标题： 快速灵活的量子启发微分方程求解器与数据集成 显示英文标题

标题： Fast and Flexible Quantum-Inspired Differential Equation Solvers with Data Integration

摘要： 准确求解高维偏微分方程（PDEs）仍然是计算数学中的一个核心挑战。传统数值方法虽然在低维情况下或使用粗网格时效果良好，但往往难以在实际应用中提供所需的精度。最近基于机器学习的方法提供了灵活性，但在准确性与可靠性方面常常不足，特别是在工业环境中。在这项工作中，我们探索了一种基于量化张量列车（QTT）的量子启发方法，能够在各种具有挑战性的情况下实现高效且精确的PDE求解。通过几个代表性实例，我们展示了QTT方法能够使线性和非线性PDE的内存和计算成本以对数尺度增长。此外，我们引入了一种在量子启发框架内的数据驱动学习新方法，结合了神经网络的适应性以及增强的准确性和减少的训练时间。 显示英文摘要

摘要： Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often struggle to deliver the precision required in practical applications. Recent machine learning-based approaches offer flexibility but frequently fall short in terms of accuracy and reliability, particularly in industrial contexts. In this work, we explore a quantum-inspired method based on quantized tensor trains (QTT), enabling efficient and accurate solutions to PDEs in a variety of challenging scenarios. Through several representative examples, we demonstrate that the QTT approach can achieve logarithmic scaling in both memory and computational cost for linear and nonlinear PDEs. Additionally, we introduce a novel technique for data-driven learning within the quantum-inspired framework, combining the adaptability of neural networks with enhanced accuracy and reduced training time.