The grid size is $K^2=2^{11}\times 2^{11}$, Re = 24 000 and results are obtained with the MPS algorithm. (b) Same as in (a) but focusing on the region where the initial vortex forms. (c) Same as in (b) but for $K^2=2^9\times 2^9$. (d) Same as in (b) but for $K^2=2^{10}\times 2^{10}$.

Figure 4

Analysis of the bond dimension as a function of time and grid size. Black crosses [red circles] correspond to the flow in Fig. 3 [Fig. 5], and solid lines are a guide to the eye. (a) Bond dimension $\chi$ versus time on a logarithmic scale. (b) The ratio between the NVPS and the total number of grid points $K^2$ in percent and as a function of time. (c) Bond dimension ${\chi }_{\text{max}}$ at $t/t_0=3$ as a function of grid size. (d) Temporally averaged bond dimension $\overline{\chi }$ as a function of grid size.

Figure 5

(a) Boundary conditions corresponding to the doubly driven cavity where the upper [bottom] lid moves at constant velocity $u_0 [-u_0]$ in $x$ direction. (b) Contour plot of the velocity magnitude $s=\sqrt{u^2+v^2}$ at $t/t_0=3$ for the doubly driven cavity on a $K^2=2^{11}\times 2^{11}$ grid with Re = 24 000 and evaluated with the MPS algorithm.

Figure 6

(a) Ratio of the average times $T_{\text{MPS}}$ for completing a single iteration of the MPS algorithm and $T_{\text{DNS}}$ for completing a single iteration of the DNS algorithm as a function of Reynolds number Re. Black crosses [red circles] correspond to the lid-driven [doubly driven] cavity. Averages are taken up to $t/t_0=3$. $T_{\text{DNS}}$ for $\mathrm{Re}=24\text{k}$ ($\mathrm{Re}=60.5\text{k}$) is only taken for $t/t_0\le 1$ ($t/t_0\le 0.1$) due to the large run times, and $T_{\text{MPS}}$ for the doubly driven cavity and $\mathrm{Re}=60.5\text{k}$ is evaluated for $t/t_0=2.1$. The grid spacing for each Re is chosen such that it matches the corresponding microscale $\eta /L\approx {\text{Re}}^{-3/4}$. For data points above (below) the horizontal blue dashed line, the MPS (DNS) algorithm runs faster than its DNS (MPS) counterpart. Solid lines are a guide to the eye. (b) Time-averaged bond dimensions $\overline{\chi }(\text{LD)} [\overline{\chi }\left(\text{DD)}\right]$ corresponding to the lid-driven [doubly driven] cavity for different Reynolds numbers.