对一维Burgers方程的量子和经典算法进行基准测试：QTN、HSE和PINN
Benchmarking Quantum and Classical Algorithms for the 1D Burgers Equation: QTN, HSE, and PINN
AI总结：
本文比较了QTN、HSE和PINN在模拟一维Burgers方程中的性能，发现QTN在精度和运行时间上表现优异，但量子方法在无故障容忍情况下不优于经典求解器。
摘要：
我们展示了量子张量网络（QTN）、流体薛定谔方程（HSE）和物理信息神经网络（PINN）在模拟一维Burgers方程中的比较基准。评估这些新兴范式与经典GMRES和谱基线方法相比，我们分析了在网格分辨率从N=4到N=128范围内解的精度、运行时间缩放和资源开销。我们的结果揭示了明显的性能等级。QTN求解器在精度（L2≈10^-7）方面表现优异，具有显著的近常数运行时间缩放，有效利用纠缠压缩来捕捉激波前沿。相比之下，虽然有限差分HSE实现仍然稳健，但谱HSE方法在高分辨率下遭受灾难性数值不稳定性，显著发散于N=128时。PINNs展示了作为无网格求解器的灵活性，但在较低精度等级（L2≈10^-1）时停滞，受限于与网格方法相比的频谱偏倚。最终，虽然量子方法为低分辨率流体动力学提供了新的表示优势，但本研究确认，目前在无故障容忍或重大算法突破处理非线性反馈的情况下，量子方法在计算上并不优于经典求解器。
英文摘要：
We present a comparative benchmark of Quantum Tensor Networks (QTN), the Hydrodynamic Schrödinger Equation (HSE), and Physics-Informed Neural Networks (PINN) for simulating the 1D Burgers' equation. Evaluating these emerging paradigms against classical GMRES and Spectral baselines, we analyse solution accuracy, runtime scaling, and resource overhead across grid resolutions ranging from $N=4$ to $N=128$. Our results reveal a distinct performance hierarchy. The QTN solver achieves superior precision ($L_2 \sim 10^{-7}$) with remarkable near-constant runtime scaling, effectively leveraging entanglement compression to capture shock fronts. In contrast, while the Finite-Difference HSE implementation remains robust, the Spectral HSE method suffers catastrophic numerical instability at high resolutions, diverging significantly at $N=128$. PINNs demonstrate flexibility as mesh-free solvers but stall at lower accuracy tiers ($L_2 \sim 10^{-1}$), limited by spectral bias compared to grid-based methods. Ultimately, while quantum methods offer novel representational advantages for low-resolution fluid dynamics, this study confirms they currently yield no computational advantage over classical solvers without fault tolerance or significant algorithmic breakthroughs in handling non-linear feedback.
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