发表机构
Texas Tech University; FPT University; Aalborg University; University of Porto（德克萨斯理工大学; 富国大学; 奥尔堡大学; 波尔图大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出统一嵌入框架及两种新嵌入（LQNN-TE-QPINN和AdaFreq-QPINN），在Burgers方程上验证了嵌入设计显著影响量子物理信息神经网络的逼近与优化性能。
AI 中文摘要
量子物理信息神经网络（QPINNs）通过训练参数化量子电路以匹配基于物理的残差来求解偏微分方程（PDEs），然而将坐标映射到量子态的嵌入的作用仍未得到充分理解。在本研究中，我们引入了一个统一的嵌入框架，将嵌入形式化为一种功能变换，塑造可供变分电路使用的特征表示，并在此框架内提出了两种嵌入。线性量子神经网络可训练嵌入QPINN（LQNN-TE-QPINN）使用辅助量子电路生成数据相关的特征，并将其添加到输入坐标中，而不是像我们之前的公式那样将它们相乘。自适应频率QPINN（AdaFreq-QPINN）将arccos-Chebyshev编码的频率缩放因子提升为可训练参数。我们在一维和二维Burgers方程上，将这两种嵌入与直接和固定解析频率编码以及替代混合架构进行了评估。在一维情况下，LQNN-TE-QPINN实现了0.0916的相对$L_2$误差，具有720个可训练参数，而直接编码QPINN的误差为0.4501，局部自适应经典物理信息神经网络的误差为0.1233，但其参数数量约为前者的十一倍；同时AdaFreq-QPINN在解析嵌入中实现了最低误差，仅增加了四个参数。在二维情况下，LQNN-TE-QPINN实现了最低的训练目标，尽管其解误差与数据重上传QPINN没有明显区分。在模拟硬件噪声下，LQNN-TE-QPINN保持了最低的绝对导数误差，但相对于其无噪声基线退化最严重。这些结果表明，嵌入设计显著影响QPINNs的逼近和优化行为。
英文摘要
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 $L_2$ 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.
Comments23 pages
