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General implementation of quantum physics-informed neural networks

Shashank Reddy Vadyala, Sai Nethra Betgeri

Published Mar 15, 2023• 17 citations
Publisher

Abstract

Recently, a novel type of Neural Network (NNs): the Physics-Informed Neural Networks (PINNs), was discovered to have many applications in computational physics. By integrating knowledge of physical laws and processes in Partial Differential Equations (PDEs), fast convergence and effective solutions are obtained. Since training modern Machine Learning (ML) systems is a computationally intensive endeavour, using Quantum Computing (QC) in the ML pipeline attracts increasing interest. Indeed, since several Quantum Machine Learning (QML) algorithms have already been implemented on present-day noisy intermediate-scale quantum devices, experts expect that ML on reliable, large-scale quantum computers will soon become a reality. However, after potential benefits from quantum speedup, QML may also entail reliability, trustworthiness, safety, and security risks. To solve the challenges of QML, we combine classical information processing, quantum manipulation, and processing with PINNs to accomplish a hybrid QML model named quantum based PINNs.

DOI
10.1016/j.array.2023.100287

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Recently, a novel type of Neural Network (NNs): the Physics-Informed Neural Networks (PINNs), was discovered to have many applications in computational physics. By integrating knowledge of physical laws and processes in Partial Differential Equations (PDEs), fast convergence and effective solutions are obtained. Since training modern Machine Learning (ML) systems is a computationally intensive endeavour, using Quantum Computing (QC) in the ML pipeline attracts increasing interest. Indeed, since several Quantum Machine Learning (QML) algorithms have already been implemented on present-day noisy intermediate-scale quantum devices, experts expect that ML on reliable, large-scale quantum computers will soon become a reality. However, after potential benefits from quantum speedup, QML may also entail reliability, trustworthiness, safety, and security risks. To solve the challenges of QML, we combine classical information processing, quantum manipulation, and processing with PINNs to accomplish a hybrid QML model named quantum based PINNs.
This paper proposes a novel approach for solving nonlinear partial differential equations (PDEs) with a quantum computer, the trainable embedding quantum physics informed neural network (TE-QPINN). We combine quantum machine learning (QML) with physics informed neural networks (PINNs) in a hybrid approach. By leveraging the advantages of classical and quantum computers, we can create algorithms that have a potential to be run on noisy intermediate-scale quantum devices (NISQ). We use feedforward neural networks (FNN) as problem-agnostic embedding functions, giving the used quantum circuit greater expressibility than previously introduced embedding. This expressibility allows us to solve a wide range of problems without using a problem specific ansatz. Additionally, we introduce a hybrid backpropagation algorithm that allows efficient updates of the used weights and biases in the FNN embedding functions. In this paper we showcase the capabilities of TE-QPINNs of a wide range of problems, including the two-dimensional Poisson, Burgers and Navier-Stokes equations. In direct comparison with classical PINNs, this approach showed an ability to achieve superior results while using the same number of parameters, highlighting their potential for more efficient optimization in high-dimensional parameter spaces, which could be transformative for future 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 (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.
Quantum machine learning (QML) has emerged as a promising paradigm for scientific computing, particularly when combined with Physics-Informed Neural Networks (PINNs) to solve partial differential equations. In this work, we introduce a Quantum Physics-Informed Neural Network (QPINN) framework to solve the Helmholtz equation in the context for scattered-wavefield. To overcome the numerical instabilities associated with point-source singularities commonly encountered in classical PINNs, we adopt the Lippmann–Schwinger integral formulation. The proposed hybrid architecture couples a classical neural-network encoder with a variational quantum circuit, enabling the learning of expressive, dynamically generated feature embeddings. Numerical experiments on a heterogeneous velocity model demonstrate that the proposed QPINN achieves high solution accuracy while requiring significantly fewer trainable parameters than analogous classical neural network architectures. Our results highlight the potential of hybrid quantum-classical approaches as efficient and scalable alternatives for wavefield simulation and for solving wave-based inverse problems.
Partial differential equations (PDEs) form the backbone of simulations of many natural phenomena, for example in climate modeling, material science, and even financial markets. The application of physics-informed neural networks to accelerate the solution of PDEs is promising, but not competitive with numerical solvers yet. Here, we show how quantum computing can improve the ability of physics-informed neural networks to solve partial differential equations. For this, we develop hybrid networks consisting of quantum circuits combined with classical layers and systematically test them on various non linear PDEs and boundary conditions in comparison with purely classical networks. We demonstrate that the advantage of using quantum networks lies in their ability to achieve an accurate approximation of the solution in substantially fewer training epochs, particularly for more complex problems. These findings provide the basis for targeted developments of hybrid quantum neural networks with the goal to significantly accelerate numerical modeling.
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.