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IEEE International Conference on Quantum Computing and Engineering (QCE)

Benchmarking Quantum-Assisted PINN (QA-PINN) for Computational Fluid Dynamics

Jay Shah, Rut Lineswala, Abhishek Chopra

Published Sep 15, 2024• 1 citation
Publisher

Abstract

Physics-Informed Neural Network (PINN) is a powerful method in computer-aided engineering (CAE), especially for design space exploration studies (like surrogate modeling and design of experiments), that combines deep learning with physics-based modeling. Though PINN is in a nascent stage, its success hinges on two key factors - generalizability and training efficiency. One common aspect of these factors is trainable parameters. The goal of this study is to improve these two factors by the introduction of Quantum Machine Learning in a classical PINN framework. This study proposes a Quantum-Assisted PINN (QA-PINN) to reduce the trainable parameters while maintaining accuracy. The study utilized a standard forward problem of simulating partial differential equations (PDEs) from the field of Computational Fluid Dynamics (CFD), a well-understood area with classical PINN. A comparative analysis was performed between QA-PINN, equivalent classical vanilla PINN (c-PINN), and state-of-the-art classical PINN - Globally Adaptive Activation Function PINN (GAAF-PINN). All methods were benchmarked against the ground-truth analytical solution for 1D Burger's equation. The analysis is conducted for 3, 4, and 5-qubit QA-PINN architectures. In comparison to c-PINN and GAAF-PINN, 4 and 5-qubit QA-PINN is shown to reduce the number of trainable parameters of the network by 20%, and while maintaining the accuracy. The reduced number of trainable parameters indicates the reduced complexity of the model and the reduced risk of overfitting, making it more generalizable. Thus, the proposed approach can have a significant impact on PINNs enhancing generalization and improving training efficiency for deploying the model for real-world applications.

