Multi-stream physics hybrid networks for solving Navier-Stokes equations

TL;DR

The Multi-stream Physics Hybrid Network is proposed, a novel neural architecture that integrates quantum and classical layers in parallel to improve the accuracy of solving fluid dynamics equations, namely''Kovasznay flow''problem.

Abstract

Understanding and solving fluid dynamics equations efficiently remains a fundamental challenge in computational physics. Traditional numerical solvers and physics-informed neural networks struggle to capture the full range of frequency components in partial differential equation solutions, limiting their accuracy and efficiency. Here, we propose the Multi-stream Physics Hybrid Network, a novel neural architecture that integrates quantum and classical layers in parallel to improve the accuracy of solving fluid dynamics equations, namely ''Kovasznay flow'' problem. This approach decomposes the solution into separate frequency components, each predicted by independent Parallel Hybrid Networks, simplifying the training process and enhancing performance. We evaluated the proposed model against a comparable classical neural network, the Multi-stream Physics Classical Network, in both data-driven and physics-driven scenarios. Our results show that the Multi-stream Physics Hybrid Network achieves a reduction in root mean square error by 36% for velocity components and 41% for pressure prediction compared to the classical model, while using 24% fewer trainable parameters. These findings highlight the potential of hybrid quantum-classical architectures for advancing computational fluid dynamics.

Figures (4)

• Figure 1: Overview of the Multi-stream Hybrid Network architecture. Spatial coordinates $(x, y)$ act as as input of three identical, but separate PHN layers. Each of these layers is responsible for predicting one component of the solution vector $(v_x, v_y, p)$. This is a data-driven model, so once the solution is predicted, the error (MSE) between exact and predicted solutions is calculated and used to update model weights at each training iteration. PHN layer architecture: input coordinates $(x, y)$ are passed through two parallel layers: quantum and classical. Classical layer is a 1-hidden-layer MLP. Quantum layer is a parameterized two qubit circuit. The $R_X$ gates describe rotations about the $X$-axis on the Bloch sphere and are parameterized by the incoming coordinates, the $\text{Rot}(\bm \theta) = R_Z(\theta^1) R_Y(\theta^2) R_Z(\theta^3)$ gates describe arbitrary rotations and are parameterized by three trained weights $\bm{\theta}$. At the end of the circuit, both qubits are measured by the $\sigma_z$ operator. The measurement results of both qubits are passed through the "Output" layer to yield a single real value.
• Figure 2: Training results of multi-stream network models after 100 epochs in data-driven learning. Here Exact, MCN and MHN denote the exact solution, classical multi-stream and hybrid multi-stream networks respectively, and $v_x, v_y, p$ are velocity and pressure projections. The MCN model does well in predicting the $p$ function, but cannot approximate the periodic functions $v_x$ and $v_y$. The MHN model approximates all solution functions equally well, with even fewer trainable parameters.
• Figure 3: Training results of the FNN, MPCN, and MPHN models after 1000 epochs in physics-driven learning. Here $v_x, v_y, p$ are velocity and pressure projections. The FNN model, with a large number of parameters, was able to reproduce the exact solution with good accuracy. MPHN was able to learn the periodic nature of the solution for $v_x, v_y$, but has problems with the velocity decaying to zero along the $x$ axis. The MPCN did not remotely succeed in predicting the velocities. All networks were able to learn the pressure $p$ distribution quite well.
• Figure 4: Overview of the MPHN architecture. It is identical to data-driven architecture in Fig. \ref{['fig:data_driven_architecture']}. The only difference is in the loss function used for training. In MPHN, PDE and boundary conditions residuals are used as a loss, the model does not have any knowledge about the analytical solution. PHN layers architecture stays exactly the same.