We investigate the performance of classical, quantum, and hybrid quantum neural networks in solving partial differential equations through both supervised and unsupervised learning. The performance of each type of network is evaluated in solving the damped harmonic oscillator, the Einstein field equations, and the time-independent Schrödinger equation. We employed physics-informed supervised neural networks to solve the damped harmonic oscillator and the Einstein field equations, and developed an unsupervised neural network algorithm based on the shooting method for the time-independent Schrödinger equation. For the quantum neural networks, we introduced three variational parameterized quantum feature maps and two quantum circuits for the hybrid quantum neural networks. Each network was examined in three different configurations and evaluated across multiple random seeds. Under favorable parameter initializations, the hybrid quantum neural networks achieve higher accuracy than the classical neural networks in most cases across all tested problems. In contrast, the quantum neural networks attain the best accuracy for the damped harmonic oscillator and also perform well in the Schrödinger equation problem. Both the quantum and the hybrid quantum neural networks require fewer parameters than the classical neural networks and converge faster during optimization. All three models exhibit sensitivity to parameter initialization, with the quantum neural networks displaying the highest variability across different problems.
We investigate the performance of classical, quantum, and hybrid quantum neural networks in solving partial differential equations through both supervised and unsupervised learning. The performance of each type of network is evaluated in solving the damped harmonic oscillator, the Einstein field equations, and the time-independent Schrödinger equation. We employed physics-informed supervised neural networks to solve the damped harmonic oscillator and the Einstein field equations, and developed an unsupervised neural network algorithm based on the shooting method for the time-independent Schrödinger equation. For the quantum neural networks, we introduced three variational parameterized quantum feature maps and two quantum circuits for the hybrid quantum neural networks. Each network was examined in three different configurations and evaluated across multiple random seeds. Under favorable parameter initializations, the hybrid quantum neural networks achieve higher accuracy than the classical neural networks in most cases across all tested problems. In contrast, the quantum neural networks attain the best accuracy for the damped harmonic oscillator and also perform well in the Schrödinger equation problem. Both the quantum and the hybrid quantum neural networks require fewer parameters than the classical neural networks and converge faster during optimization. All three models exhibit sensitivity to parameter initialization, with the quantum neural networks displaying the highest variability across different problems.
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.
The Schrodinger equation in one spatial dimension admits a small set of exactly solvable potentials that serve as natural proving grounds for any new eigenvalue solver. We formulate Physics-Informed Neural Networks (PINNs) and Physics-Informed Quantum Neural Networks (PIQNNs) for the time-independent Schrodinger equation and apply them to three of these benchmarks: the harmonic oscillator, the infinite square well, and the finite square well. In each case a composite loss encodes the differential-equation residual, the normalization condition, the boundary behavior, and the orthogonality between eigenstates, so that the trial wave function is driven toward a genuine eigenfunction without supervised data. The eigenvalues and wave functions returned by both methods are compared against the exact spectra and against three classical references: the matrix Numerov method, the finite difference method, and the shooting method. For the smooth oscillator the two neural solvers reproduce the lowest four eigenvalues to parts per million, while for the square wells they recover the analytic levels with comparable fidelity even where the potential is discontinuous. The quantum circuit, built as a layered angle-embedding ansatz with strongly entangling blocks, converges more reliably than its classical counterpart on the higher excited states, where the loss landscape of the classical network becomes harder to navigate.
Physics-informed neural networks (PINNs) often struggle on nonlinear partial differential equations (PDEs) with sharp gradients, stiff dynamics, high-frequency content, or multiscale structure. Such limitations, rooted in spectral bias, ill-conditioned optimization, and unstable convergence, restrict PINN accuracy in regimes where advanced solvers are most needed. In this work, we develop a hybrid quantum-classical physics-informed neural network (HQPINN) that integrates a classical neural-network backbone with a parameterized quantum circuit (PQC) to enrich the solution representation. The framework is benchmarked against a classical PINN on three representative nonlinear PDEs: Burgers' equation, the Allen-Cahn equation, and the Korteweg-de Vries (KdV) equation. The framework is further examined through a systematic sensitivity analysis of qubit count, circuit depth, PQC placement, collocation density, and classical-network width. Across all benchmarks, HQPINNs exhibit smoother training dynamics, reduced loss oscillations, and improved final accuracy, with the largest gains occurring in stiff and multiscale regimes. Relative L2 error decreases by about fourfold for Burgers' equation and fivefold for the Allen-Cahn equation, while improvements for the KdV equation are more moderate. Overall, the results demonstrate that carefully co-designed hybrid quantum-classical architectures can mitigate key limitations of classical PINNs and provide practical design guidance for near-term quantum-enhanced PDE solvers.
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.
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.
Abstract Many real-world problems, like modelling environment dynamics, physical processes, time series etc involve solving partial differential equations (PDEs) parameterised by problem-specific conditions. Recently, a deep learning architecture called Fourier neural operator (FNO) proved to be capable of learning solutions of given PDE families for any initial conditions as input. However, it results in a time complexity linear in the number of evaluations of the PDEs while testing. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we exploit the running efficiency offered by these and propose quantum algorithms inspired by the classical FNO, which result in time complexity logarithmic in the number of evaluations and are expected to be substantially faster than their classical counterpart. At their core, we use the unary encoding paradigm and orthogonal quantum layers and introduce a new quantum Fourier transform in the unary basis. We propose three different quantum circuits to perform a quantum FNO. The proposals differ in their depth and their similarity to the classical FNO. We also benchmark our proposed algorithms on three PDE families, namely Burgers’ equation, Darcy’s flow equation and the Navier–Stokes equation. The results show that our quantum methods are comparable in performance to the classical FNO. We also perform an analysis on small-scale image classification tasks where our proposed algorithms are at par with the performance of classical convolutional neural networks, proving their applicability to other domains as well.