DOI
10.1109/qce60285.2024.00199

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Physics-Informed Neural Network (PINN) is a powerful method in computer-aided engineering (CAE), especially for design space exploration studies (like surrogate modeling and design of experiments), that combines deep learning with physics-based modeling. Though PINN is in a nascent stage, its success hinges on two key factors - generalizability and training efficiency. One common aspect of these factors is trainable parameters. The goal of this study is to improve these two factors by the introduction of Quantum Machine Learning in a classical PINN framework. This study proposes a Quantum-Assisted PINN (QA-PINN) to reduce the trainable parameters while maintaining accuracy. The study utilized a standard forward problem of simulating partial differential equations (PDEs) from the field of Computational Fluid Dynamics (CFD), a well-understood area with classical PINN. A comparative analysis was performed between QA-PINN, equivalent classical vanilla PINN (c-PINN), and state-of-the-art classical PINN - Globally Adaptive Activation Function PINN (GAAF-PINN). All methods were benchmarked against the ground-truth analytical solution for 1D Burger's equation. The analysis is conducted for 3, 4, and 5-qubit QA-PINN architectures. In comparison to c-PINN and GAAF-PINN, 4 and 5-qubit QA-PINN is shown to reduce the number of trainable parameters of the network by 20%, and while maintaining the accuracy. The reduced number of trainable parameters indicates the reduced complexity of the model and the reduced risk of overfitting, making it more generalizable. Thus, the proposed approach can have a significant impact on PINNs enhancing generalization and improving training efficiency for deploying the model for real-world applications.
Abstract 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 PINNs (QCPINNs) 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 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.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding governing physical laws directly into the training objective. Recent advances in quantum machine learning have motivated hybrid quantum-classical extensions aimed at enhancing representational capacity while remaining compatible with near-term quantum hardware. In this work, we investigate trainable embedding strategies within quantum-assisted PINNs for solving parabolic PDEs, using one- and two-dimensional heat equations as canonical benchmarks. We introduce two quantum-assisted architectures that differ in their embedding components. In the first approach, a classical feed-forward neural network generates trainable feature maps for quantum data encoding (FNN-TE-QPINN). In the second, the embedding stage is realized entirely by a parameterized quantum circuit (QNN-TE-QPINN), yielding a fully quantum feature map. Our findings emphasize the critical role of embedding design and support hybrid quantum-classical approaches for parabolic PDE modeling in the NISQ era.
The steady incompressible Navier--Stokes equations pose significant computational challenges due to their nonlinear convective terms and pressure--velocity coupling. Physics-informed neural networks (PINNs) provide a mesh-free framework for approximating such systems, but classical PINNs can experience optimization difficulties in nonlinear flow regimes. In this work, we propose a quantum physics-informed neural network (QPINN) framework with a quantum neural network (QNN)-based trainable embedding for the lid-driven cavity problem. The proposed approach uses a QNN to learn data-adaptive quantum feature maps that encode spatial coordinates before they are processed by a variational quantum circuit within a physics-informed loss formulation. Numerical experiments show that the proposed QNN-TE-QPINN exhibits stable training behavior and competitive solution accuracy compared with classical PINNs and hybrid quantum models using classical embeddings, while requiring significantly fewer trainable parameters. Rather than claiming computational speedup, these results highlight the potential of trainable quantum embeddings for parameter-efficient physics-informed learning. The findings suggest that embedding design plays an important role in quantum-assisted PDE solvers and support further investigation of QNN-based trainable embeddings for nonlinear fluid dynamics benchmarks.
Recently, physics-informed neural networks (PINNs) have aroused an upsurge in the field of scientific computing including solving partial differential equations (PDEs), which convert the task of solving PDEs into an optimization challenge by adopting governing equations and definite conditions or observation data as loss functions. Essentially, the underlying logic of PINNs is based on the universal approximation and differentiability properties of classical neural networks (NNs). Recent research has revealed that quantum neural networks (QNNs), known as parameterized quantum circuits, also exhibit universal approximation and differentiability properties. This observation naturally suggests the application of PINNs to QNNs. In this work, we introduce a physics-informed quantum neural network (PI-QNN) by employing the QNN as the function approximator for solving forward and inverse problems of PDEs. The performance of the proposed PI-QNN is evaluated by various forward and inverse PDE problems. Numerical results indicate that PI-QNN demonstrates superior convergence over PINN when solving PDEs with exact solutions that are strongly correlated with trigonometric functions. Moreover, its accuracy surpasses that of PINN by two to three orders of magnitude, while requiring fewer trainable parameters. However, the computational time of PI-QNN exceeds that of PINN due to its operation on classical computers. This limitation may improve with the advent of commercial quantum computers in the future. Furthermore, we briefly investigate the impact of network architecture on PI-QNN performance by examining two different QNN architectures. The results suggest that increasing the number of trainable network layers can enhance the expressiveness of PI-QNN. However, an excessive number of data encoding layers significantly increases computational time, rendering the marginal gains in performance insufficient to compensate for the shortcomings in computational efficiency.
Physics-Informed Neural Networks (PINN) emerged as a powerful tool for solving scientific computing problems, ranging from the solution of Partial Differential Equations to data assimilation tasks. One of the advantages of using PINN is to leverage the usage of Machine Learning computational frameworks relying on the combined usage of CPUs and co-processors, such as accelerators, to achieve maximum performance. This work investigates the design, implementation, and performance of PINNs, using the Quantum Processing Unit (QPU) co-processor. We design a simple Quantum PINN to solve the one-dimensional Poisson problem using a Continuous Variable (CV) quantum computing framework. We discuss the impact of different optimizers, PINN residual formulation, and quantum neural network depth on the quantum PINN accuracy. We show that the optimizer exploration of the training landscape in the case of quantum PINN is not as effective as in classical PINN, and basic Stochastic Gradient Descent (SGD) optimizers outperform adaptive and high-order optimizers. Finally, we highlight the difference in methods and algorithms between quantum and classical PINNs and outline future research challenges for quantum PINN development.