# QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs

arXiv: https://arxiv.org/abs/2503.16678 · HTML: https://mlsanity.com/paper/2503.16678/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 112 references that are in the map.
- First posted on arXiv 20 Mar 2025.

## Builds on (most cited first, 27 of 27)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [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)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Quantum machine learning: a classical perspective](https://mlsanity.com/paper/1707.08561/) (2017)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [TensorFlow Quantum: A Software Framework for Quantum Machine Learning](https://mlsanity.com/paper/2003.02989/) (2020)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces](https://mlsanity.com/paper/2009.00298/) (2020)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs](https://mlsanity.com/paper/2311.15283/) (2023)
- [Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes](https://mlsanity.com/paper/2304.11247/) (2023)
- [Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2411.15111/) (2024)
- [On Physics-Informed Neural Networks for Quantum Computers](https://mlsanity.com/paper/2209.14754/) (2022)
- [Solving partial differential equations on near-term quantum computers](https://mlsanity.com/paper/2208.05805/) (2022)
- [Self-Adaptive Physics-Informed Quantum Machine Learning for Solving Differential Equations](https://mlsanity.com/paper/2312.09215/) (2023)
- [Unsupervised Random Quantum Networks for PDEs](https://mlsanity.com/paper/2312.14975/) (2023)

## Built on by (most cited first, 9 of 9)

- [Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs](https://mlsanity.com/paper/2606.20326/) (2026)
- [Learning Fluid-Structure Interaction with Physics-Informed Machine Learning and Immersed Boundary Methods](https://mlsanity.com/paper/2505.18565/) (2025)
- [Wavelet-Accelerated Physics-Informed Quantum Neural Network for Multiscale Partial Differential Equations](https://mlsanity.com/paper/2512.08256/) (2025)
- [Trainability-Oriented Hybrid Quantum Regression via Geometric Preconditioning and Curriculum Optimization](https://mlsanity.com/paper/2601.11942/) (2026)
- [Quantum AS-DeepOnet: Quantum Attentive Stacked DeepONet for Solving 2D Evolution Equations](https://mlsanity.com/paper/2603.02261/) (2026)
- [Variational Quantum Physics-Informed Neural Networks for Hydrological PDE-Constrained Learning with Inherent Uncertainty Quantification](https://mlsanity.com/paper/2604.09374/) (2026)
- [Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guarantees](https://mlsanity.com/paper/2605.00330/) (2026)
- [Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations](https://mlsanity.com/paper/2608.08782/) (2026)
- [Variational Parameter Calibration with Physics-Aware Latent-Space Surrogates](https://mlsanity.com/paper/2608.11435/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum-classical physics-informed neural network (QCPINN) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy while using only 10-30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L_2 error for Helmholtz, Klein-Gordon, and Convection-diffusion equations, with a reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics-informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality.QCPINN presents a promising pathway to address the computational challenges associated with solving PDEs.