Quantum physics informed neural networks for multi-variable partial differential equations (2503.12244v1)

Abstract: Quantum Physics-Informed Neural Networks (QPINNs) integrate quantum computing and machine learning to impose physical biases on the output of a quantum neural network, aiming to either solve or discover differential equations. The approach has recently been implemented on both the gate model and continuous variable quantum computing architecture, where it has been demonstrated capable of solving ordinary differential equations. Here, we aim to extend the method to effectively address a wider range of equations, such as the Poisson equation and the heat equation. To achieve this goal, we introduce an architecture specifically designed to compute second-order (and higher-order) derivatives without relying on nested automatic differentiation methods. This approach mitigates the unwanted side effects associated with nested gradients in simulations, paving the way for more efficient and accurate implementations. By leveraging such an approach, the quantum circuit addresses partial differential equations -- a challenge not yet tackled using this approach on continuous-variable quantum computers. As a proof-of-concept, we solve a one-dimensional instance of the heat equation, demonstrating its effectiveness in handling PDEs. Such a framework paves the way for further developments in continuous-variable quantum computing and underscores its potential contributions to advancing quantum machine learning.

Quantum Physics-Informed Neural Networks for Multi-Variable Partial Differential Equations

The paper "Quantum physics informed neural networks for multi-variable partial differential equations" explores the integration of quantum computing with machine learning to develop quantum physics-informed neural networks (QPINNs). This research aims to enhance the capabilities of classical Physics-Informed Neural Networks (PINNs) by leveraging the computational potential of quantum neural networks (QNNs) to solve partial differential equations (PDEs). By combining the principles of quantum mechanics and machine learning, QPINNs represent a novel approach to addressing complex differential equations that have traditionally posed significant challenges to classical computational methods.

Central to the paper is the demonstration of QPINNs' ability to solve PDEs using a continuous-variable quantum computing (CVQC) architecture. This architecture is particularly well-suited for implementing quantum versions of classical neural networks due to its capacity to encode information in the continuous degrees of freedom of the electromagnetic field. The authors propose a new methodology to compute higher-order derivatives within this framework without resorting to nested automatic differentiation, a common source of computational inefficiency and inaccuracy.

The theoretical formulation of QPINNs allows for the incorporation of established physical laws directly into the training process. These physical constraints guide the neural network towards solutions that are consistent with the governing equations and boundary conditions, thereby enhancing the accuracy and reliability of the solution. By extending previous work in quantum computing applications for differential equations, this research demonstrates a significant step forward in the practical implementation of QPINNs.

The paper provides proof-of-concept results by solving a one-dimensional instance of the heat equation, showcasing the framework's capability in addressing real-world PDE problems. The developed QPINN architecture exhibited excellent numerical performance, with the Normalized Mean Squared Error (NMSE) for the one-dimensional Poisson equation reaching values as low as $1.39 \times 10^{-5}$, illustrating its potential for high-precision solutions.

Furthermore, the paper highlights the integration of TensorFlow and Strawberry Fields, quantum machine learning libraries, for efficient simulation and training of QPINNs. This integration facilitates the deployment of optimization algorithms essential for training the network, offering a robust environment for developing advanced quantum machine learning models.

The implications of this research are both practical and theoretical. Practically, the proposed framework opens new pathways for utilizing quantum computing in scientific computing and engineering, where solutions to complex PDEs are crucial. Theoretically, it extends our understanding of how quantum machine learning models can surpass classical approaches by embedding physical laws into the computational framework, promising more efficient solutions to inherently complex problems.

Future work could explore scaling these methods to more complex, higher-dimensional PDEs and evaluating the performance of QPINNs on actual quantum hardware. Additionally, developing hybrid models that synergize classical and quantum computations could further enhance the applicability and effectiveness of this approach. As quantum technology continues to evolve, QPINNs hold promise for revolutionizing computational approaches to solving PDEs across various scientific disciplines.