A QPINN Framework with Quantum Trainable Embeddings for the Lid-Driven Cavity Problem

TL;DR

This paper introduces a quantum physics-informed neural network framework with trainable quantum embeddings for solving the lid-driven cavity problem, demonstrating stable training and competitive accuracy with fewer parameters.

Contribution

The work proposes a novel QPINN framework using QNN-based trainable embeddings, highlighting their potential for efficient physics-informed learning in nonlinear fluid dynamics.

Findings

QNN-TE-QPINN shows stable training behavior.

Achieves competitive accuracy with fewer parameters.

Highlights importance of embedding design in quantum PDE solvers.

Abstract

The steady incompressible Navier--Stokes equations pose significant computational challenges due to their nonlinear convective terms and pressure--velocity coupling. Physics-informed neural networks (PINNs) provide a mesh-free framework for approximating such systems, but classical PINNs can experience optimization difficulties in nonlinear flow regimes. In this work, we propose a quantum physics-informed neural network (QPINN) framework with a quantum neural network (QNN)-based trainable embedding for the lid-driven cavity problem. The proposed approach uses a QNN to learn data-adaptive quantum feature maps that encode spatial coordinates before they are processed by a variational quantum circuit within a physics-informed loss formulation. Numerical experiments show that the proposed QNN-TE-QPINN exhibits stable training behavior and competitive solution accuracy compared with classical PINNs and hybrid quantum models using classical embeddings, while requiring significantly fewer trainable parameters. Rather than claiming computational speedup, these results highlight the potential of trainable quantum embeddings for parameter-efficient physics-informed learning. The findings suggest that embedding design plays an important role in quantum-assisted PDE solvers and support further investigation of QNN-based trainable embeddings for nonlinear fluid dynamics benchmarks.