quasi

Quantum Machine Learning for Enhancing Fluid Dynamic Simulations

Idea

Leveraging quantum machine learning techniques to enhance fluid dynamic simulations using

Objectives

Methodology

1. Classical PINNs

• Implement baseline PINNs using standard deep learning frameworks such as TensorFlow and PyTorch.
• Solve benchmark fluid dynamics problems, including the Navier-Stokes equations for incompressible flow.
• Evaluate accuracy and computational performance on varying problem scales.

2. Quantum PINNs (QPINNs)

• Utilize hybrid quantum-classical platforms like TensorFlow Quantum or Qiskit or PennyLane to design QPINNs.
• Employ PQCs to encode problem features and leverage variational quantum algorithms for optimization.
• Train models using quantum simulators and available quantum hardware (e.g., IBM Quantum).

3. Attention-Enhanced PINNs

• Integrate multi-head attention mechanisms to focus on critical regions of the solution domain.
• Test attention layers for improving convergence rates and accuracy in learning complex dynamics.

4. Physically-Informed Quantum Neural Networks (PI-QNNs)

• Develop quantum neural networks with loss functions explicitly encoding PDE constraints.
• Explore the use of entanglement to represent coupled physical systems.
• Validate models on multi-scale and high-dimensional fluid dynamics problems.

5. Benchmarking Framework

• Evaluate classical, attention-enhanced, and quantum models on metrics such as:
  • Accuracy: Compare predicted solutions against analytical or high-fidelity numerical solutions.
  • Efficiency: Assess computational cost on quantum and classical hardware.
  • Scalability: Examine performance for increasing dimensionality and complexity.

Quantum Circuits for Quantum PINNs

At the core are parameterized quantum circuits (PQCs), which offer a way to model complex, high-dimensional functions with fewer parameters compared to classical neural networks. These circuits enable the encoding of PDE solutions as quantum states, with optimization performed using hybrid quantum-classical approaches.

• Architecture: QPINNs employ hardware-efficient ansatz designs tailored to specific fluid dynamics problems, incorporating gates that respect the problem’s symmetries.
• Attention Mechanisms: Attention layers are integrated within PQCs to focus computational resources on critical fluid features, such as vortices or shockwaves, enhancing solution accuracy.
• Optimization: Loss functions based on the residuals of governing PDEs are minimized using hybrid gradient methods, such as the Parameter Shift Rule or Quantum Natural Gradient Descent.

Tensor Networks and Matrix Product States

Tensor networks, including Matrix Product States (MPS), play a crucial role in encoding the high-dimensional states of fluid dynamics simulations. MPS are particularly efficient for problems with localized interactions, allowing scalable representation of complex flow fields.

1. Decomposition: The computational domain is divided into overlapping subdomains, each represented as a tensor, enabling localized modeling of fluid behavior.
2. Quantum Embedding: Tensor network states are mapped onto quantum circuits to exploit quantum entanglement for enhanced representational power.
3. Integration with QPINNs: MPS-based data compression is combined with PQCs, ensuring efficient handling of large simulation domains.

QMERA for Multi-Scale Fluid Dynamics

Multi-scale Entanglement Renormalization Ansatz (QMERA) extends tensor network techniques to hierarchical modeling of fluid systems. QMERA is particularly suited for multi-scale phenomena, such as turbulence, where both fine-grained and coarse-grained features must be captured efficiently.

• Hierarchical Encoding: QMERA constructs a renormalized representation of fluid states, iteratively coarse-graining fine details while preserving global features.
• Renormalization Layers: These layers reduce the effective dimensionality of the problem, enabling efficient quantum optimization.
• Applications: Turbulent flows and multi-phase interactions have been effectively simulated using QMERA-enhanced QPINNs, demonstrating superior accuracy compared to classical PINNs.

Attention-Enhanced Mechanisms

Attention mechanisms in QPINNs prioritize critical regions of the solution space, such as high-gradient areas in fluid flow. This is achieved through quantum-inspired attention layers, which:

• Dynamically reallocate computational resources during training.
• Enhance convergence rates by focusing on regions where PDE residuals are highest.
• Integrate seamlessly with tensor networks and quantum circuits for hybrid optimization.

Frameworks and Resouces

Pennylane, Nvidia PhysicsNeMo, DGX Quantum, Qiskit, IBM Quantum https://pennylane.ai/ https://developer.nvidia.com/physicsnemo https://quantum.ibm.com/

References

Some reference papers are in the Quantum and Physics-Informed AI models

Afrah Farea, Saiful Khan, & Mustafa Serdar Celebi. (2025). QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs. arXiv:2503.16678 [quant-ph]. https://arxiv.org/abs/2503.16678

Hybrid Quantum PINNs for Computational Fluid Dynamics. (2025). IOP Science. https://doi.org/10.1088/2632-2153/ad43b2

Quantum Physics-Informed Machine Learning for Multiscale Simulations. (2024). arXiv:2403.08954 [quant-ph]. https://arxiv.org/abs/2403.08954

PINNSFORMER: A Transformer-Based Framework for Physics-Informed Neural Networks. (2023). arXiv:2307.11833v3 [cs.LG].

Pengpeng Xiao, Muqing Zheng, Anran Jiao, Xiu Yang, & Lu Lu. (2024). Quantum DeepONet: Neural Operators Accelerated by Quantum Computing. [Manuscript].

Siddhant Dutta, Nouhaila Innan, Sadok Ben Yahia, & Muhammad Shafique. (2024). AQ-PINNs: Attention-Enhanced Quantum Physics-Informed Neural Networks for Carbon-Efficient Climate Modeling. arXiv:2409.01626v1 [cs.LG].

Physics-Informed Neural Networks in Crystal Growth Simulations D. Kolberg, C. Anders, & S. Reke. (2023). Fast Prediction of Transport Structures in the Melt by Physics-Informed Neural Networks during ‘VMCz’ Crystal Growth of Silicon. Journal of the Institute of Metals, https://doi.org/10.1080/00219592.2023.2236656

J. Ling, Y. Zhang, & D. Chen. (2023). Simulation of Thermal‑Fluid Coupling in Silicon Single Crystal Growth Based on Gradient Normalized Physics‑Informed Neural Network. Physics of Fluids. https://doi.org/10.1063/5.0123811 (Code repository: https://github.com/callmedrcom/SIPINN)

Z. Wang, F. Chen, & M. Li. (2024). RF-PINNs: Reactive Flow Physics-Informed Neural Networks for Field Reconstruction of Laminar and Turbulent Flames. Journal of Computational Physics, 113698. https://doi.org/10.1016/j.jcp.2024.113698

Y. Zhao, H. Liu, & J. Xu. (2025). Research on the Thermal‑Fluid Coupling in the Growth Process of Czochralski Silicon Single Crystals Based on an Improved Physics‑Informed Neural Network. AIP Advances, 15(10), 105202. https://doi.org/10.1063/5.0271778

Contact

Name: A Seshaditya

E-Mail: aditya.a.sesh@alumni.tu-berlin.de, aditya@quasi.digital

Website: https://quasi.digital

LinkedIn: https://www.linkedin.com/in/a-seshaditya-7180822a5/

License

This project is licensed under the MIT License. See the LICENSE file for details.