For a nonlinear dynamical system depending 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 the proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, the 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 multi-linear algebra tools allows to incorporate the information about the parameter dependence of the system into the reduced model and leads to a POD--DEIM type ROM which (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 non-linear dynamical systems.

翻译：针对含参数的非线性动力系统，本文提出一种新型张量降阶模型（TROM）。该降阶模型基于投影方法，对于无参数系统，其形式类似于本征正交分解（POD）结合离散经验插值方法（DEIM）。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——后者是POD与DEIM中的关键降维技术。为此，本文考虑了三种主流低秩张量压缩格式：典型多部、塔克与张量链。通过引入多线性代数工具，可将系统参数依赖信息纳入降阶模型，形成POD-DEIM型ROM，其具有以下特征：(i) 参数特异性（局部化），能预测未参与训练的未见参数值下的系统动态；(ii) 缓解高维参数空间的不利影响；(iii) 在线计算成本仅依赖于张量压缩秩，而与全阶模型规模无关；(iv) 相较于传统POD-DEIM ROM，能实现更低的降阶空间维度。本文阐述该方法原理，分析其预测能力，并通过两个含参非线性动力系统实例评估其性能。