AI 中文总结
研究高维非线性偏微分方程数值解中维度诅咒问题，提出基于纤维依赖消除的框架，将TT交叉方法扩展到隐式格式，解决封闭性问题，结合牛顿迭代和秩适应性，实现高效隐式积分，数值实验验证了方法有效性。
AI 中文摘要
张量列车（TT）表示已成为缓解高维张量微分方程数值解中维度诅咒的有效框架。现有方法中，TT交叉方法很有吸引力，因其仅需控制方程的逐点评估，能自然处理任意非线性，避免切空间投影及近奇异低秩因子相关数值困难。但现有TT交叉秩截断方法限于显式时间积分。将TT交叉方法扩展到隐式格式存在障碍，本文引入一个有原则的纤维依赖消除框架，通过交叉插值恒等式将相邻纤维表示为交叉选择纤维的线性组合来解决此障碍，得到封闭配置系统，保留TT交叉方法主要优点。该框架适用于线性和非线性高维偏微分方程，与牛顿迭代和秩适应性自然结合。数值实验表明依赖消除迭代快速收敛，保留隐式多步格式的时间精度，能有效隐式积分含高达\(10^{55}\)自由度的全阶离散高维非线性问题。
英文摘要
Tensor-train (TT) representations have emerged as an effective framework for mitigating the curse of dimensionality in the numerical solution of high-dimensional tensor differential equations. Among existing approaches, TT-cross methods are particularly attractive because they require only pointwise evaluations of the governing equations, naturally accommodate arbitrary nonlinearities, and avoid tangent-space projections and the numerical difficulties associated with nearly singular low-rank factors. However, existing TT-cross rank-truncation methods have been restricted to explicit time integration. Extending TT-cross methods to implicit schemes presents an obstacle: the collocation equations associated with the cross-selected fibers depend on neighboring fibers that are not part of the unknown set. Consequently, the resulting nonlinear system is not closed, preventing the direct application of standard implicit solvers. In this work, we introduce a principled fiber-dependency elimination framework that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities. The resulting formulation produces a closed collocation system while preserving the principal advantages of TT-cross methods. The proposed framework applies to both linear and nonlinear high-dimensional partial differential equations and is naturally combined with Newton iterations and rank adaptivity. Numerical experiments demonstrate rapid convergence of the dependency-elimination iterations, preservation of the temporal accuracy of implicit multistep schemes, and efficient implicit integration of high-dimensional nonlinear problems with full-order discretizations containing up to $10^{55}$ degrees of freedom.
