用量子循环网络解高维非线性偏微分方程，效率更高更稳定。
Quantum Recurrent Neural Networks with Encoder-Decoder for Time-Dependent Partial Differential Equations
- 将量子电路嵌入GRU/LSTM，构建量子循环网络框架。
- 在4类方程上实现高维时空数据压缩与稳定演化，精度优异。
- 适合需高效求解复杂动态系统的科研人员使用。
非线性时变偏微分方程在多领域建模中至关重要，但计算复杂度高，尤其在高维情形下挑战巨大。本文探索基于编码器-解码器架构的量子循环神经网络，将变分量子电路集成至门控循环单元和长短期记忆网络中。该模型能高效压缩高维时空数据至紧凑潜在空间，提升时间演化效率。我们在汉密尔顿-雅各比-贝尔曼方程、伯格斯方程、格雷-斯科特反应-扩散系统及三维迈克利斯-门顿反应-扩散方程上验证算法性能。结果表明，基于量子的方法在捕捉非线性动力学、处理高维空间和提供稳定解方面表现卓越，展现出解决复杂系统问题的创新潜力。
原文摘要 · Abstract (English)
Nonlinear time-dependent partial differential equations are essential in modeling complex phenomena across diverse fields, yet they pose significant challenges due to their computational complexity, especially in higher dimensions. This study explores Quantum Recurrent Neural Networks within an encoder-decoder framework, integrating Variational Quantum Circuits into Gated Recurrent Units and Long Short-Term Memory networks. Using this architecture, the model efficiently compresses high-dimensional spatiotemporal data into a compact latent space, facilitating more efficient temporal evolution. We evaluate the algorithms on the Hamilton-Jacobi-Bellman equation, Burgers' equation, the Gray-Scott reaction-diffusion system, and the three dimensional Michaelis-Menten reaction-diffusion equation. The results demonstrate the superior performance of the quantum-based algorithms in capturing nonlinear dynamics, handling high-dimensional spaces, and providing stable solutions, highlighting their potential as an innovative tool in solving challenging and complex systems.
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