Wavelet-Accelerated Physics-Informed Quantum Neural Network for Multiscale Partial Differential Equations

TL;DR

This paper introduces a wavelet-accelerated quantum neural network framework that efficiently solves complex multiscale partial differential equations by reducing computational costs and enhancing accuracy without relying on automatic differentiation.

Contribution

It presents a novel wavelet-based quantum neural network architecture that improves multiscale PDE solving efficiency and accuracy over existing PINNs and quantum PINNs.

Findings

Achieves superior accuracy with less than 5% of trainable parameters of classical wavelet PINNs.

Provides a 3-5x speedup over existing quantum PINNs.

Reduces training time by eliminating automatic differentiation.

Abstract

This work proposes a wavelet-based physics-informed quantum neural network framework to efficiently address multiscale partial differential equations that involve sharp gradients, stiffness, rapid local variations, and highly oscillatory behavior. Traditional physics-informed neural networks (PINNs) have demonstrated substantial potential in solving differential equations, and their quantum counterparts, quantum-PINNs, exhibit enhanced representational capacity with fewer trainable parameters. However, both approaches face notable challenges in accurately solving multiscale features. Furthermore, their reliance on automatic differentiation for constructing loss functions introduces considerable computational overhead, resulting in longer training times. To overcome these challenges, we developed a wavelet-accelerated physics-informed quantum neural network that eliminates the need for automatic differentiation, significantly reducing computational complexity. The proposed framework incorporates the multiresolution property of wavelets within the quantum neural network architecture, thereby enhancing the network's ability to effectively capture both local and global features of multiscale problems. Numerical experiments demonstrate that our proposed method achieves superior accuracy while requiring less than five percent of the trainable parameters compared to classical wavelet-based PINNs, resulting in faster convergence. Moreover, it offers a speedup of three to five times compared to existing quantum PINNs, highlighting the potential of the proposed approach for efficiently solving challenging multiscale and oscillatory problems.