In this work, we introduce the Quantum-Classical Hybrid Physics-Informed Neural Network with Multiplicative and Additive Couplings (QPINN-MAC): a novel hybrid architecture that integrates the framework of Physics-Informed Neural Networks (PINNs) with that of Quantum Neural Networks (QNNs). Specifically, we prove that through strategic couplings between classical and quantum components, the QPINN-MAC retains the universal approximation property, ensuring its theoretical capacity to represent complex solutions of ordinary differential equations (ODEs). Simultaneously, we demonstrate that the hybrid QPINN-MAC architecture actively mitigates the barren plateau problem, regions in parameter space where cost-function gradients decay exponentially with circuit depth, a fundamental obstacle in QNNs that hinders optimization during training. Furthermore, we prove that these couplings prevent gradient collapse, ensuring trainability even in high-dimensional regimes. Thus, our results establish a new pathway for constructing quantum-classical hybrid models with theoretical convergence guarantees, which are essential for the practical application of QPINNs.
Comments
21 pages, 4 figures, includes theoretical proofs of universal approximation and barren plateaus mitigation
In this work, we introduce the Quantum-Classical Hybrid Physics-Informed Neural Network with Multiplicative and Additive Couplings (QPINN-MAC): a novel hybrid architecture that integrates the framework of Physics-Informed Neural Networks (PINNs) with that of Quantum Neural Networks (QNNs). Specifically, we prove that through strategic couplings between classical and quantum components, the QPINN-MAC retains the universal approximation property, ensuring its theoretical capacity to represent complex solutions of ordinary differential equations (ODEs). Simultaneously, we demonstrate that the hybrid QPINN-MAC architecture actively mitigates the barren plateau problem, regions in parameter space where cost-function gradients decay exponentially with circuit depth, a fundamental obstacle in QNNs that hinders optimization during training. Furthermore, we prove that these couplings prevent gradient collapse, ensuring trainability even in high-dimensional regimes. Thus, our results establish a new pathway for constructing quantum-classical hybrid models with theoretical convergence guarantees, which are essential for the practical application of QPINNs.
General operator learning for parametric partial differential equations (PDEs) is a fundamental challenge at the intersection of artificial intelligence and physics-based modeling. Physics-informed Deep Operator Networks (PI-DeepONets) incorporate governing equations into learning, but face optimization difficulties in high dimensions. Quantum extensions offer additional representational capacity, yet their trainability can be hindered by barren plateaus. Here, we introduce QPI-DeepONet-MAC, a hybrid quantum-classical architecture that integrates parameterized quantum circuits into PI-DeepONets through a multiplicative-and-additive coupling (MAC) of classical representations and quantum expectation values. We prove analytically that the proposed architecture retains universal operator approximation capabilities. Furthermore, we derive bounds on quantum parameter gradients, establishing explicit conditions to avoid barren plateaus alongside a scaling criterion for trainability. Additionally, we derive a lower bound on the number of sensors required for a target accuracy, directly linking sampling requirements to input function regularity, domain dimension, and PDE dynamical growth. Finally, we propose an informational regularization scheme based on quantum coherence and state fidelity dynamics to prevent coherence loss and preserve quantum structure during training. Together, these results establish a theoretical foundation for quantum-enhanced physics-informed operator learning and position QPI-DeepONet-MAC as a scalable, stable framework for solving parametric PDEs.
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) 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.
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.
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) universal approximation theorem for this architecture, achieving a convergence rate of 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.
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.