Quantum Physics-Informed Neural Networks for Maxwell's Equations: Circuit Design, "Black Hole" Barren Plateaus Mitigation, and GPU Acceleration

TL;DR

This paper introduces a Quantum PINN framework combining quantum circuits and classical neural networks to solve Maxwell's equations, achieving higher accuracy with fewer parameters and improved training stability through energy conservation constraints.

Contribution

The work develops a novel Quantum PINN approach with GPU-accelerated quantum simulation, demonstrating enhanced accuracy and stability in solving 2D Maxwell's equations compared to classical PINNs.

Findings

QPINN achieves up to 19% higher accuracy than classical PINN.

Adding energy conservation stabilizes training and mitigates barren plateau issues.

GPU-accelerated quantum simulation enables efficient end-to-end training.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving partial differential equations (PDEs) by embedding the governing physics into the loss function associated with a deep neural network. In this work, a Quantum PINNs (QPINN) framework is proposed to solve two-dimensional (2D) time-dependent Maxwell's equations. Our approach utilizes a parametrized quantum circuit in conjunction with the classical neural network architecture and enforces physical laws, including a global energy conservation principle, during training. A quantum simulation library, TorQ, was developed to efficiently compute circuit outputs and derivatives by leveraging GPU acceleration based on PyTorch, enabling end-to-end training of the QPINN. The method was evaluated on two 2D electromagnetic wave propagation problems: one in free space (vacuum) and the other has an added dielectric medium. Multiple quantum circuit ans\"atze, input scales, and an added loss term were compared in a thorough ablation study. Furthermore, recent techniques to enhance PINN convergence, including random Fourier feature embeddings and adaptive time weighting, have been incorporated. Our results demonstrate that the QPINN achieves accuracy comparable to, and even greater than, the classical PINN baseline, while using a significantly smaller number of trainable parameters. This study also shows that adding an energy conservation term to the loss stabilizes training and improves the physical fidelity of the solution in the lossless free-space case. This added term helps mitigate a new kind of barren plateau (BP) related phenomenon - ``black hole'' (BH) loss landscape for the quantum experiments in that scenario. By optimizing the quantum-circuit ansatz and embedding energy-conservation constraints, our QPINN achieves up to a 19% higher accuracy on 2D Maxwell benchmark problems compared to a classical PINN.