量子辅助可训练嵌入物理信息神经网络用于抛物型偏微分方程
Quantum-Assisted Trainable-Embedding Physics-Informed Neural Networks for Parabolic PDEs
AI总结：
本文提出量子辅助可训练嵌入的物理信息神经网络，用于求解抛物型偏微分方程，通过两种不同嵌入策略提升模型性能。
摘要：
物理信息神经网络（PINNs）已发展为一种强大的框架，用于通过将支配物理定律直接嵌入到训练目标中来求解偏微分方程（PDEs）。最近在量子机器学习中的进展推动了混合量子-经典扩展，旨在增强表示能力，同时与近期量子硬件保持兼容。在本工作中，我们研究了量子辅助PINNs中的可训练嵌入策略，用于求解抛物型PDEs，使用一维和二维热方程作为标准基准。我们引入了两种量子辅助架构，其嵌入组件存在差异。在第一种方法中，经典前馈神经网络为量子数据编码生成可训练特征图（FNN-TE-QPINN）。在第二种方法中，嵌入阶段完全由参数化量子电路实现（QNN-TE-QPINN），产生完全量子特征图。我们的发现强调了嵌入设计的关键作用，并支持在NISQ时代用于抛物型PDE建模的混合量子-经典方法。
英文摘要：
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.
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