Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes

TL;DR

This work addresses the computational burden of CFD in complex geometries by leveraging physics-informed neural networks (PINNs) and extending them with a hybrid quantum PINN (HQPINN) to solve steady incompressible Navier–Stokes flows in 3D geometries like a Y-shaped mixer. The authors detail the PINN formulation, training workflow, and architecture, compare classical PINNs against transfer-learning approaches, and demonstrate a 21% improvement in PDE/BC satisfaction for HQPINN over the classical model in 3D tests. They show that PINNs generalize across viscosity within a trained range but struggle with extrapolation, and that HQPINN benefits from quantum depth-infused layers, albeit with slower training on simulators. The study highlights the potential of hybrid quantum models to enhance accuracy in complex CFD tasks and outlines avenues for scalable quantum hardware-backed solvers and advanced architectures for 3D shape optimization.

Abstract

Finding the distribution of the velocities and pressures of a fluid by solving the Navier-Stokes equations is a principal task in the chemical, energy, and pharmaceutical industries, as well as in mechanical engineering and the design of pipeline systems. With existing solvers, such as OpenFOAM and Ansys, simulations of fluid dynamics in intricate geometries are computationally expensive and require re-simulation whenever the geometric parameters or the initial and boundary conditions are altered. Physics-informed neural networks are a promising tool for simulating fluid flows in complex geometries, as they can adapt to changes in the geometry and mesh definitions, allowing for generalization across fluid parameters and transfer learning across different shapes. We present a hybrid quantum physics-informed neural network that simulates laminar fluid flows in 3D Y-shaped mixers. Our approach combines the expressive power of a quantum model with the flexibility of a physics-informed neural network, resulting in a 21% higher accuracy compared to a purely classical neural network. Our findings highlight the potential of machine learning approaches, and in particular hybrid quantum physics-informed neural network, for complex shape optimization tasks in computational fluid dynamics. By improving the accuracy of fluid simulations in complex geometries, our research using hybrid quantum models contributes to the development of more efficient and reliable fluid dynamics solvers.

Figures (6)

• Figure 1: Scheme for the implementation of the HQPINN's training process and its architecture. HQPINN takes the $(x, y, z)$ coordinates of the points, sampled in the geometrical domain (purple circles), and yields $(\bm v, p)$ as output, where $\bm v$ is the velocity vector (green circles) and $p$ is the pressure (orange circle). The neural network itself begins with a classical part, which is a multilayer perceptron with $l$ layers of $n$ neurons. Then, there is a Parallel Hybrid Network, which consists of a Quantum depth-infused layer (a variational quantum circuit) and a multilayer perceptron. A "Filter" mechanism divides input points into four groups—fluid domain, walls, inlets, and outlets—each with its specific constraint (a boundary condition or a PDE). The error for each constraint is meticulously calculated and incorporated into the total loss, which is then minimized via gradient descent to refine the model's predictions. The process iteratively adjusts the network's weights using full-batch gradient descent, ensuring that every point in the geometrical domain contributes to a holistic solution that respects both the fluid dynamics within the domain and the conditions at its boundaries. The training process is described in Sec. \ref{['sec:classic_details']}.
• Figure 2: Isometric view of a slice of the cylinder. Color denotes the velocity magnitude for clarity.
• Figure 3: (a), (c), (e) Ground truth pressure and velocity projections. (b), (d), (f) PINN predicted pressure and velocity projections. (g), (h) Relative pressure and velocity magnitude errors between PINN and ground truth (absolute error divided by maximum pressure or velocity magnitude across the cylinder).
• Figure 4: 1) Classical model. (a) Loss curve. The classical PINN managed to learn a solution near the entry point well. However, it vanishes to zero further down the mixer. 2) Transfer learned models. (a) Loss curve. The transfer-learned models trained better with each iteration, surpassing the original model. (b, c, d) Distributions of the fluid pressure and velocities for the last model with $\alpha = 35^\circ$.
• Figure 5: Hybrid quantum model. (a) Loss curve. (b, c, d) Distribution of the fluid pressure and velocities for the HQPINN at ($\alpha = 30^\circ$). Similarly to the classical model, a non-trivial solution near the entry point is present. However, there is an asymmetry between the left and the right pipe, which should not be there. This could be an effect of the data encoding strategy.