A Classical-Quantum Hybrid Architecture for Physics-Informed Neural Networks

TL;DR

This paper introduces a hybrid quantum-classical neural network architecture that combines physics-informed and quantum neural networks, ensuring universal approximation and improved trainability for solving complex differential equations.

Contribution

It presents a novel hybrid architecture with proven universal approximation and mitigation of barren plateau issues, advancing the theoretical foundation of quantum-augmented physics-informed neural networks.

Findings

Retains universal approximation property

Mitigates barren plateau problem in training

Prevents gradient collapse in high-dimensional regimes

Abstract

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.