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

arXiv: https://arxiv.org/abs/2512.08256 · HTML: https://mlsanity.com/paper/2512.08256/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 49 references that are in the map.
- First posted on arXiv 9 Dec 2025.

## Builds on (most cited first, 14 of 14)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces](https://mlsanity.com/paper/2009.00298/) (2020)
- [A divide-and-conquer algorithm for quantum state preparation](https://mlsanity.com/paper/2008.01511/) (2020)
- [Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes](https://mlsanity.com/paper/2304.11247/) (2023)
- [A practical PINN framework for multi-scale problems with multi-magnitude loss terms](https://mlsanity.com/paper/2308.06672/) (2023)
- [QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2503.16678/) (2025)
- [Quantum Machine Learning for Finance](https://mlsanity.com/paper/2109.04298/) (2021)
- [On Physics-Informed Neural Networks for Quantum Computers](https://mlsanity.com/paper/2209.14754/) (2022)
- [Investigations on convergence behaviour of Physics Informed Neural Networks across spectral ranges and derivative orders](https://mlsanity.com/paper/2301.02790/) (2023)

## Abstract (arXiv)

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.