Quantum-Assisted Trainable-Embedding Physics-Informed Neural Networks for Parabolic PDEs

TL;DR

This work investigates trainable embedding strategies within quantum-assisted PINNs for solving parabolic PDEs, using one- and two-dimensional heat equations as canonical benchmarks and introduces two quantum-assisted architectures that differ in their embedding components.

Abstract

Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding governing physical laws directly into the training objective. Recent advances in quantum machine learning have motivated hybrid quantum-classical extensions aimed at enhancing representational capacity while remaining compatible with near-term quantum hardware. In this work, we investigate trainable embedding strategies within quantum-assisted PINNs for solving parabolic PDEs, using one- and two-dimensional heat equations as canonical benchmarks. We introduce two quantum-assisted architectures that differ in their embedding components. In the first approach, a classical feed-forward neural network generates trainable feature maps for quantum data encoding (FNN-TE-QPINN). In the second, the embedding stage is realized entirely by a parameterized quantum circuit (QNN-TE-QPINN), yielding a fully quantum feature map. Our findings emphasize the critical role of embedding design and support hybrid quantum-classical approaches for parabolic PDE modeling in the NISQ era.

Figures (9)

• Figure 1: A hybrid classical quantum trainable embedding physics-informed neural networks architecture. Classical embedding maps collocation points into angle parameters for a parameterized quantum circuit. The expectation values are used to compute the residual loss, which is minimized using a classical optimizer.
• Figure 2: Variational quantum circuit of the Heat Equation.
• Figure 3: Embedding quantum circuit for the Heat equation.
• Figure 4: Exact solution solved by the RK45 method for the one-dimensional Heat equation..
• Figure 5: Performance comparison results among PINN, FNN-TE-QPINN, and QNN-TE-QPINN Models for one-dimensional Heat equation after 150 training epochs. (a) & (b) The loss function of the PINN model and the absolute error between its solution and the RK45 reference solution. (c) & (d) The loss function and absolute error of the FNN-TE-QPINN model. (e) & (f) The loss function and absolute error of the QNN-TE-QPINN model.