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arXiv:2606.26865

Quantum Physics-Informed Neural Networks for Solving Integro and Fractional PDEs

Deepak Gupta, Ratikanta Behera

math.APquant-phSubmitted Jun 25, 2026• Announced Jun 26, 2026• 0 citations
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Abstract

Quantum neural networks have emerged as powerful models for approximating nonlinear functions. Yet their use in solving integro-differential equations (IDEs) and fractional integro-partial differential equations (FIPDEs), which involve inherently nonlocal operators, remains unexplored. This work introduces a quantum physics-informed neural network (QPINN) framework that combines a quantum neural network with the governing equations of general nonlinear IDEs and FIPDEs. The proposed quantum network uses an affine feature map and variational quantum circuits to produce trial solutions with explicit trigonometric structure. We prove a quantitative $L^{2}(μ)$ universal approximation theorem for this architecture, achieving a convergence rate of $\mathcal{O}(n^{-1/2})$. This extends classical Fourier approximation theory to quantum circuits for physics-informed learning. We propose two QPINN variants: the numerical-quadrature QPINN (N-QPINN), which handles nonlocal integrals and fractional operators via high-order numerical quadrature while computing local derivatives through automatic differentiation of quantum trial solutions; and the auxiliary-function QPINN (A-QPINN), which eliminates numerical quadrature by introducing auxiliary variables that reformulate each integro-differential equation as an equivalent coupled system of partial differential equations, enabling a multi-output quantum neural network to simultaneously represent the solution and its associated variables. A series of numerical experiments demonstrates that the proposed QPINN framework accurately captures the behavior of nonlinear IDEs and FIPDEs and outperforms classical physics-informed neural networks.

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Quantum neural networks have emerged as powerful models for approximating nonlinear functions. Yet their use in solving integro-differential equations (IDEs) and fractional integro-partial differential equations (FIPDEs), which involve inherently nonlocal operators, remains unexplored. This work introduces a quantum physics-informed neural network (QPINN) framework that combines a quantum neural network with the governing equations of general nonlinear IDEs and FIPDEs. The proposed quantum network uses an affine feature map and variational quantum circuits to produce trial solutions with explicit trigonometric structure. We prove a quantitative L2(μ)L^{2}(μ)L2(μ) universal approximation theorem for this architecture, achieving a convergence rate of O(n−1/2)\mathcal{O}(n^{-1/2})O(n−1/2). This extends classical Fourier approximation theory to quantum circuits for physics-informed learning. We propose two QPINN variants: the numerical-quadrature QPINN (N-QPINN), which handles nonlocal integrals and fractional operators via high-order numerical quadrature while computing local derivatives through automatic differentiation of quantum trial solutions; and the auxiliary-function QPINN (A-QPINN), which eliminates numerical quadrature by introducing auxiliary variables that reformulate each integro-differential equation as an equivalent coupled system of partial differential equations, enabling a multi-output quantum neural network to simultaneously represent the solution and its associated variables. A series of numerical experiments demonstrates that the proposed QPINN framework accurately captures the behavior of nonlinear IDEs and FIPDEs and outperforms classical physics-informed neural networks.
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.
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Quantum physics-informed neural networks (QPINNs) solve partial differential equations (PDEs) by training parameterized quantum circuits against physics-based residuals, yet the role of the embedding that maps coordinates into quantum states remains insufficiently understood. In this research, we introduce a unified embedding framework that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and we propose two embeddings within it. The Linear Quantum Neural Network Trainable Embedding QPINN (LQNN-TE-QPINN) generates data-dependent features with an auxiliary quantum circuit and adds them to the input coordinates, instead of multiplying them as in our previous formulation. The adaptive-frequency QPINN (AdaFreq-QPINN) promotes the frequency-scaling factors of an arccos-Chebyshev encoding to trainable parameters. We evaluate both against direct and fixed analytical frequency encodings and alternative hybrid architectures on one- and two-dimensional Burgers equations. In one dimension, LQNN-TE-QPINN achieves a relative L2L_2L2​ error of 0.0916 with 720 trainable parameters, compared with 0.4501 for direct-encoding QPINN and 0.1233 for a locally adaptive classical physics-informed neural network with approximately eleven times more parameters, while AdaFreq-QPINN attains the lowest error among the analytical embeddings with only four additional parameters. In two dimensions, LQNN-TE-QPINN attains the lowest training objective, although its solution error is not clearly separated from that of a data re-uploading QPINN. Under simulated hardware noise, LQNN-TE-QPINN retains the lowest absolute derivative error while degrading most relative to its noiseless baseline. These results demonstrate that embedding design substantially influences the approximation and optimization behavior of QPINNs.
Physics-informed neural networks (PINNs) and hybrid quantum-classical extensions provide a promising framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. In this work, we study embedding strategies for trainable embedding quantum physics-informed neural networks (TE-QPINNs) in the context of nonlinear reaction-diffusion (RD) systems. We introduce an extended TE-QPINN (x-TE-QPINN) architecture that supports both classical and fully quantum embeddings, enabling a controlled comparison between feedforward neural network-based feature maps and parameterized quantum circuit embeddings. The first architecture is the classical embedding feed-forward neural network-based TE-QPINN (FNN-TE-QPINN), while the latter variant is a purely quantum one, referred to as quantum embedding neural network-based TE-QPINN (QNN-TE-QPINN). The proposed framework employs hardware-efficient variational quantum circuits and species-specific readout operators to approximate coupled multi-field dynamics while enforcing governing equations, boundary conditions, and initial conditions through a physics-informed loss function. By isolating the embedding mechanism while keeping the variational ansatz, loss formulation, and optimization procedure fixed, we analyze the impact of embedding design on gradient structure, parameter scaling, and quantum resource requirements. Numerical experiments on one- and two-dimensional RD equations demonstrate that quantum embeddings can replace classical embeddings without degradation in solution accuracy and, in certain regimes, exhibit improved optimization behavior compared to classical PINNs and hybrid quantum models with fixed embeddings. These results provide architectural insight into hybrid quantum PDE solvers and inform the design of resource-efficient quantum physics-informed learning methods.
Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics-Informed Neural Network (QPINN) and a Quantum-inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation can be represented as a polynomial that incorporates tensor rank decomposition. Numerical experiments are conducted on the Hamilton--Jacobi--Bellman (HJB) PDE arising from the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. The results show that our quantum models achieve lower losses and approximation errors than a classical fully connected PINN while using substantially fewer trainable parameters. Our quantum models further outperform a classical PINN constructed to share a similar inductive bias, providing experimental evidence of quantum-induced improvement in the tested settings and highlighting a resource-efficient pathway toward classical and near-term quantum solvers for PDEs with exploitable solution structure.
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.