We introduce a new tensor integration method for time-dependent PDEs that controls the tensor rank of the PDE solution via time-dependent diffeomorphic coordinate transformations. Such coordinate transformations are generated by minimizing the normal component of the PDE operator relative to the tensor manifold that approximates the PDE solution via a convex functional. The proposed method significantly improves upon and may be used in conjunction with the coordinate-adaptive algorithm we recently proposed in JCP (2023) Vol. 491, 112378, which is based on non-convex relaxations of the rank minimization problem and Riemannian optimization. Numerical applications demonstrating the effectiveness of the proposed coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

翻译：我们提出了一种新的时变偏微分方程张量积分方法，该方法通过时变微分同胚坐标变换控制PDE解的张量秩。这类坐标变换通过最小化PDE算子相对于近似PDE解的张量流形的法向分量来生成，该过程借助凸泛函实现。所提方法显著改进了我们近期在JCP (2023) Vol. 491, 112378中提出的坐标自适应算法（该算法基于秩最小化问题的非凸松弛和黎曼优化），并可与该算法联合使用。通过典型Liouville和Fokker-Planck方程的数值算例，展示并讨论了所提坐标自适应张量积分方法的有效性。