In this paper, we investigate tensor based nonintrusive reduced-order models (ROMs) for parametric cross-diffusion equations. The full-order model (FOM) consists of ordinary differential equations (ODEs) in matrix or tensor form resulting from finite difference discretization of the differential operators by taking the advantage of Kronecker structure. The matrix/tensor differential equations are integrated in time with the implicit–explicit (IMEX) Euler method. The reduced bases, relying on a finite sample set of parameter values, are constructed in form of a two-level approach by applying higher-order singular value decomposition (HOSVD) to the space–time snapshots in tensor form, which leads to a large amount of computational and memory savings. The nonintrusive reduced approximation for an arbitrary parameter value is obtained through tensor product of the reduced basis by the parameter dependent core tensor that contains the reduced coefficients. The reduced coefficients for new parameter values are computed with the radial basis functions. The efficiency of the proposed method is illustrated through numerical experiments for two-dimensional Schnakenberg and three-dimensional Brusselator cross-diffusion equations. The spatiotemporal patterns are accurately predicted by the reduced-order models with speedup factors of orders two and three over the full-order models.

在本文中，我们研究了用于参数交叉扩散方程的基于张量的非侵入式降阶模型 (ROM)。全阶模型 (FOM) 由矩阵或张量形式的常微分方程 (ODE) 组成，这些方程是利用 Kronecker 结构对微分算子进行有限差分离散化而产生的。矩阵/张量微分方程与隐式-显式 (IMEX) Euler 方法在时间上积分。简化的基，依赖于参数值的有限样本集，通过将高阶奇异值分解 (HOSVD) 应用于张量形式的时空快照，以两级方法的形式构建，这导致了一个大的计算量和内存节省量。任意参数值的非侵入式约简近似是通过包含约简系数的参数相关核心张量的约简基的张量积获得的。使用径向基函数计算新参数值的缩减系数。通过二维 Schnakenberg 和三维 Brusselator 交叉扩散方程的数值实验说明了所提出方法的效率。时空模式由降阶模型准确预测，其加速因子为全阶模型的二阶和三阶。通过二维 Schnakenberg 和三维 Brusselator 交叉扩散方程的数值实验说明了所提出方法的效率。时空模式由降阶模型准确预测，其加速因子为全阶模型的二阶和三阶。通过二维 Schnakenberg 和三维 Brusselator 交叉扩散方程的数值实验说明了所提出方法的效率。时空模式由降阶模型准确预测，其加速因子为全阶模型的二阶和三阶。