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

TL;DR

The paper introduces WPIQNN, a framework that merges wavelet-based solution representations with physics-informed quantum neural networks to tackle multiscale PDEs featuring sharp gradients and oscillations. By representing the solution in a wavelet basis and using quantum encodings with fixed wavelet matrices, the method avoids automatic differentiation in the residual loss, reducing computational cost while maintaining high accuracy. Empirical results across heat conduction, Helmholtz, Klein–Gordon, and Maxwell problems show WPIQNN achieving superior accuracy with only a small fraction of the parameters of WPINN and delivering 3–5x faster training than existing PIQNNs. The approach offers a scalable, efficient pathway for solving challenging multiscale and oscillatory PDEs with potential applicability to broader scientific computing tasks on quantum-enabled platforms.

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.

Figures (12)

• Figure 1: PIQNN Architecture: It consists of an encoding layer followed by variational layers composed of trainable parameters and entangling gates. Final measurements yield the predicted output $\hat{\mathcal{U}}(\boldsymbol{x}; \theta)$. The figure illustrates a 4-qubit setup with two variational layers.
• Figure 2: Architecture of the proposed WPIQNN framework. The main components include the input layer comprising spatial-temporal coordinates $(x,t)$, followed by a QNN 1 with angle encoding to extract quantum features. The feature layer is subsequently encoded using amplitude encoding for processing by a QNN 2. The output is passed through classical post-processing layers to produce wavelet coefficients. These coefficients are combined with precomputed wavelet matrices to construct the final loss function.
• Figure 3: Loss plots with iterations for PIQNN-I (left) and WPIQNN (right) for solving the heat conduction equation \ref{['heat_conduct']} with $\varepsilon = 0.15$. Solid lines represent the mean loss, and the shaded areas indicate the corresponding loss variance across 10 independent runs.
• Figure 4: Left: Relative $\mathcal{L}_2$-error with relative $\mathcal{L}_\infty$-error WPIQNN. Right: Comparison of relative $\mathcal{L}_2$-error of different methods. Solid lines represent the mean relative $\mathcal{L}_2$-error, and the shaded areas indicate the corresponding error variance across 10 independent runs.
• Figure 5: Comparison of exact and predicted solutions for the heat conduction equation \ref{['heat_conduct']} with $\varepsilon = 0.15$. Top row: From left to right-exact solution, WPIQNN prediction, and corresponding point-wise absolute error. Bottom row: From left to right-pointwise absolute errors for the PIQNN-I, PIQNN-II, and WPINN methods, respectively.

Theorems & Definitions (4)

• Example 1: High gradient heat conduction problem
• Example 2: Helmholtz equation in high-frequency regime
• Example 3: Klein-Gordan equation
• Example 4: Maxwell equation