用量子启发的张量方法高效求解高维微分方程，精度与速度兼得。
Fast and Flexible Quantum-Inspired Differential Equation Solvers with Data Integration
- 基于量化张量列车（QTT）构建量子启发求解器，降低计算复杂度。
- 线性和非线性方程均实现内存与计算成本的对数级增长。
- 融合神经网络数据驱动学习，提升精度且缩短训练时间。
高维偏微分方程（PDEs）的精确求解仍是计算数学的核心挑战。传统数值方法在低维或粗网格下有效，但在实际应用中难以保证所需精度。近年来基于机器学习的方法虽具灵活性，但常因精度和可靠性不足而受限于工业场景。本文探索一种基于量化张量列车（QTT）的量子启发方法，可在多种复杂场景中实现高效、准确的PDE求解。通过多个代表性例子表明，该方法对线性和非线性PDE均实现内存与计算成本的对数级缩放。此外，提出一种新型数据驱动学习技术，在量子启发框架内结合神经网络的适应性与更高精度及更短训练时间。
原文摘要 · Abstract (English)
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.
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