For a nonlinear dynamical system that depends on parameters, the paper introduces a novel tensorial reduced-order model (TROM). The reduced model is projection-based, and for systems with no parameters involved, it resembles proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, TROM employs low-rank tensor approximations in place of truncated SVD, a key dimension-reduction technique in POD with DEIM. Three popular low-rank tensor compression formats are considered for this purpose: canonical polyadic, Tucker, and tensor train. The use of multilinear algebra tools allows the incorporation of information about the parameter dependence of the system into the reduced model and leads to a POD-DEIM type ROM that (i) is parameter-specific (localized) and predicts the system dynamics for out-of-training set (unseen) parameter values, (ii) mitigates the adverse effects of high parameter space dimension, (iii) has online computational costs that depend only on tensor compression ranks but not on the full-order model size, and (iv) achieves lower reduced space dimensions compared to the conventional POD-DEIM ROM. The paper explains the method, analyzes its prediction power, and assesses its performance for two specific parameter-dependent nonlinear dynamical systems.

翻译：针对含参数的非线性动力系统，本文提出了一种新颖的张量降阶模型（TROM）。该降阶模型基于投影方法，对于无参数系统，其形式类似于结合离散经验插值方法（DEIM）的本征正交分解（POD）。对于参数化系统，TROM采用低秩张量近似替代截断奇异值分解（SVD），而SVD是POD-DEIM方法中的关键降维技术。为此，本文考虑了三种主流的低秩张量压缩格式：典型多面分解、Tucker分解和张量列分解。利用多重线性代数工具，可将系统参数依赖关系的信息融入降阶模型，从而构建一种POD-DEIM类型的ROM，该模型：(i) 具有参数特异性（局部性），能够预测训练集外（未见）参数值下的系统动力学；(ii) 缓解高维参数空间的不利影响；(iii) 在线计算成本仅取决于张量压缩秩，而与全阶模型规模无关；(iv) 相比传统POD-DEIM ROM，实现了更低的降维空间维度。本文详细阐述了该方法，分析了其预测能力，并针对两个特定参数依赖的非线性动力系统评估了其性能。