We propose a multilevel tensor-train (TT) framework for solving nonlinear partial differential equations (PDEs) in a global space-time formulation. While space-time TT solvers have demonstrated significant potential for compressed high-dimensional simulations, the literature contains few systematic comparisons with classical time-stepping methods, limited error convergence analyses, and little quantitative assessment of the impact of TT rounding on numerical accuracy. Likewise, existing studies fail to demonstrate performance across a diverse set of PDEs and parameter ranges. In practice, monolithic Newton iterations may stagnate or fail to converge in strongly nonlinear, stiff, or advection-dominated regimes, where poor initial guesses and severely ill-conditioned space-time Jacobians hinder robust convergence. We overcome this limitation by introducing a coarse-to-fine multilevel strategy fully embedded within the TT format. Each level refines both spatial and temporal resolutions while transferring the TT solution through low-rank prolongation operators, providing robust initializations for successive Newton solves. Residuals, Jacobians, and transfer operators are represented directly in TT and solved with the adaptive-rank DMRG algorithm. Numerical experiments for a selection of nonlinear PDEs including Fisher-KPP, viscous Burgers, sine-Gordon, and KdV cover diffusive, convective, and dispersive dynamics, demonstrating that the multilevel TT approach consistently converges where single-level space-time Newton iterations fail. In dynamic, advection-dominated (nonlinear) scenarios, multilevel TT surpasses single-level TT, achieving high accuracy with significantly reduced computational cost, specifically when high-fidelity numerical simulation is required.

翻译：我们提出了一种多级张量链框架，用于在全局时空表述下求解非线性偏微分方程。尽管时空张量链求解器已展现出压缩高维模拟的巨大潜力，但现有文献中缺乏与经典时间步进方法的系统性比较，误差收敛分析有限，且对张量链舍入如何影响数值精度的定量评估不足。同样，现有研究未能展示该方法在多样化偏微分方程和参数范围内的性能。在实践中，对于强非线性、刚性或对流主导的体系，整体牛顿迭代可能停滞或不收敛，其中不良的初始猜测和严重病态的时空雅可比矩阵阻碍了鲁棒收敛。我们通过引入一个完全嵌入张量链格式的从粗到细多级策略来克服这一限制。每一级在通过低秩延拓算子传递张量链解的同时，细化空间和时间分辨率，从而为后续牛顿求解提供鲁棒的初始化。残差、雅可比矩阵和传递算子均直接以张量链表示，并采用自适应秩DMRG算法求解。针对包括Fisher-KPP方程、粘性Burgers方程、sine-Gordon方程和KdV方程在内的一组非线性偏微分方程的数值实验，涵盖了扩散、对流和色散动力学，结果表明多级张量链方法在单级时空牛顿迭代失败的情况下能够持续收敛。在动态、对流主导的（非线性）场景中，多级张量链超越了单级张量链，以显著降低的计算成本实现了高精度，特别是在需要高保真数值模拟时。